Infinity, need help

[Kosh3]

Alright, basically I need a second opinion, as im not a mathematician.

Basically my argument was that one infinity added to another does not increase its cardinality, that the cardinality of infinity remains constant. It came about because we were arguing about the size of the universe, and I said that if the universe was infinately large it could not expand, because there is no place it could go that it not at already. We note that the universe is expanding, therefore cannot be infinate. He said because one infinity can increase the size of another, an infinately large universe could expand by an infinite degree more.



imagine two sets

1,2,3,4,5,6,.....

1,2,4,16,32,64.....

where they are both infinately long. Both have an equal number of members. They are both the same size, even though the first looks like it would have more than the second.

The guy I was arguing with said "There are different scales of infinity. Anyhow, from this, you can follow that something infinite can expand... it just becomes a greater infinity."

Basically - that one infinity added to another can make it bigger. He also said:

"The set of all integers is equal to the set of positive integers (though you'd think it's twice as big, it's not). However, the set of all real numbers, rational and irrational, is greater than the set of integers, though they are both infinite. There is no debate on this issue. "

Well, im not a mathematician (so please talk as though explaining to a lamen). So I ask you guys: is he right about infinities, and is he right that an infinately large universe can expand.

Thx

PS - of course, this all raises the issue of whether infinity is just a purely mathematical concept and does not reside anywhere except maths. just like paradoxes :)

[JohannGoodflag]

Hello Kosh,

Alright, basically I need a second opinion, as im not a mathematician.

Basically my argument was that one infinity added to another does not increase its cardinality, that the cardinality of infinity remains constant.

This is true if the two infinities are of the same "nature", very roughly speaking. The thing is that there is more than one size of infinity.

The infinity that you are familiar with, and that you appeal to in your particular examples of infinite sets, is the "countable" infinity. The idea of a countable infinity put in common day terms is this: that you can create an (infinitely long) list, and each element of the set will appear at some finite position on the list. So, for example, while the natural numbers is an infinitely large set, you can eventually reach any particular natural number by starting from zero, and counting up one at a time. Here, the list is the natural numbers written in increasing order, and the position of each number N on the list is just N+1 (assuming you start counting from one).

So, the countable infinity is the one you probably know and love, but there are larger kinds of infinities, which we call "uncountable" infinities. One of the simplest examples would be the set of all points on the real line. This kind of infinity is usually called (the infinity of) the continuum. Another example which you might run across is the set of all sets of natural numbers: this also happens to have the cardinality of the continuum.

If you add two infinities to one another, the result is whichever of the two infinities was larger. If you add two infinities of the same size --- two countable infinities, for instance --- you get the same infinity back: this is the result you are familiar with. However, if
you add an uncountable infinity (such as the continuum) to a countable infinity, you will get that uncountable infinity as a result.

BTW, there are infinitly many (!) kinds of infinity --- there are more than just two. The countable infinity and the continuum are the two most commonly encountered in practise.

The guy I was arguing with said "There are different scales of infinity. Anyhow, from this, you can follow that something infinite can expand... it just becomes a greater infinity."

The last sentence doesn't really make a lot of sense from the point of view of cosmology. When cosmologists say that space is expanding, I'm fairly sure that they don't mean that it is possibly becoming "more infinite" in the mathematical sense.

I am a mathematician, but I am not a cosmologist. However, I will hazard a guess as to the motivation for the idea of expanding space.

If I'm not mistaken, the underlying reason why we say this at all is because we observe that distant objects seem to be travelling away from us, and the further away an object is, the faster it is receding. If we assume that we are not the center of the universe, i.e. that these observations are not peculiar to where we're observing from, we would guess that all objects are receeding from every other object, and that each object does not observe any sort of accelerating force locally. A logical possibility would then be that space itself is stretching or "expanding", so that objects already in it are pulled apart from one another. If the rate of "expansion" of space is uniform in a certain way --- a region of space expands in proportion to the size of that space ---we would expect objects to reced from one another at a rate which is proportional to their distance from one another.

This says nothing of any reason why space should be expanding, but I think this is the naive version (at least) of how the theory goes. For a non-naive version, you'd have to get a response from a cosmologist. :)

This can still make some sense (in principle: mathematician, not cosmologist!) if space is infinite. You wouldn't think of the "boundaries" of the space being pushed further out, because the space wouldn't have boundaries. However, if each region of space is expanding, you could reasonably say the space itself is expanding. This is like considering how the function f(x) = 2x acts on the real line: because f(x) "stretches" every region of the real line, you could reasonably say that it stretches the real line itself. The end result is the same, of course --- the space is the same size: it's infinite, and it's the same kind of infinity --- but the gaps in between the objects lying in the space has expanded. In a meaningful sense, one may still say that the space is being stretched, or that it is expanding.

I hope this is helpful!

JohannGoodflag

P.S. I would argue that paradoxes only occur in language, not math: but because we develop math with language, math sometimes shows us where our language errs, and forces us to be more careful. But I guess that's neither here nor there.

[ex-xian]

Originally posted by JohannGoodflag
P.S. I would argue that paradoxes only occur in language, not math: but because we develop math with language, math sometimes shows us where our language errs, and forces us to be more careful. But I guess that's neither here nor there.
I've been studying infinite sets lately, so I was looking forward to responding to the OP. Imagine my distress when you beat me to the punch!

I just have one point of disagreement with your last comment. Gödel proved that we can't be entirely sure the completeness of our systems, since we choose consistency over completeness. Paradoxes can occur in mathematics, but we revise our math to cover them for the sake of consistency. For example, set theory was revised to exclude any sets that contain themselves when Russell discovered the paradox that bears his name.

[Kosh3]

Thanks JohannGoodFlag, that helped a lot.

I would just add - I think mathematics and language can describe paradoxes (create them if you will), but that none actually exist. And certainly I think more will occur in language than in maths, for it is a far less rational system.

PS - a point brought up was that Cantor used completed infinite sets rather than potential ones. Does that change anything?

[wiploc]

Originally posted by Kosh3
Alright, basically I need a second opinion, as im not a mathematician.

I'm not a mathematician either, but I still like to share my opinion. :)



Basically my argument was that one infinity added to another does not increase its cardinality, that the cardinality of infinity remains constant.

I'm not even sure of what "cardinality" means, but I still have my rash opinion.

Suppose X is a particular infinity, let's say the number of integers. Then if you multiply X by 2, you get a bigger number; it's still infinity, but it's bigger. Let's call the bigger infinity Y, as in X*2=Y.

That lets us know that Y/X=2. Saying, "infinity divinded by infinity equals 2'" might sound like nonsense, but in this case it works because we know that one of the infinities is exactly twice the size of the other.

I put this theory to some physics students once. One of them cheerfully agreed with me, and the other looked confused. So I'll be pleased to see what the mathematicians on this thread say.

So, I'm thinking that if "cardinality" means size, adding one to infinity does increase its cardinality. On the other hand, if "cardinality" means something like "order of magnitude," then adding one to infinity does not increase its cardinality. That is, if you add one to an aleph 1 infinity, you don't get an aleph 2 infinity. Or, put another way, X/(X+1)=1.

Since I wrote the above, I got a call from a physicist. I put my theory to him, and he says I've got it all wrong. He says if X equals infinity, then 2X=X and 2X/X is undefined. It's as if, once you start dealing with trans-finite numbers, you are forced to round everthing off.


It came about because we were arguing about the size of the universe, and I said that if the universe was infinately large it could not expand, because there is no place it could go that it not at already.

It helps me to think of infinity as being something like "outdoors." If you go outdoors, and then go ten feet farther, you are still outdoors.



"The set of all integers is equal to the set of positive integers (though you'd think it's twice as big, it's not). However, the set of all real numbers, rational and irrational, is greater than the set of integers, though they are both infinite. There is no debate on this issue. "

This is true. 2X may not be considered bigger than X because they are both aleph 1 infinities, but X to the power of X is bigger than X, because that's an aleph 2 infinity. So even thou 2X isn't bigger than X, there are still things that are bigger than X. It's not like infinity is the highest number, a number that you can't get bigger than.

crc

[ex-xian]

Originally posted by Kosh3
Thanks JohannGoodFlag, that helped a lot.

I would just add - I think mathematics and language can describe paradoxes (create them if you will), but that none actually exist. And certainly I think more will occur in language than in maths, for it is a far less rational system.

PS - a point brought up was that Cantor used completed infinite sets rather than potential ones. Does that change anything?
What about the statements,

The sentence below is false.
The sentence above is true.

Or, more succinctly,
This sentence is false.

As I said above, Russell discovered a paradox in set theory, Russell's Paradox (I seem to recall reading that someone discovered it before Russell, but I can't remember who it was).

Russell's Paradox comes about like this:
For Ax, P(x) implies x=x. Let S={x: P(x) is a true statement}. Note that S is the set of all sets--for otherwise, x would not be equal to x. Further, note that S is an element of S.

Now consider the set T={x: x is not in x}. That is, T is the set that contains all sets that are not included in themselves. Assume T is in T, then T is not in T, a contradiction. Assume T is not in T, then T is in T, again, a contradiction. Then T cannot be a set at all.

This was a clear and present paradox in set theory before it was axiomatized. To confront this paradox, ZFC (Zermelo-Fraenkel-Axiom of Choice) set theory includes the axiom, no set is a member of itself.

[ex-xian]

Originally posted by wiploc

I'm not even sure of what "cardinality" means, but I still have my rash opinion.
The cardinaltiy of a set is the number of members in a set. Let S={1, 3, 5}. The cardinaltiy of S is 3. Cantor defined the cardinality of the set of the natural numbers as aleph-null.

Suppose X is a particular infinity, let's say the number of integers. Then if you multiply X by 2, you get a bigger number; it's still infinity, but it's bigger. Let's call the bigger infinity Y, as in X*2=Y.
Normal algebra doesn't work with transfinite numbers. Consider the natural numbers, 1, 2, 3, 4, ..... And consider the even numbers, 2, 4, 6, 8, .... It's obvious that for any natural number, there is a corresponding even number. Therefore the cardinality of the natural numbers is the same as the cardinality of even numbers.

That lets us know that Y/X=2. Saying, "infinity divinded by infinity equals 2'" might sound like nonsense, but in this case it works because we know that one of the infinities is exactly twice the size of the other.

I put this theory to some physics students once. One of them cheerfully agreed with me, and the other looked confused. So I'll be pleased to see what the mathematicians on this thread say.

Infinity divided by infinity is an undefined operation, the same of 0/0.

So, I'm thinking that if "cardinality" means size, adding one to infinity does increase its cardinality. On the other hand, if "cardinality" means something like "order of magnitude," then adding one to infinity does not increase its cardinality. That is, if you add one to an aleph 1 infinity, you don't get an aleph 2 infinity. Or, put another way, X/(X+1)=1.

Since I wrote the above, I got a call from a physicist. I put my theory to him, and he says I've got it all wrong. He says if X equals infinity, then 2X=X and 2X/X is undefined. It's as if, once you start dealing with trans-finite numbers, you are forced to round everthing off. [/b][/quote]
I don't know about the rounding off part, but that's correct.

It helps me to think of infinity as being something like "outdoors." If you go outdoors, and then go ten feet farther, you are still outdoors.
Yes, but don't forget that there are countable and uncountable infinites. To wit,
Originally posted by JohannGoodFlag
So, the countable infinity is the one you probably know and love, but there are larger kinds of infinities, which we call "uncountable" infinities. One of the simplest examples would be the set of all points on the real line. This kind of infinity is usually called (the infinity of) the continuum. Another example which you might run across is the set of all sets of natural numbers: this also happens to have the cardinality of the continuum.

[jpbrooks]

Originally posted by Kosh3


I would just add - I think mathematics and language can describe paradoxes (create them if you will), but that none actually exist.



I agree. "Real" paradoxes cannot exist.



PS - a point brought up was that Cantor used completed infinite sets rather than potential ones. Does that change anything?

I would say that it does because completed infinite sets, unlike potentially infinite ones, are not obtained by "adding together" (or juxtaposing?) finite quantities or things, even though such completed infinite sets (the set of real numbers in any line segment with a length greater than zero, for example), once obtained, can be subdivided into finite subsets. The set of all individual finite objects in the universe cannot be "added to" one another to form a completed infinite set. Thus propositions that are true only for completed infinite sets cannot be presumed to be applicable to the entire collection of finite things that make up the universe.

But a related point is that it is just not clear (to me, at least) how any argument concerning infinities of cardinality greater than aleph-1 could apply to the universe of real "objects". What actual collection of physical, material, or corporeal things in the universe has a cardinality greater than that of the real numbers?

[wiploc]

Originally posted by jpbrooks
But a related point is that it is just not clear (to me, at least) how any argument concerning infinities of cardinality greater than aleph-1 could apply to the universe of real "objects". What actual collection of physical, material, or corporeal things in the universe has a cardinality greater than that of the real numbers?

If space is contiguous, then there are as many places on a one inch line as there are numbers between zero and one.

On the other hand, if space is not contiguous, then the circumference of the universe has jaggies.
crc

[jpbrooks]

Originally posted by wiploc
If space is contiguous, then there are as many places on a one inch line as there are numbers between zero and one.

On the other hand, if space is not contiguous, then the circumference of the universe has jaggies.
crc

If space is not contiguous, what exists in the "gaps"?

But how does any of this alter the "cardinality of space"? The cardinality of "places" ("points", etc.) in space, as of "points" on a number line, does not depend on how it is segmented, (and again, cannot be greater than aleph-1).

[wiploc]

Originally posted by jpbrooks
If space is not contiguous, what exists in the "gaps"?

I'm clearly in over my head here, but it seems to me that the answer has to be: nothing. And that's a way nothing too, not just emptiness. It has to be a nothingness such that there aren't even any places in it.



But how does any of this alter the "cardinality of space"? The cardinality of "places" ("points", etc.) in space, as of "points" on a number line, does not depend on how it is segmented, (and again, cannot be greater than aleph-1).

The cardinality of a one inch line is aleph-1. If you could somehow remove all the real points, leaving only the integer points, then it would have a cardinality of aleph-null. If you removed everything but the end points, you would have a cardinality of 2.

It's not a matter of segmenting a contiguous line. It's a matter of making a line out of a reduced number of points. I know that doesn't make sense. We think of space as contiguous, so it doesn't seem sensible to imagine non-contiguous lines.

But, we have to make a choice. We have an either/or situation. Only one thing is true. Either "infinity" describes a real world situation (the number of places there are) or space is not contiguous (there are not an infinite number of places).

If you ever find yourself on stage debating William Lane Craig; and he employs the argument from audience incredulity to show that the world is not infinitely old because there are no infinities in the real world; then it becomes appropriate for you to show the audience the incredible implications of his claim.

For instance, if space is not contiguous, then there is no such thing as movement from one place to another. That is an illusion. All "movment" actually consists of a series of teleportations.

crc

[JohannGoodflag]

First, some corrections:

Attributed to me by ex-xian
The set of all integers is equal to the set of positive integers (though you'd think it's twice as big, it's not). However, the set of all real numbers, rational and irrational, is greater than the set of integers, though they are both infinite. There is no debate on this issue.
I did not actually say this, but I mostly stand behind this statement.

And then, in my post:
I would argue that paradoxes only occur in language, not math: but because we develop math with language, math sometimes shows us where our language errs, and forces us to be more careful. But I guess that's neither here nor there.
As pointed out by a few posters, this is not quite correct (e.g. the existence of Russell's paradox). Of course, paradoxes discovered in the past have forced refinements of mathematics, so clearly it can arise in some models of mathematics. I have only recently (in the past two months) begun a transition in my philosophy of mathematics from an absolutist stance to a pragmatist stance, so my statement was a relic of my previous world-model. Also, as ex-xian pointed out, Godel showed that we can never prove that mathematics is correct, so my statement was unsupported to begin with.

What I should have said is this: I postulate that there is a useful model of mathematics which is free of contradictions (although one could never prove it to be so). Thus, in my view, it is not necessarily true that mathematics contains contradictions, although some "inferior" models of math certainly do.

wiploc:
As ex-xian points out, you may find that when you try to treat infinities like "normal" numbers, you run into all kinds of problems. For instance, the fact that X/(X + 1) is undefinined if X is an infinite cardinal (that is, an infinity in the sense of "infinite set" introduced by Cantor). Because the infinite is so much different from the finite in some respects, something in the usual rules of algebra have to give way when you add infinities to your number-collection.

Having said this, you may be interested in investigating the topic of "nonstandard numbers", which includes formalised concepts of infinite numbers and infinitesimals. These infinitie(simal)s are things you can do algebra with, and indeed use to build most of the practical results from calculus ("nonstandard analysis"). They're valid mathematical concepts, although unpopular for some reason. Also, the infinite non-standard numbers don't correspond well with the idea of the size of a set, or sums of the form (1 + 1 + 1 + ...), AFAIK, and expressions such as 1/0 are still undefined in non-standard analysis.

jrpbrooks:
[...]completed infinite sets, unlike potentially infinite ones, are not obtained by "adding together" (or juxtaposing?) finite quantities or things, even though such completed infinite sets (the set of real numbers in any line segment with a length greater than zero, for example), once obtained, can be subdivided into finite subsets. The set of all individual finite objects in the universe cannot be "added to" one another to form a completed infinite set. Thus propositions that are true only for completed infinite sets cannot be presumed to be applicable to the entire collection of finite things that make up the universe.

If one supposes the universe is infinite, then the objects in the universe already form an infinite set, so I'm not so sure your statement holds. Of course, if the universe is finite, then juxtaposing them cannot form an infinite set, but for a much less interesting reason: because there is only finitely many of them. (I have no informed opinion one way or the other as to whether it is actually infinite.)


But a related point is that it is just not clear (to me, at least) how any argument concerning infinities of cardinality greater than aleph-1 could apply to the universe of real "objects". What actual collection of physical, material, or corporeal things in the universe has a cardinality greater than that of the real numbers?
[...]
The cardinality of "places" ("points", etc.) in space, as of "points" on a number line, does not depend on how it is segmented, (and again, cannot be greater than aleph-1).

First, a brief note. The cardinality of points in space can (in principle) exceed aleph-1. The continuum is not necessarily the same as aleph-1: it could be aleph-2, aleph-3, or indeed, aleph-17. The idea that the continuum is aleph-1 is the continuum hypothesis, and is independent of set theory (it cannot be proven or disproven in Zermelo-Frankel). It must be assumed if it is to be used.

As to what collection of objects could exceed the cardinality of the continuum: we don't know, but not knowing is not a strong argument. The idea that one could have more than one kind of infinity was itself mind-blowing at Cantor's time. I will admit that it is very difficult to concieve of how anything larger than the continuum could have any practical relevence for physics, but that does not mean that there might not be some relevance.

If space is not contiguous, what exists in the "gaps"?
It depends on whether the gaps are actually there. We have learned that matter is not (necessarily) infinitely divisible: at the very least, it gets very chunky at a small level. Most certainly it is not a smooth continuum. It is possible, in principle, that the same is true of spacetime. I'm not qualified to say in what way, but you are almost certainly aware of at least one discrete space --- a chess-board. If space is discrete, a question then of what exists in the gaps in space would be like asking what exists between the squares on a chess board (or asking what integer comes between 1 and 2).

Kosh3:
I am unaware of a clear, mathematical definition as to what a "completed infinite" is, so I know of nothing that can actually be said about them. The philosophical issue of actual vs. potential infinites is a matter of metaphysics IMO, and so is unrelated to mathematics, except in the sense that one could try to treat them mathematically by providing definitions.

Infinities as they occur in modern set-theory, with Cantor's definition of cardinality, are not actual or potential. They are just infinities as they occur in modern set theory.

JohannGoodflag

(edited for minor errata)

[ex-xian]

Originally posted by JohannGoodflag
First, some corrections:
Attributed to me by ex-xian
The set of all integers is equal to the set of positive integers (though you'd think it's twice as big, it's not). However, the set of all real numbers, rational and irrational, is greater than the set of integers, though they are both infinite. There is no debate on this issue.

I did not actually say this, but I mostly stand behind this statement.

Oops! I accidentally pasted the wrong paragraph. I've edited that post and put in what I had wanted to say. Now the last part actually makes sense.

Originally posted by JohannGoodflag
I have only recently (in the past two months) begun a transition in my philosophy of mathematics from an absolutist stance to a pragmatist stance, so my statement was a relic of my previous world-model.
What a coincidence...I've done the same thing. My xian upbringing influenced me into being a mathematical absolutist/realist. I was fond of saying that mathematics is the languange of god.

I'm finishing up a BS in math, and will start grad school in the spring, but I also have a minor in philosophy (and physics :) ), so I did an independent study class in analytic philosophy. Studying the analytic tradition, from Russell through Rorty, caused me to question my previous conception of what mathematics really is. I've also began to take a quasi-pragmatist stance regarding mathematics (very much agaisnt my will). I now say that mathematics isn't the language of god, but is, instead, one of his hobbies.

[JohannGoodflag]

I'm finishing up a BS in math, and will start grad school in the spring, but I also have a minor in philosophy (and physics ), so I did an independent study class in analytic philosophy. Studying the analytic tradition, from Russell through Rorty, caused me to question my previous conception of what mathematics really is. I've also began to take a quasi-pragmatist stance regarding mathematics (very much agaisnt my will). I now say that mathematics isn't the language of god, but is, instead, one of his hobbies.
Funny, based on your nick, I supposed you had different religious beliefs. Or, is this a pop-sci distillation you present to others?

My former absolutist stance for mathematics was probably also due to a christian upbringing, or at the very least, was strongly reinforced by it.

I'm curious: by quasi-pragmatist, what do you mean? When I say pragmatist, I mean "This seems to provide a wide variety of mental tools, which seem to represent interesting patterns or solve interesting problems, many of which I meet in the real world". I also use standard formal logic (as opposed to a logic with multiple values, say, or a non-standard collection of rules of inference) as a basis for math, which I also choose for the same pragmatic reasons.

JohannGoodflag

[spacer1]

I have never been able to grasp how one infinity could be "larger" than another.

On the issue of "countable" vs. "uncountable" infinities, what makes the set of points in a continuum any larger than the set of natural numbers? If the set of even numbers is the same "size" or cardinality as the set of all integers, why shouldn't the same logic apply to the continuum? They all just keep going on forever, don't they?

On reading back, perhaps the answer to my questions lies in the fact that a continuum does not have any "points", which is what makes it "uncountable"? However, I have long intuited that this continuity is a necessary feature of infinity.

If a set is countable, mustn't it also be finite?

[JohannGoodflag]

Hello spacer1,
I'm going to reply to your concerns in reverse order.

If a set is countable, mustn't it also be finite?
This is a matter of terminology: some people use the word differently. The concept itself, however, is standard. By "countable", I just mean a set where there's a way of listing the elements so that any one element can be reached in a finite amount of time. (Then, a finite set will also countable, but there are contable sets which are infinitely large --- such as the natural numbers.)

On reading back, perhaps the answer to my questions lies in the fact that a continuum does not have any "points", which is what makes it "uncountable"? However, I have long intuited that this continuity is a necessary feature of infinity. Well, a continuum definitely contains points (of the zero-extent mathematical variety): it contains infinitely many of them. Just because they aren't in isolation does not mean they aren't there. However, it's possible to have an infinite number of points without a continuum.

Here, I'm going to use a colloquial definition of continuum: a curve which has no gaps. (What I'm actually proving here, in mathematical terms, is that these sets are disconnected in the real line: we use the word "continuum" in a different way than the physical, materialist sense.)

Some common examples:
The natural numbers themselves are obviously infinite in quantity, but they do not form a continuum: there's a lot of space between any two of them.
The sequence (1, 1/2, 1/4, 1/8, ...) has infinitely many points which end up crowding near zero, but there is no continuity: each point in this collection can be isolated from all of the others, so this is clearly not a coninuum.
Finally, consider the rational numbers. There are infinitely many of them, but you can easily concieve of lengths which are constructible by ruler-and-compass (such as the hypothenuse of a right-triangle, with the base and the height of unit 1) which lie on the real line, but which cannot be represented by a rational number. So, there are points on the line which are not represented by rational numbers: the set of rational numbers has "gaps". (These "gaps" are "infinitely small", but they are nonetheless there.) So, the rationals also do not form a continuum.
So, there are infinite sets around which lack the property of continuity.

On the issue of "countable" vs. "uncountable" infinities, what makes the set of points in a continuum any larger than the set of natural numbers? If the set of even numbers is the same "size" or cardinality as the set of all integers, why shouldn't the same logic apply to the continuum? They all just keep going on forever, don't they?Well, the same logic does apply to the continuum and the countably infinite --- but because they are not exactly the same thing, the logic produces different results.

I will try to briefly present the mathematical treatment of the infinite. I'm going to get a little bit more technical, but not too much. We say (in math) that a set is B "at least as big" as another set A, if you can come up with a function f: A --> B which maps each of the elements of A to unique elements of B. For all x and y, we need f to be such that f(x) = f(y) if and only if x = y. Such a function f is said to be one-to-one, or injective.
Two sets are said to be the same size if A is at least as big as B, and B is also at least as big as A. So, there are injective functions from A to B, and from B to A as well.
To make things simpler, it is possible to prove that two sets are the same size (in the sense just above) if there is a bijection from A to B: that is, a function which is one-to-one, and which exhausts all of the elements of B --- for every b in B, there is some a in A such that b = f(a).
We say that B is bigger than A if B is at least as big as A, but B is not the same size as A.(We use this formulation to define "countable" and "infinite" in mathematics: we say that a set S is countable if the set of natural numbers N is at least as big as S, and we say S is infinite if S is at least as big as N.)

All finite sets are countable in this terminology, as I mentioned before. If you have the set S = {a,b,c}, you can form a function f: S --> N which maps

f(a) = 0 , f(b) = 1 , f(c) = 2

which is one-to-one. Then, N is at least as big as S, so S is countable.

When we say that the (natural) even numbers E have the same size as the natural numbers N, we're just saying that we can come up with the mappings required in the definitions above. f(x) = 2x will map from N to E, and is one-to-one (injective); g(x) = x/2 will map from E to N, and is also injective. So, we know N and E have the same cardinality. (Alternatively, we can observe that f by itself also exhausts all of the even numbers, so it's a bijection --- then, the existence of f by itself shows that N and E have the same size.)

What we mean when we say that the continuum (for instance, P(N), the set of sets of natural numbers) is bigger than the set of natural numbers N, is that
There does exist an injective function f: N --> P(N),
There does not exist an injective function g: P(N) --> N.
Equivalently, you can replace the second point with: "There does not exist a bijective function f: N --> P(N)."

So, that's what we mean when we say that the continuum is bigger than the countably infinite. What this means to you is that there is no way to come up with a list of all sets of integers, with a finite position for each set on the list. If you try, you will be missing some of the sets of natural numbers, and therefore the list will fail to exhaust all of the sets of natural numbers.

If you are interested in a proof of the fact that P(N) is bigger than N, or (equivalently) that the real numbers R are bigger than N, try looking up Cantor's diagonalization argument in your local library or on the web. (Some presentations are better than others, but it is possible to give a short and fairly simple presentation of it.)

JohannGoodflag

[spacer1]

Hello Johann,

Thank you for your well-presented reply. Unfortunately, I still have some doubts.
By "countable", I just mean a set where there's a way of listing the elements so that any one element can be reached in a finite amount of time.
What do you mean by "reached"? Where do we start in order to "reach" any element, and by what process do we do this reaching? That is, what steps are involved which take up "a finite amount of time"?
(Then, a finite set will also countable, but there are contable sets which are infinitely large --- such as the natural numbers.)
What makes the natural numbers infinite in the first place? Is it merely a matter of convention to designate the natural numbers as infinity (aleph null)?

I sense it may be somewhat analagous to "rounding off" decimal places. What is the point at which the natural numbers end and infinity begins? 7 billion? 3 trillion?

What is the distinction between infinite and finite here?
Well, a continuum definitely contains points (of the zero-extent mathematical variety): it contains infinitely many of them. Just because they aren't in isolation does not mean they aren't there.
What do you mean by "in isolation"? How are the natural numbers "in isolation" in a way which these supposed "points" in a continuum are not?
The natural numbers themselves are obviously infinite in quantity, but they do not form a continuum: there's a lot of space between any two of them.
Despite the "space" between them, couldn't the same isomorphic mapping over the naturals occur with the natural numbers multiplied by 10000? i.e., 1, 2, 3, 4,..... has a one-to-one correspondence with 10000, 20000, 30000, 40000,.....

My point here is that if you allow for there to be "points" in a continuum, then they can be treated just as the natural numbers are treated. Why even bother with the notion of a continuum (or infinity) then?
So, there are infinite sets around which lack the property of continuity.
But if these non-continuous infinite sets, such as the naturals, are open-ended (in order to be infinite), what stops them from being injective with a continuum of numbers? It seems no different than saying that the naturals and evens have the same cardinality.

It appears to me that we could do away with either the notion of infinity, or the notion of a continuum (or both).


*My apologies for not replying to all of your points. I have had serious computer problems lately and have had to type this on a friend's computer. I hope to return soon to read and respond further.

[JohannGoodflag]

Warning: there is a difference between the "colloquial" descriptions of infinity, and formal ones.

Originally posted by spacer1
What do you mean by "reached"? Where do we start in order to "reach" any element, and by what process do we do this reaching? That is, what steps are involved which take up "a finite amount of time"?
I'm talking here about the colloquial description of having a (possibly infinitely long) list of the elements of the set, and counting through them one at a time until you reach the item you're looking for. In this description, I assume that it takes a fixed amount of time to go through each item (maybe, to recognise that it is or is not the item you're looking for): so, it takes the same amount of time to go from the 1st item to the 2nd, as it does to go from the 1,000,001st item to the 1,000,002nd.

By "reached", I just mean to find the item in the list where the element you're looking for is situated.

What makes the natural numbers infinite in the first place? Is it merely a matter of convention to designate the natural numbers as infinity (aleph null)?
Well, the definition of "finite set" is basically that the number of elements in the set can be represented by a natural number.

This cannot be the case for N (the natural numbers themselves), because we hold that for any natural number m, m+1 is also a natural number. So, if we have any finite set of natural numbers, we can always find another natural number that is not in the set. So, any finite set is not the set of all natural numbers. To put it another way, the set of all natural numbers is infinite (= "not finite").

There is no "point" at which the finite ends and the infinite begins. They are not regions which abut one another. You can explore "finite numbers" by adding and subtracting finite numbers, multiplying and dividing by finite numbers, squaring, cubing, exponentiating, whatever you like --- and you will never be any closer to "the infinite", because "the infinite" is not a place per se. We just define infinities, use them, and in time learn to recognise them when they appear.

The relationhip between the finite and the infinite is almost, in a way, like the relationship between triangles and squares.A square is a square, and you cannot come close to having a square by exploring possible triangles. (Like all analogies, though, this one is imperfect.)

What do you mean by "in isolation"? How are the natural numbers "in isolation" in a way which these supposed "points" in a continuum are not?
In these examples concerning subsets of the real line, for a set S (a subset of the real numbers in this case) and a point p in S, I use the following definition: p is isolated from other elements of S if there exists a line segment L, such that p lies in the line segment L, and is the only element of S to do so. More colloquially, p is isolated if you can "fence off" p from the other elements of S.

You can't do this for the continuum: one of the basic ideas of the continuum is that every neighborhood of p contains infinitely many points. (This property is called density. This also holds true of the rational numbers, even though they do not form a continuum.) In this sense, the elements of a continuous set are not isolated from one another.

Despite the "space" between them, couldn't the same isomorphic mapping over the naturals occur with the natural numbers multiplied by 10000? i.e., 1, 2, 3, 4,..... has a one-to-one correspondence with 10000, 20000, 30000, 40000,.....

My point here is that if you allow for there to be "points" in a continuum, then they can be treated just as the natural numbers are treated. Why even bother with the notion of a continuum (or infinity) then?

Yes, it's isomorphic. This 1:1 (and exhaustive) correspondance shows that the set of positive multiples of 10000 is the same size as the positive integers. The amount of space between them isn't fundamental to the cardinality of the sets: I was only illustrating how the integers failed to form a continuum.

Can you tell me what it means, to you, to be able to "treat numbers like the natural numbers are treated"? I'm not sure what to say to your question.

One thing I can tell you is that you can identify many, many points in a continuous line segment (although not all), by considering the points with rational distances from one of end points. Take, for example, a line segment from 0 to 1. We know that 1/2 is a point in this segment. So are 1/4 and 3/4, and so are 1/8, 3/8, 5/8, and 7/8. The line segment contains its' half-way point, which defines two smaller line segments: these line segments also contain their half-way points, and so on. A la Zeno, we have an infinite number of points, and they are all contained in the unit interval. These and many more points are the points that may exist within a continuum. This, to answer how one could "allow" for there to be points in a continuum.

Why bother with the concept of infinity? Because we recognise that although humans will never actually consider an infinite number of things individually, we can consider collections of objects which may have more elements than any natural number represents. The cardinality of some collections may exceed any particular natural number. This is the concept of what it is to be infinite again.

Why bother with the concept of the continuum? Well, that's bounded up in calculus (where the concept of continuity is extremely important and useful). As for the infinity of the continuum, this is just the result when you use Cantor's definition of cardinality (the technical one I presented earlier): there will be more than one kind of infinity, and the infinity of the number of points on the real line is an infinity which is larger than the infinity of the natural numbers. (See my previous post.)

But if these non-continuous infinite sets, such as the naturals, are open-ended (in order to be infinite), what stops them from being injective with a continuum of numbers? It seems no different than saying that the naturals and evens have the same cardinality.

It appears to me that we could do away with either the notion of infinity, or the notion of a continuum (or both).

Nothing stops the integers from being injective with the continuum. That just means that the reals are at least as big as the naturals (see my previous post). However, you also have to have an injection from the reals to the naturals in order for the reals and the naturals to have the same size.

In your example with the naturals and the even numbers, the detail you have glossed over is that you not only have an injection, you have a bijection: not only do you have a 1:1 correspondance, but you exhaust every even number as well. It is this which you cannot do with a map from the integers to the reals.

The presence of an injective function just means that the domain is smaller than or equal to the range, not that it is the same size. In order to have the same size, a bijection is required: one-to-one, and onto.

My apologies for not replying to all of your points. I have had serious computer problems lately and have had to type this on a friend's computer. I hope to return soon to read and respond further.
No worries about not responding to all my points (yet)... I don't exactly help by writing such long posts. :) I do help this is helpful though, and hope I can clarify more.

JohannGoodflag

[Bellarmino]

Maybe you can help me with a question that I have wondered about: I understand why there are "more" real numbers than whole numbers. Are there any sets with "more" than the real numbers?

[Demosthenes]

When you represent the cardinality of an infinite set with transfinite numbers such for example aleph-null is assigned to the cardinality of the set of natural numbers. Larger transfinite numbers can be constructed and in fact there are infinite transfinite numbers. I imagine one can also futher construct even higher transfinite numbers representing the set of transfinite numbers.

The cardinality of the set of real number is known to be larger than the set of natural numbers, but we don't know where exactly in the transfinite hierarchy the real numbers' cardinality is located. That is basically the continuum hypothesis proposed by Cantor who originally developed the transfinite arithmetics. In more depth, the hypothesis states that every uncountable set has at least as many elements as the set of real numbers. We think it's aleph-1, the next number after aleph-null. Unfortunately it's been discovered that the continuum hypothesis is not decideable(sofar as I understand it). Nevertheless, the hypothesis remain an interesting area of study.

ex-xian, I'm curious to what approach of quantum gravity you'll be studying...string theory or loop quantum gravity? If I were to get into that field, I'd go into loop quantum gravity for sure :D

I still have a year and half before I finish my mathematics BS. There's so much to learn I sometimes despair of ever comprehending the entire breadth of today's mathematics.

[JohannGoodflag]

Originally posted by Bellarmino
Maybe you can help me with a question that I have wondered about: I understand why there are "more" real numbers than whole numbers. Are there any sets with "more" than the real numbers?First, two questions:
Do you also understand why the set P(N) of sets of whole numbers is larger than the set N of whole numbers?
Are you looking for any old set, or are you looking for a set of "numbers" which is bigger than the set of real numbers?

[spacer1]

Hello Johann, thanks again for your fine response.
Warning: there is a difference between the "colloquial" descriptions of infinity, and formal ones.
I'm not sure that I am familiar with either, except for my online readings in trying to understand Cantor's concepts, as well as my own thoughts on the subject.
I'm talking here about the colloquial description of having a (possibly infinitely long) list of the elements of the set, and counting through them one at a time until you reach the item you're looking for.
Colloquially then, at least, the concept of time is bound up in the concept of infinity. In principle, it would take an infinite amount of time to count an infinite amount of numbers. Hence, my analogy to "rounding off".

I am unsure whether formal descriptions are also reliant on this time factor?
Well, the definition of "finite set" is basically that the number of elements in the set can be represented by a natural number.
But isn't any set able to be represented by a natural number? This touches upon my earlier point that once you define points along a line, then it becomes necessarily finite. I maintain that infinity cannot be "pinned down" (or, that you cannot define points on a continuum, for it to remain a continuum).
This cannot be the case for N (the natural numbers themselves), because we hold that for any natural number m, m+1 is also a natural number.
Don't you mean "the set of natural numbers themselves"? If so, then by saying "N is a set" would seem to necessarily enclose that set and make it finite.

This is not to say that I disagree that the naturals, or bijectives thereof, are infinite. My concerns are more related to higher cardinalities of infinity.
...p is isolated from other elements of S if there exists a line segment L, such that p lies in the line segment L, and is the only element of S to do so.

...one of the basic ideas of the continuum is that every neighborhood of p contains infinitely many points....In this sense, the elements of a continuous set are not isolated from one another.
If you allow for there to be points on a continuum, though, then wouldn't any selected element be the only element to lie on that particular point? If you say that 0.536 lies on the continuum, I don't see how there could be any overlap with any other point on that line.

I still don't quite see the distinction.
Can you tell me what it means, to you, to be able to "treat numbers like the natural numbers are treated"? I'm not sure what to say to your question.
To treat them as finite numbers with definite values. To say they are infinite is merely the logical deduction that, as you say, for any natural number m, we can always find m+1. It is as though the numbers get ahead of us too fast for us to be able to pin them all down.
Why bother with the concept of the continuum? Well, that's bounded up in calculus (where the concept of continuity is extremely important and useful).
However, calculus can only ever be an approximation to continuity, with its reliance upon (finitely defined) differentials.
In your example with the naturals and the even numbers, the detail you have glossed over is that you not only have an injection, you have a bijection: not only do you have a 1:1 correspondance, but you exhaust every even number as well. It is this which you cannot do with a map from the integers to the reals.
Thanks for the clarification of terms. I did intend bijection.

My point here was that with both sets of naturals and evens, there would need to be twice as many naturals for a bijection with the evens (e.g., no. of naturals from 1 to 10 = no. of evens from 1 to 20). Given the open-endedness of these sets, couldn't the same relationship apply between the natural and the real numbers?

I don't see a need for higher cardinalities here. Higher cardinalities merely appear to be a mathematical analogue to self-referential paradoxes. I only see a need for one infinity.

[Bookwyrm]

originally posted by spacer1
Given the open-endedness of these sets, couldn't the same relationship apply between the natural and the real numbers?

No, and here is a sort of proof constructed with the help of the Discrete Mathematics textbook I didn't get around to selling back to the college book shop:

Consider a pairing of elements from the set of Natural numbers and the set of Real numbers between 1 and 0. (this will work for any pairing you care to come up with, so I'll just choose one arbitrarily)

For example:

0 <--> .10101...
1 <--> .12312...
2 <--> .12412...
3 <--> .13913...
4 <--> .24624...

and so on, using your imagination to pretend that someone has assigned a Real number with a value between one and zero to each natural number.

Now, it can be shown that this pairing omits at least one real number with a value between 0 and 1.

arrange the Real numbers which have been paired off on a table like the one below, and circle digits in them according to this rule: the nth number has its nth digit after the decimal circled.


.(0)1010...
.1(2)312...
.12(4)12...
.139(1)3...
.2462(4)...
. .
. .
. .


now, we will generate a Real number and see if its one of the ones we've paired off. The rule that will generate this number is "for each circled digit on the table above, in order, we will write a 1 if the circled digit is a 0, otherwise we'll write a 0."

Using the table above we get the number .10000...

So now to see if its already on our list. Well its obviously not any of the numbers I have already written down, nor can it match any of the other numbers on the list, because our rule for generating this number guaranteed that the nth number in our pairing will have a different value for the nth digit after its decimal place than the new number has in that digit. As this rule when applied to any proposed one-to-one correspondence between the Natural numbers and the Real numbers will generate a number not in the correspondence, there can be no bijection between the Naturals and the Reals.

[JohannGoodflag]

JohannGoodflag
Warning: there is a difference between the "colloquial" descriptions of infinity, and formal ones.

spacer1
I'm not sure that I am familiar with either, except for my online readings in trying to understand Cantor's concepts, as well as my own thoughts on the subject.
I was posting this disclaimer as a defensive measure. I've already posted the formal definition of infinity --- in response to your first post, in fact. I'll summarize it again here, to reduce clutter and to spare you a thread search.
Definition of "infinite" and "countable"
We say that a set is B at least as big as another set A, if there exists an injective function f: A --> B. Two sets A and B are said to be the same size if A is at least as big as B, and B is also at least as big as A. Equivalently, A is the same size as B if there is a bijection from A to B. We say that B is bigger than A if B is at least as big as A, but B is not the same size as A.
Define the set N of natural numbers. Then, we use this formulation to define "countable" and "infinite" in mathematics: S is countable if N is at least as big as S.
S is infinite if S is at least as big as N.As you can see, no concept of time is involved.

But isn't any set able to be represented by a natural number? No. Why do you think so? This is only true of finite sets.

I don't want to sound condescending, but here goes anyay: you seem to be confusing some parts of the idea of infinity with other parts. Each natural number, and each real number, is finite: each natural number is finite in that it is the cardinality of a set which is not infinite (taking "infinite" from the definition above), and each real number is finite in the sense that its absolute value is less than or equal to some natural number. However, this says nothing about the how many natural numbers or real numbers there are. There are infinitely many of both, so we say that both the set of natural numbers N and real numbers R are infinite.

By the argument written by Bookwrym above (known as Cantor's diagonalization argument), there cannot be a bijection f: N --> R. Therefore, by the definition I gave above, the set of reals R is bigger than the set of naturals N: that is, there are strictly more of them.

A set may have infinitely many elements: this causes no contradictions. In particular: the set of natural numbers N is itself infinite (in the informal sense), because there are infinitely many natural numbers, as you acknowledge. (I say in the informal sense, because N is infinite trivially in the formal sense: N is at least as big as N because of the identity function.)

JohannGoodflag
[...] one of the basic ideas of the continuum is that every neighborhood of p contains infinitely many points [...] In this sense, the elements of a continuous set are not isolated from one another.

spacer1
If you allow for there to be points on a continuum, though, then wouldn't any selected element be the only element to lie on that particular point? If you say that 0.536 lies on the continuum, I don't see how there could be any overlap with any other point on that line.
True: but not matter how small a 'fence' you try to draw around any point, there will always be infinitely many points which are closer. There is no overlap, but the points of the line crowd in around any one point you choose, and indeed crowd 'infinitely close'. No single point is infinitely close to the point p, but taken as a collection, they certainly do come infinitely close.

Two questions:

Do you recognise the sequence (1, 1/2, 1/4, ...) as having infinitely many elements?
What would you say about the distance of the members of this sequence from the number 0?


[...] calculus can only ever be an approximation to continuity, with its reliance upon (finitely defined) differentials.
Calculus isn't an approximation: the measurements that you make in the real world are the approximations. Calculus is the bit of math that we use to deal with the things that we are measuring. We use calculus on the measurements because the measurements are the only information we have, and we get approximate answers. So, the input is approximate, and the output is approximate. The tool itself, however, is not an approximation: it is a tool to deal with the approximations we make when measuring.

The same can be said about all of mathematics: every last bit. For calculus, the only difference is that the fact that the measurements are approximate is more obvious. This is not a strike against calculus: the fact that it works so well despite the fact that it only ever gets used for obviously-approximate input data actually is a point strongly in its' favour.

My point here was that with both sets of naturals and evens, there would need to be twice as many naturals for a bijection with the evens (e.g., no. of naturals from 1 to 10 = no. of evens from 1 to 20). But here, you're not counting all of the evens/natural numbers. You're not even counting them in the same domain. The concept you're putting forward is certainly an intuitive sort of idea, but intuition is mostly built from prior experience, and I would wager that you haven't dealt with infinite collections of things in real life.

The concept you're talking about is something more like population density than total population: you can take a sample space, and notice that there are about half as many even numbers in that space as there are natural numbers. The even numbers are half as common, in a sense, as the natural numbers. But even though they are spread out more, there are exactly the same total number of even numbers as natural numbers.

If you are still having difficulty with the concepts, perhaps you could try articulating the reasoning you have done yourself concerning infinity. Some of the ideas that you seem to hold (based on your writing here) which I find puzzling are:

That a set may only contain finitely many elements
That a set can only be described by coming up with an exhaustive list of its' elements
That a set of "definite finite values" can only have finitely many elements

A shot in the dark: from your point of view, is mathematics is only true to the extent that it refers to obviously recognisable circumstances in the physical world? If so, this would seem to explain your position, and why our positions differ.

JohannGoodflag

[spacer1]

Disclaimer to Bookwyrm, Johann and lurkers:
Although I may risk looking like a fool by questioning what I assume to be well-accepted mathematical principles, this questioning is pretty much what I view philosophy to be. Even if I am wrong, it will hopefully help me to correct my errors and also help to strengthen your own views.

Bookwyrm:
The rule that will generate this number is "for each circled digit on the table above, in order, we will write a 1 if the circled digit is a 0, otherwise we'll write a 0."
Why not just call the number 3 the number 4, and say that 3 x 3 = 16? (Apologies, but I have limited computer access and haven't fully grasped the diagonal argument, but I want to reply while I have the opportunity).
Well its obviously not any of the numbers I have already written down, nor can it match any of the other numbers on the list, because our rule for generating this number guaranteed that the nth number in our pairing will have a different value for the nth digit after its decimal place than the new number has in that digit.
I find it easier just to pair up the first natural with 0.00....01, the second with 0.00....02, etc. I don't think it matters how many zeroes those dots represent since, if the naturals extend on forever, so will the reals. Without the full list in your example, I find it hard to see why 0.1000... could not be on it, and therefore, cannot comprehend how there cannot be the same number of reals as naturals.

Johann,
S is infinite if S is at least as big as N.

As you can see, no concept of time is involved.

Do you not define N as being infinite in the first place because of time constraints, regardless of S?
By the argument written by Bookwrym above...
See my response to him.
True: but not matter how small a 'fence' you try to draw around any point, there will always be infinitely many points which are closer. There is no overlap, but the points of the line crowd in around any one point you choose, and indeed crowd 'infinitely close'. No single point is infinitely close to the point p, but taken as a collection, they certainly do come infinitely close.
But we weren't speaking of a range of numbers, we were speaking of specific points being "in isolation".
Do you recognise the sequence (1, 1/2, 1/4, ...) as having infinitely many elements?
Yes.
What would you say about the distance of the members of this sequence from the number 0?
That they can be infinitely close to (or far from) zero, but cannot be zero.
Calculus isn't an approximation: the measurements that you make in the real world are the approximations. Calculus is the bit of math that we use to deal with the things that we are measuring.
I disagree. I don't see any approximation in 2 + 2 = 4, but I do believe (from memory of my high school studies over ten years ago) that, regardless of any real world applications, calculus only gives approximate answers, due to the use of the delta/change function. (I trust you'll be able to correct me if I am wrong.)
The concept you're talking about is something more like population density than total population: you can take a sample space, and notice that there are about half as many even numbers in that space as there are natural numbers. The even numbers are half as common, in a sense, as the natural numbers. But even though they are spread out more, there are exactly the same total number of even numbers as natural numbers.
I agree that there are the same total number, and it is because I assume that they have different domains. It is because of this open-endedness that they can have the same total number of elements. The same, therefore, could be said of the reals, right?
[Explain:] That a set may only contain finitely many elements
This probably relates to my understanding of a "set", which may not be in accordance with the mathematical definition.

I would define a set as being some specified collection of elements, which is held fixed for some communicative purpose at some specific time. I have no problem with saying that, logically, the natural numbers extend on infinitely (or that, if we were to physically count the naturals it would take a very long time which may only be limited by death, or by our not having descriptions for certain very large numbers), but the concept of a "set" suggests to me finiteness and boundedness.
[Explain:] That a set can only be described by coming up with an exhaustive list of its' elements
This relates to my previous response, but I don't find any real problem with describing a set as infinite, bearing in mind that the so-called set isn't the finite collection of elements that I would normally define a set to be.
[Explain:] That a set of "definite finite values" can only have finitely many elements
Since we could put such values into a one-to-one correspondence with the naturals (or a bijection thereof), then there should be a specific finite natural number which could be associated with each element, despite how many elements there are. You may say that there are an infinite number of such elements, but infinity is not a number. It is a logical, shorthand extension.
A shot in the dark: from your point of view, is mathematics is only true to the extent that it refers to obviously recognisable circumstances in the physical world? If so, this would seem to explain your position, and why our positions differ.
It's a tough question. I'm not sure.

[JohannGoodflag]

spacer1
Do you not define N as being infinite in the first place because of time constraints, regardless of S?
The short answer is yes. Is this a problem?

In the spirit of creating an abstracted ideal, I see no problem with having an infinite set. The fact that we only ever need to use a finite subset of the natural numbers is not important: by making N infinitely large in the informal sense, we don't put any restrictions on ourselves. After all, although we will only ever use a finite number of the natural numbers, we don't know how many of the natural numbers we will need. By not adding some prespecified ceiling, we can make use of all the numbers that we may need.

We do assume N is infinite, so that we make N the standard against which other infinite sets are measured. If you like, you can define sets for yourself to only ever be finite: but then you'll be doing a different (and in my opinion, less powerful and less useful) form of mathematics than I usually do. Also, you'll have to cope with concepts such as "the largest number" which, depending on your philosophical bent, may be awkward.

[...]we weren't speaking of a range of numbers, we were speaking of specific points being "in isolation". True. But in isolation from what? When I broached the topic, I meant "in isolation from all the other real numbers". There is no way to isolate any point from all the other real numbers, because they get packed in infinitely close.

JohannGoodflag Do you recognise the sequence (1, 1/2, 1/4, ...) as having infinitely many elements? [...] What would you say about the distance of the members of this sequence from the number 0?

spacer1
That they can be infinitely close to (or far from) zero, but cannot be zero.

Here I may have an opening to explore your viewpoint. Would you agree that individual points in the sequence are always a finite, non-zero distance from 0? Would you agree that the sequence as a whole (taken as a single "object", so to speak, consisting of many parts) is infinitely close to 0? Would you not describe "infinitely close" as being "at zero distance"? If not, would you describe it in terms of infinitesimals, or some different concept?

The point I'm putting across is that the infinite collections of points can have different properties than finite collections of points. Perhaps you avoid this by only dealing with finite sets, and although it is you perogative to do so, I hope you see it is not necessary to avoid infinite sets for this reason.

If you like, we can drop this part of the discussion. The "points in isolation" topic was just one approach I was trying to put across the standard-mathematical view of infinity, but if it doesn't help, there's no sense beating it to death.

I don't see any approximation in 2 + 2 = 4, but I do believe (from memory of my high school studies over ten years ago) that, regardless of any real world applications, calculus only gives approximate answers, due to the use of the delta/change function. (I trust you'll be able to correct me if I am wrong.) 2 + 2 = 4 is indeed exact. My question is: 2 and 2 of what? How do you decide when there is two of something before you? How do you decide when there are two groups of two objects before you? What if its 2 litres of water and 2 litres of alcohol --- do you expect to obtain 4 litres of 100 proof? There are measurements involved which may be approximate, and there are judgements as to whether merely adding the two measurements together is the appropriate operation.

I will state that calculus is exactly as precise as grade-school addition. The way that it is often taught is as the limit of a sequence of approximations (e.g. using narrow rectangles to fill a curved area for integration), but the limit is not itself an approximation: it is the exact answer. For some integrals that we do not yet know how to solve symbolically, we can always fall back on approximation to get partial answers. But there are big areas in calculus where everything can be done exactly.

I do not know just now how to convince you that calculus is exact, if you wish to think of it as being approximate. However, if all you're looking for is an educated opinion, this is mine: like all of mathematics, calculus is exact, although you can get an approximate answer by estimating, rounding off, and so on. It's a matter of how you use it. Calculus is no more an approximation than addition is: you can round off or approximate in calculus, but you can do that when calculating the tip for your restaurant bill too. That doesn't make arithmetic inherently approximate: it makes it flexible and fault-tolerant.

I agree that there are the same total number, and it is because I assume that they have different domains. It is because of this open-endedness that they can have the same total number of elements. The same, therefore, could be said of the reals, right? No, for reasons explained by Bookwyrm (which I will try to make clearer further below).

I would define a set as being some specified collection of elements, which is held fixed for some communicative purpose at some specific time. I have no problem with saying that, logically, the natural numbers extend on infinitely (or that, if we were to physically count the naturals it would take a very long time which may only be limited by death, or by our not having descriptions for certain very large numbers), but the concept of a "set" suggests to me finiteness and boundedness. It depends on what you mean by "to specify". In mathematics, a set is defined by a property P: some predicate which is satisfied by all the members of the set, and which is not satisfied by anything which is not a member of the set. If there are an infinite number of objects which staisfy a given property, the set which corresponds to that property will be infinite. Of course, if by "to specify" you mean to produce an exhaustive list within your lifetime, then a set will be finite. So, it depends on what you actually want to do with sets. In standard mathematics, the first approach is the one taken.

Since we could put such values into a one-to-one correspondence with the naturals (or a bijection thereof), then there should be a specific finite natural number which could be associated with each element, despite how many elements there are. You may say that there are an infinite number of such elements, but infinity is not a number. Of course, by claiming you can make a bijective correspondance with the naturals, you assume at the start that every set is countable at least, if not finite --- but that's neither here nor there for the point I want to make. The thing is that if you accept that there are infinitely many natural numbers, then you don't have to associate "infinity" with any of the elements in the set. You already have an infinite supply of numbers (each of which is finite and with a definite value) with which to associuate elements of the set. To say a set is infinite does not mean that "infinity" is a member of the set.



Bookwyrm
The rule that will generate this number is "for each circled digit on the table above, in order, we will write a 1 if the circled digit is a 0, otherwise we'll write a 0."

spacer1
Why not just call the number 3 the number 4, and say that 3 x 3 = 16? (Apologies, but I have limited computer access and haven't fully grasped the diagonal argument, but I want to reply while I have the opportunity).
You have gravely misunderstood Bookwyrm here. He's not stating anything contrafactual here: he's describing a way of producing a real number which is not in the range of the function ("not on the list").

Let's try to put it in terms of functions. Let f: N --> R be a function from the naturals to the reals, and define the function g: N x N --> R by the description:

g(a,n) = the n-th decimal digit of f(a), to the right of the decimal point.

Let q(a) = 0 if a is non-zero, and q(a) = 1 if a is 1. Then, define the real number

z = q(g(1,1)) x 0.1 + q(g(2,2)) x 0.01 + q(g(3,3)) x 0.001 + ...

The number z is a real number, and for every natural number n, it differs in at least the n-th decimal place from the value f(n), which is easily verified. As a result, z is a number which is not "covered" by the function f. Then, f cannot be a bijection. However, the only assumption we made about f was that it was a function from N to R. So, we know that there is no bijection from N to R --- they are not the same size.

I find it easier just to pair up the first natural with 0.00....01, the second with 0.00....02, etc. I don't think it matters how many zeroes those dots represent since, if the naturals extend on forever, so will the reals. Without the full list in your example, I find it hard to see why 0.1000... could not be on it, and therefore, cannot comprehend how there cannot be the same number of reals as naturals.
In 0.00....01, how many zeroes are there? Are you using infinitesimals, here?

[EDIT: sorry, I just realised that you said that you didn't think it was important! Must read more carefully...]

If it isn't important how many zeroes there are, let's fix it at 16 zeroes (for instance). Then, the number 0.000 000 000 000 000 001 (seventeen zeroes) will be missing from your list, although it's also a rational number. This is assuming, of course, that your enumeration never backtracks (your description above suggests an increasing sequence).

It sounds to me as though you are expressing the futility of trying to enumerate the rationals at all, just because there are infinitely many of them. This seems to be the general thrust of your approach to infinity: if you can't consider them all in a finite amount of time, there's no point in dignifying all of them by recognising their existence. (I am probably slightly exaggerating your position, but not deliberately.)

I would hazard a guess that you have a great distaste for the concept of an "actual infinity" in mathematics, yes?

JohannGoodflag

[Bellarmino]

Originally posted by Demosthenes
When you represent the cardinality of an infinite set with transfinite numbers such for example aleph-null is assigned to the cardinality of the set of natural numbers. Larger transfinite numbers can be constructed and in fact there are infinite transfinite numbers. I imagine one can also futher construct even higher transfinite numbers representing the set of transfinite numbers.

The cardinality of the set of real number is known to be larger than the set of natural numbers, but we don't know where exactly in the transfinite hierarchy the real numbers' cardinality is located. That is basically the continuum hypothesis proposed by Cantor who originally developed the transfinite arithmetics. In more depth, the hypothesis states that every uncountable set has at least as many elements as the set of real numbers. We think it's aleph-1, the next number after aleph-null. Unfortunately it's been discovered that the continuum hypothesis is not decideable(sofar as I understand it). Nevertheless, the hypothesis remain an interesting area of study.

ex-xian, I'm curious to what approach of quantum gravity you'll be studying...string theory or loop quantum gravity? If I were to get into that field, I'd go into loop quantum gravity for sure :D

I still have a year and half before I finish my mathematics BS. There's so much to learn I sometimes despair of ever comprehending the entire breadth of today's mathematics.
Wow, you're ahead of where I was at when I was a Junior math major.

[ex-xian]

Originally posted by JohannGoodflag quote:
Funny, based on your nick, I supposed you had different religious beliefs. Or, is this a pop-sci distillation you present to others?
I'm not sure what you mean. My nick is "ex-xian" as in I'm an ex-christian.

MORE
I'm curious: by quasi-pragmatist, what do you mean? When I say pragmatist, I mean "This seems to provide a wide variety of mental tools, which seem to represent interesting patterns or solve interesting problems, many of which I meet in the real world". I also use standard formal logic (as opposed to a logic with multiple values, say, or a non-standard collection of rules of inference) as a basis for math, which I also choose for the same pragmatic reasons.

JohannGoodflag

I suppose I used "quasi" to differentiate from those who would say that mathematics is no more or less certain than literature or language.

Originally posted by Demosthenes
ex-xian, I'm curious to what approach of quantum gravity you'll be studying...string theory or loop quantum gravity? If I were to get into that field, I'd go into loop quantum gravity for sure :D

I still have a year and half before I finish my mathematics BS. There's so much to learn I sometimes despair of ever comprehending the entire breadth of today's mathematics.
Oh, loop theory all the way. I get so frustrated with people like Mickio Kaku who publish books as if string-theory was empirically proven, and THE GUT. It motivates all the would-be physicists who couldn't integrate a nested trig function if you let them use a computer to spout off abouts things they know not of.

I have one semester to go; I'll be starting grad school in the fall (still not sure where). I'd like to spend some time studying set theory to really get into these issues, but I think I'll end up studying algebra, since group theory has applications in QM.

Originally posted by Bellarmino
Wow, you're ahead of where I was at when I was a Junior math major.
(I know this was directed at Demo, but I'm pretending it was at me too). Well, us math majors these days...we'll all Gausses-in-training. :D

[JohannGoodflag]

ex-xian
I'm not sure what you mean. My nick is "ex-xian" as in I'm an ex-christian. That's what I thought. My comment was prompted by your statement:
I now say that mathematics isn't the language of god, but is, instead, one of his hobbies. Implicit in this statement is a recognition of God as an entity which can have a hobby. This, coupled with the present tense. I was confused that an "ex-christian" (in the context of IIDB, meaning probably atheist) would continue to say such a thing. But then, I live in a place where admitting being an atheist is not such a big deal: maybe I was being overly presumptive of your environment. I would never talk about god colloquially in conversation, except for the sake of humour.

All of which is neither here nor there for the purposes of a mathematical discussion, of course.

I suppose I used "quasi" to differentiate from those who would say that mathematics is no more or less certain than literature or language. Alright, interesting. It actually never occurred to me (until this moment) that some people might lump me in with "the other side" of the science wars for my stance: but then, as a pragmatist, I think the track record of mathematics as a tool is proof enough that there's more to math than convenient fiction.

To me, pragmatism is to absolutism as rationalism is to handed-down-from-above religion: that the ideal is to do things because they work and they make sense, not because they are The One True Way. So I guess I'm not concerned with being grouped with postmodernists.

[ex-xian]

Originally posted by JohannGoodflag
Implicit in this statement is a recognition of God as an entity which can have a hobby. This, coupled with the present tense. I was confused that an "ex-christian" (in the context of IIDB, meaning probably atheist) would continue to say such a thing. But then, I live in a place where admitting being an atheist is not such a big deal: maybe I was being overly presumptive of your environment. I would never talk about god colloquially in conversation, except for the sake of humour.

All of which is neither here nor there for the purposes of a mathematical discussion, of course.
Oh, I see how I caused some confusion. I just say that for rhetorical purposes to stress that fact that mathematics isn't somehow inherent in our universe.

Alright, interesting. It actually never occurred to me (until this moment) that some people might lump me in with "the other side" of the science wars for my stance: but then, as a pragmatist, I think the track record of mathematics as a tool is proof enough that there's more to math than convenient fiction.

To me, pragmatism is to absolutism as rationalism is to handed-down-from-above religion: that the ideal is to do things because they work and they make sense, not because they are The One True Way. So I guess I'm not concerned with being grouped with postmodernists.
Well, calling yourself a pragmatist certainly can cause some confusion (as it did with me). At best, they'll lump you with Peirce...at worst they'll think you hold with Rorty.

[theophilus]

[QUOTE]Originally posted by jpbrooks
I agree. "Real" paradoxes cannot exist.

I think this may be technically incorrect.

A paradox is an "apparent contradiction," i.e., the contradiction has not been solved but that does not mean it is real.

An "antinomy" is a real contradiction (or contrarity?), i.e, a violation of the second law of logic.

Real antinomies "cannot" exist by definition.

I think this is correct. Any comments.

[theophilus]

The math stuff is all fascinating (I"m not being sarcastic- I never went beyond second year algebra and nearly failed that) but may be missing the point.

In speaking of the "infinity" of the universe and, therefore, the impossibility of it expanding, we are dealing with a metaphysical issue.

First, in the sense of time, infinity has no beginning, i.e., no place to start counting. If it did, it would not be infinite (this is a point of confusion among Christians and non regarding God's infinity, i.e., he hasn't been around for a "long time").

Regarding space, if the universe is infinite then it must be "everywhere," so there can be no "where" into which it can expand (or it would not be infinite). Even if you say that it "creates" more space as it expands, that still means there was some "place" where it was not.

This is a problem I see with the BB. If it "began" as an infinitely small singularity (it had no space) then it was no "where." If the universe is all there is, then where was it when it was no place?

I'm sure there are issues of Relativity and Quantum theory which I don't pretend to understand, but I think this is more of the direction the discussion needs to go.

[JohannGoodflag]

Originally posted by ex-xian
Well, calling yourself a pragmatist certainly can cause some confusion (as it did with me). At best, they'll lump you with Peirce...at worst they'll think you hold with Rorty.
Good to know: I'm not up on philosophico-poltical jargon. I picked "pragmatist" for what it literally means, not as a reference to a specific school (or clustering) of thought. Come to think of it though, I don't know that I don't fall in with that lot. I have little to no awareness of the philosophical spectrum: I'm only an amateur philosopher, and one who has done all his philosophising as a solo venture.

All I know about my philosophical position is this: that the only absolutes I have are my self-awareness, the stimuli I recieive, and my bodily reactions to the world. I can try to dismiss everyday worldviews as being castles built on air, but that does not negate the fact that my finger burns if I put it in a candle flame, and that I would probably experience mortal fear if I was confronted by a hungry tiger. If only as a game, the world has a sort of structure.

Even if math lacks any absolute "truth", it nonetheless stands that math is really good at providing ways of talking about the structure of the world, to the point where we can actually predict what we will observe, or develop reliable tools for affecting our environment in useful ways.

My stance is that this is all I could reasonably ask for from math; that any stronger demand will delve into metaphysical quandaries, but at the same time, that I have no real reason to demand anything more if it does work that well.

[JohannGoodflag]

Originally posted by theophilus
In speaking of the "infinity" of the universe and, therefore, the impossibility of it expanding, we are dealing with a metaphysical issue.

First, in the sense of time, infinity has no beginning, i.e., no place to start counting. If it did, it would not be infinite (this is a point of confusion among Christians and non regarding God's infinity, i.e., he hasn't been around for a "long time").

Regarding space, if the universe is infinite then it must be "everywhere," so there can be no "where" into which it can expand (or it would not be infinite). Even if you say that it "creates" more space as it expands, that still means there was some "place" where it was not.

This is a problem I see with the BB. If it "began" as an infinitely small singularity (it had no space) then it was no "where." If the universe is all there is, then where was it when it was no place?

I'm sure there are issues of Relativity and Quantum theory which I don't pretend to understand, but I think this is more of the direction the discussion needs to go.
First, you can (in principle) have an infinity with a beginning, but no end: so no, "infinite time" doesn't necessarily have a beginning. The counting numbers are an example of such an infinity: they start at 0 or 1 (depending on your preference), and never end. The set of all positive real numbers is another sort, and a continuum as well, which is maybe more the sort of infinity you might be interested in. Sure, it isn't an infinitely long time already passed and gone, but I don't think that's important to the debate.

Second, I already addressed the idea of an expanding infinite space (the second post of the thread, IIRC). As you note, the space as a whole does not expand: but it can make sense to say "space is expanding" if the general motion of clusters of matter w.r.t. time can also be described by a dilation of the original co-ordinate system. The word "stretching" may be better suited to this than "expanding", but then we're just getting into semantics. Do you have an objection to this description?

As for your question, "If the universe is all there is, then where was it when it was no place" --- what meaning do you suppose this question has? For that matter, is there any "when" without the universe, if that's all that there is?

Your ideas do hark back to the original post of the thread, which is good: but for the purposes of discussing the OP, an understanding of infinity is very important. Some people here wish for a better understanding of infinity, or at least to discuss different ways that infinity may be understood: I think it is apropos to discuss it in this thread. Under one interpretation, it's also what this thread is actually about: the topic just happened to come up because someone used the standard notions of infinity to provide a weird answer to an astronomy question.

[theophilus]

[QUOTE]Originally posted by JohannGoodflag
As for your question, "If the universe is all there is, then where was it when it was no place" --- what meaning do you suppose this question has? For that matter, is there any "when" without the universe, if that's all that there is?

I have confessed (several times, in fact) to being no physicist. In my simple minded way, if something material "IS," it must "BE" someplace. I don't think we have any experience to justify the contrary.

So, I have always had this fundamental question (stupid, I'm sure) of "where" the universe qua all time & space things "IS." My understanding (mis?) of the BB is that it began as a dimensionless point of pure energy. So,
1. All time/space/mater things must be "someplace."
2. The universe constitutes ALL time/space/matter.
3. The universe "resulted" from the BB.
4. The BB resulted from a dimensionless "singularity" of energy

"Where" was the dimensionless point before the universe (all space/time/material things) began expanding?

I'm sure this is stupid, and I'm not trying to make a point.

Your ideas do hark back to the original post of the thread, which is good: but for the purposes of discussing the OP, an understanding of infinity is very important. Some people here wish for a better understanding of infinity, or at least to discuss different ways that infinity may be understood: I think it is apropos to discuss it in this thread. Under one interpretation, it's also what this thread is actually about: the topic just happened to come up because someone used the standard notions of infinity to provide a weird answer to an astronomy question.

My comment was not intended as a rebuke. I simply thought the discussion was getting a little "escoteric" for the original question.

I still don't see how a mathematically infinite "set" which has a beginning is analogous to the infinity of the universe which is claimed to have no beginning but is a recurring expansion and collapse.

[spacer1]

Hello Johann,

spacer1: Do you not define N as being infinite in the first place because of time constraints, regardless of S?

JohannGoodflag: The short answer is yes. Is this a problem?
A problem? No. However, it does seem to suggest that time is implicitly bound up in the concept of infinity. Would you agree?
True. But in isolation from what? When I broached the topic, I meant "in isolation from all the other real numbers". There is no way to isolate any point from all the other real numbers, because they get packed in infinitely close.
In isolation from any other point. Pick any real number and you will notice that it is not any other real number. This relates to my earlier statement that you cannot define points on a continuum.
Would you not describe "infinitely close" as being "at zero distance"?
No. "At zero distance" isn't "close" at all. It is the same point. 2.999999999999999999999999 is not 3, no matter how many 9's you have, just as 2,999,999,999,999,999 is not 3,000,000,000,000,000, or just as 9 is not 10.

It may be "close enough" for many purposes, but it is not the same.
If you like, we can drop this part of the discussion. The "points in isolation" topic was just one approach I was trying to put across the standard-mathematical view of infinity, but if it doesn't help, there's no sense beating it to death.
I believe I have a reasonable understanding of the "standard" view. I am just trying to gain a deeper understanding, and I appreciate the opportunity to bounce my ideas off someone who is defending the standard view.
The way that it is often taught is as the limit of a sequence of approximations (e.g. using narrow rectangles to fill a curved area for integration), but the limit is not itself an approximation: it is the exact answer.
I'll take your word for it.
Let q(a) = 0 if a is non-zero, and q(a) = 1 if a is 1.

The number z is a real number, and for every natural number n, it differs in at least the n-th decimal place from the value f(n), which is easily verified.
My facetious remark to Bookwyrm was to question the reasoning behind "Let q(a) = 0 if a is non-zero". Why change a non-zero to a zero? It is as if you were to say "list all the natural numbers from one to ten, but change any 7 to a 4", thus proving that the number 7 does not exist in the first ten numbers.
I would hazard a guess that you have a great distaste for the concept of an "actual infinity" in mathematics, yes?
I have no idea what you mean by "actual infinity". I believe the concept of infinity logically exists, but that you cannot define any points within it.

*Again, apologies for not responding to the entirety of your post. Time constraints and all that. :D

[wiploc]

Originally posted by theophilus
[QUOTE]I have confessed (several times, in fact) to being no physicist. In my simple minded way, if something material "IS," it must "BE" someplace. I don't think we have any experience to justify the contrary.

I'm no physicist either, so, while my explanations may help, they should perhaps be taken with a grain of salt.



So, I have always had this fundamental question (stupid, I'm sure) of "where" the universe qua all time & space things "IS."

The answer would have to be everywhere. If "universe" means "everything that exists," then it exists everywhere that exists.



My understanding (mis?) of the BB is that it began as a dimensionless point of pure energy.

Nobody believes this any more. Einsteinian physics would have taken us to that (end?) point, but quantum mechanics has us starting just "after" that, when the universe already had a bit of size. Quantum mechanics rules very small things, and --- despite the name --- the big bang was very small at first.



So,
1. All time/space/mater things must be "someplace."
2. The universe constitutes ALL time/space/matter.
3. The universe "resulted" from the BB.
4. The BB resulted from a dimensionless "singularity" of energy

"Where" was the dimensionless point before the universe (all space/time/material things) began expanding?

Had the BB been a dimentionless point at the beginning, I don't think that changes the answer: it would have been everywhere.

Imagine an early snapshot of the universe. Let's say it is six inches across. There is no space outside of it; the only places that exist are in that six inch sphere.

It is wrong to think that that six sphere was located someplace inside what is now the universe (over by Polaris, maybe). Instead, the places in the current universe were all in that little sphere. You could point to a spot in the sphere, and say, "One day, when this little universe grows up, it will be huge; and this little spot will be huge; and this little spot will contain many galaxies, including the Milky Way."

So, if the current universe contains everyplace, and everyplace in the current universe used to be in the six inch universe, and everyplace in the six inch universe was at one time stuffed all the way down into a dimentionless point, then that point would have been everywhere. There were no other wheres.



I'm sure this is stupid, and I'm not trying to make a point.

Maybe my response is stupid too, but it was fun getting to field it to see what people say.

crc

[ex-xian]

Originally posted by spacer1
In isolation from any other point. Pick any real number and you will notice that it is not any other real number. This relates to my earlier statement that you cannot define points on a continuum.
Just because they are dense doesnt' mean we can't define them. I'm not sure what you mean.

No. "At zero distance" isn't "close" at all. It is the same point. 2.999999999999999999999999 is not 3, no matter how many 9's you have, just as 2,999,999,999,999,999 is not 3,000,000,000,000,000, or just as 9 is not 10.
Actually, 2.9, with the 9 repeating is equal to 3.

[theophilus]

[QUOTE]Originally posted by wiploc
Nobody believes this any more. Einsteinian physics would have taken us to that (end?) point, but quantum mechanics has us starting just "after" that, when the universe already had a bit of size. Quantum mechanics rules very small things, and --- despite the name --- the big bang was very small at first.

Please clarify what "nobody believes" anymore:
Big Ban cosmology
The dimensionless point

Had the BB been a dimentionless point at the beginning, I don't think that changes the answer: it would have been everywhere.

See, that's where I get stuck. How could a dimensionless point be said to be anywhere, let alone everywhere. Unless by everywhere you mean there was no "where," which still leaves me with the problem of where "where" came from.

Imagine an early snapshot of the universe. Let's say it is six inches across. There is no space outside of it; the only places that exist are in that six inch sphere.

It is wrong to think that that six sphere was located someplace inside what is now the universe (over by Polaris, maybe). Instead, the places in the current universe were all in that little sphere. You could point to a spot in the sphere, and say, "One day, when this little universe grows up, it will be huge; and this little spot will be huge; and this little spot will contain many galaxies, including the Milky Way."

I think I understand this, everything in the universe is IN the universe much like everything in a baloon may be said to be in the baloon before it is inflated (except the volume of air). That still doesn't explain "where" the baloon/universe IS (even when it was only six inches around).

So, if the current universe contains everyplace, and everyplace in the current universe used to be in the six inch universe, and everyplace in the six inch universe was at one time stuffed all the way down into a dimentionless point, then that point would have been everywhere. There were no other wheres.

Maybe my response is stupid too, but it was fun getting to field it to see what people say.

crc

Certainly not stupid, but I'm still scratching my head.

Thanks.

[ex-xian]

Originally posted by theophilus
See, that's where I get stuck. How could a dimensionless point be said to be anywhere, let alone everywhere. Unless by everywhere you mean there was no "where," which still leaves me with the problem of where "where" came from.
Any point is, by defintion, dimensionless. If it had measurable dimensions, it would be a line, plane, etc, not a point.

[spacer1]

Originally poseted by ex-xian:
Just because they are dense doesnt' mean we can't define them. I'm not sure what you mean.
I thought I had explained this reasonably well in my earlier posts, but I will try to explain it again for your benefit. I am arguing that you cannot define points on a continuum, which contains an infinite number of elements, and that as soon as you do define any point, you necessarily make the set finite. As I said, "Pick any real number and you will notice that it is not any other real number." How "infinitely close" one number is to another cannot be demonstrated by giving definite values to those reals.
Actually, 2.9, with the 9 repeating is equal to 3.
I realise that this is the accepted view, but it assumes the understanding of infinity which I am currently questioning.

[ex-xian]

Originally posted by spacer1
I thought I had explained this reasonably well in my earlier posts, but I will try to explain it again for your benefit. I am arguing that you cannot define points on a continuum, which contains an infinite number of elements, and that as soon as you do define any point, you necessarily make the set finite. As I said, "Pick any real number and you will notice that it is not any other real number." How "infinitely close" one number is to another cannot be demonstrated by giving definite values to those reals.
Just out of curiousity, how much math have you had? That might make it easier to explain things.

I think I see what confuses you. We can prove that the reals are dense, even though we cannot generate all the numbers. This is what's known as an existence proof. Just because we can't define how close the numbers get to each other doesn't change the fact that they exist and the reals are dense.

I realise that this is the accepted view, but it assumes the understanding of infinity which I am currently questioning.
Well, one proof is that there is no number between 2.999... and 3, so they must be equal. But this is what you have confusion about.

There is a way to prove this with inifinite sequences, but this is a little simpler. Let x=.999...... Then 10x=9.9999...... So 10x-x=9. That is, 9x=9 and x=1. Then 2+.99999 = 2+1=3.

[theophilus]

Originally posted by ex-xian
Any point is, by defintion, dimensionless. If it had measurable dimensions, it would be a line, plane, etc, not a point.

I understand that and that is part of the problem. Doesn't a "bang," however Big, have to happen somewhere? At some point, the point had to assume/acquire dimension (even if it was infinitesimal) and that would require location, wouldn't it.
So, I'm back to my question. Once there was any amount of dimensionality in that regard, where was it located.
It can't be said it was in the universe because the universe didn't exist, right.

[theophilus]

I was confusing concepts in my first post, i.e., eternality and infinity.
I apologize.

[theophilus]

Originally posted by ex-xian
Just out of curiousity, how much math have you had? That might make it easier to explain things.

I think I see what confuses you. We can prove that the reals are dense, even though we cannot generate all the numbers. This is what's known as an existence proof. Just because we can't define how close the numbers get to each other doesn't change the fact that they exist and the reals are dense.
[b]
Well, one proof is that there is no number between 2.999... and 3, so they must be equal. But this is what you have confusion about.

There is a way to prove this with inifinite sequences, but this is a little simpler. Let x=.999...... Then 10x=9.9999...... So 10x-x=9. That is, 9x=9 and x=1. Then 2+.99999 = 2+1=3.

Would this have anything to do with irrational/incomensurable numbers?

[ex-xian]

Originally posted by theophilus
Would this have anything to do with irrational/incomensurable numbers?
Not exaclty. An irrational number is one that cannot be expressed as a ratio of integers. Here's a link that proves the existence of irrationals, http://hemsidor.torget.se/users/m/mauritz/math/num/irat.htm

These infinite decimals can be expressed as ratios, so they're rational. Incidentally, both the rationals and the irrationals are dense in R.

[jpbrooks]

This has certainly developed into a very interesting discussion since my last visit. Sorry about the delay in responding. (See my note at the bottom of this post.)


Originally posted by wiploc



:
Originally posted by jpbrooks
If space is not contiguous, what exists in the "gaps"?



I'm clearly in over my head here, ...



You're certainly not alone! That is precisely my general sentiment in many of the forums in which I participate.



but it seems to me that the answer has to be: nothing. And that's a way nothing too, not just emptiness. It has to be a nothingness such that there aren't even any places in it.



And this, if I'm understanding your response correctly, suggests to me that space, in that case, would be contiguous.




But how does any of this alter the "cardinality of space"? The cardinality of "places" ("points", etc.) in space, as of "points" on a number line, does not depend on how it is segmented, (and again, cannot be greater than aleph-1).



The cardinality of a one inch line is aleph-1. If you could somehow remove all the real points, leaving only the integer points, then it would have a cardinality of aleph-null. If you removed everything but the end points, you would have a cardinality of 2.



I agree. But my points were made in regard to what could actually be the case in the real world. In the real universe, I'm not certain how one could remove, for example, all of the points on a line that represent segments of irrational length, leaving only the points that represent rational lengths.


It's not a matter of segmenting a contiguous line. It's a matter of making a line out of a reduced number of points. I know that doesn't make sense. We think of space as contiguous, so it doesn't seem sensible to imagine non-contiguous lines.

But, we have to make a choice. We have an either/or situation. Only one thing is true. Either "infinity" describes a real world situation (the number of places there are) or space is not contiguous (there are not an infinite number of places).



Well actually, it seems more like a "both/and" than an either/or situation. I meant to make a distinction between infinities that are obtained by (sub)dividing whole (contiguous) things and those that are obtained by a process of adding together finite numbers of finite objects. In the former sense of infinity, infinity does indeed describe the (spatial) situation in the real world, while in the latter sense, it doesn't. Whether or not it does depends on the sense of infinity that is employed.



If you ever find yourself on stage debating William Lane Craig; and he employs the argument from audience incredulity to show that the world is not infinitely old because there are no infinities in the real world; then it becomes appropriate for you to show the audience the incredible implications of his claim.



The issues surrounding Craig's argument are complex. I, like him, am a Theist, and I believe that I understand the main thrust of his argument. But I'm not certain that the argument is as "airtight" as it is believed by its proponents to be. For one thing, the same thing that I said about space (above) can also be said about time, specifically that, even if it is the case that physical processes had to have a beginning, any time scale can be extended backward beyond the point that all physical processes began, into eternity past (and forward, into eternity future). So time itself need not have had a beginning (or a future end). Also, there is, on the basis of Craig's argument alone, no reason to assume that the "First Cause" of all physical processes is not a natural and/or impersonal entity.



For instance, if space is not contiguous, then there is no such thing as movement from one place to another. That is an illusion. All "movment" actually consists of a series of teleportations.

crc

My own view (which I freely admit could be mistaken) on this matter is that whether "movement" would involve teleportation depends on the way that physical phenomena are generated at "sub quantum" levels of reality. Perhaps there is no actual movement at all at a certain "sub quantum" level. There might be only changes of state in certain locations of a(n otherwise) motionless underlying medium that appears to be motion to us at the macroscopic level of reality.





Originally posted by JohannGoodflag



jrpbrooks:



[...]completed infinite sets, unlike potentially infinite ones, are not obtained by "adding together" (or juxtaposing?) finite quantities or things, even though such completed infinite sets (the set of real numbers in any line segment with a length greater than zero, for example), once obtained, can be subdivided into finite subsets. The set of all individual finite objects in the universe cannot be "added to" one another to form a completed infinite set. Thus propositions that are true only for completed infinite sets cannot be presumed to be applicable to the entire collection of finite things that make up the universe.



If one supposes the universe is infinite, then the objects in the universe already form an infinite set, so I'm not so sure your statement holds. Of course, if the universe is finite, then juxtaposing them cannot form an infinite set, but for a much less interesting reason: because there is only finitely many of them. (I have no informed opinion one way or the other as to whether it is actually infinite.)



It wasn't my original intention to defend the Cosmological Argument (since I'm not a proponent of that argument) but while it may be granted that the universe may have simply "popped" into existence (all at once) as a fully formed infinite collection of individually finite phtysical entities, the probability of that event occurring would have to be (1 / infinity or) zero. But outside of "popping" into existence as a completed infinite set of "objects, I have no idea how an actually infinite universe could come to exist. Any sort of "process" of coming into being would be a sequential set of "steps", which would seem to be precisely the kind of thing that would be subject to attack by Craig's type of Cosmological argument. So, I don't have much confidence in the assumption that the universe is a completed intinite set of physical things.



But a related point is that it is just not clear (to me, `at least) how any argument concerning infinities of cardinality greater than aleph-1 could apply to the universe of real "objects". What actual collection of physical, material, or corporeal things in the universe has a cardinality greater than that of the real numbers?
[...]
The cardinality of "places" ("points", etc.) in space, as of "points" on a number line, does not depend on how it is segmented, (and again, cannot be greater than aleph-1).



First, a brief note. The cardinality of points in space can (in principle) exceed aleph-1. The continuum is not necessarily the same as aleph-1: it could be aleph-2, aleph-3, or indeed, aleph-17. The idea that the continuum is aleph-1 is the continuum hypothesis, and is independent of set theory (it cannot be proven or disproven in Zermelo-Frankel). It must be assumed if it is to be used.



Yes. I agree that the Continuum Hypothesis has been demonstrated to be undecidable.



As to what collection of objects could exceed the cardinality of the continuum: we don't know, but not knowing is not a strong argument. The idea that one could have more than one kind of infinity was itself mind-blowing at Cantor's time. I will admit that it is very difficult to concieve of how anything larger than the continuum could have any practical relevence for physics, but that does not mean that there might not be some relevance.



True! But the point that I was trying to make was that it is not apparent at all how any collection of real physical objects can, even in principle, be arranged into more (actually existing and not merely abstract) sets than there are points in the continuum.



If space is not contiguous, what exists in the "gaps"?



It depends on whether the gaps are actually there. We have learned that matter is not (necessarily) infinitely divisible: at the very least, it gets very chunky at a small level. Most certainly it is not a smooth continuum. It is possible, in principle, that the same is true of spacetime. I'm not qualified to say in what way, but you are almost certainly aware of at least one discrete space --- a chess-board. If space is discrete, a question then of what exists in the gaps in space would be like asking what exists between the squares on a chess board (or asking what integer comes between 1 and 2).


(edited for minor errata)

But (as I mentioned above) if we simply assume that nothing exists in the "gaps" in space, I'm not certain how it would differ significally from a continuum.

I'm a little busier than I expected to be at home at this time, and many of the computer labs at UIC are still on "intersession" schedule. So my replies are likely to be very slow in coming. Sorry about that. I'll be back later.

[acronos]

I realize we are probably wearing out the mathematicians, but I desperately hope they will take time for one more fray.

The limit of x/(x+1) as x approaches infinity is 1.
The limit of 2x/x as x approaches infinity is 2.

I know that mathematicians say that it is not debatable that twice infinity is the same cardinality as infinity. However, how is such a statement useful? I don’t see it. Can any mathematicians explain to me why set theory does this? The above concepts generated by limits seem so much more useful. Why do we call the problems listed above when x=infinity “undefined” rather than just solving the limit and calling it like it is.

To me set theory infinities seem like a tool created by the same type of thinking that created Zeno’s paradox, which is idiotic (“not us