More than one size of infinite set?

[Blackbeard]

On another thread (http://www.iidb.org/vbb/showthread.php?t=204772), it was stated that there can be more than one size of infinite set. This was in response to my stating that since there are an infinite number of rational numbers and an infinite number of irrational numbers, and thus an equal number of each, that there is a 50-50 chance of a random distance being an irrational number as opposed to a rational number. In response someone who is obviously more well versed in mathematics than myself stated:

......I'm afraid that on the count of the number of irrational numbers, you're a bit off:

There are in fact more irrational number than rational numbers. There is more than one size of infinite set; and the number of irrational numbers is provably larger than the number of rational numbers, in a meaningful sense which mathematicians care about. The rational numbers are "countable" (that is, you can match them off one-by-one with the positive integers), whereas this cannot be done with the irrational numbers (there are simply too many of them to do this). ...............

So, I wanna see if I can figure this out logically. Let me start with integers. Clearly, there are an infinte amount of integers. Also clearly, between each integer, there are an infinite amount of rational numbers. So, logically one could say that there are infinity^2 rational numbers. Correct? Now I can see that it is reasonable to assume that between any two rational numbers, there are an infinite amount of irrational numbers, so there would be infinity^3 irrational numbers? Is my thinking flawed? Close? Way off?

Under the assumption that I have the right idea, all I can say is that this seems to go against my common sense. If infinity is, well, infinite, how can there be different sizes of infinite sets? If one can argue that any number times infinity equals infinity, then would that not be true of infinity as well so that infinity^2 still yields infinity? Just like any number times zero equals zero, including zero? Infinity and zero seem to share certain properties. If division by zero is "illegal", then so should division by infinity be illegal. But, I could argue that division by zero = infinity and vice verse. The problem I see here is that zero lies on our number line, smack dab in the middle, yet infinity does not. Infinity is not even considered to be a number as far as I can tell, yet zero is. This seems to me to be problematic. if there is a zero, there should be it's reciprocal, which would to me, logically, be infinity and there is no reason I can think of that it should not be treated in exactly that way, zero's reciprocal. So just as there can not be "different sizes of a zero set", there should not be able to be "different sizes of an infinite set".

Am I making sense? :huh:

An idea I had a long time ago when pondering infinity is that the whole "number line" concept is inherently wrong and not fitting to the natural order of things. A number circle to me would be more realistic, a circle with 0 on one pole, infinity on the other 180 degrees opposite, positive numbers on one hemisphere, negative on the other. So as you go up or down from zero numerically, either way you will eventually arrive at the same infinity, which like zero has neither a negative or a positive. now I know this has no impact on how mathematics is performed, but it is a much neater way to envision numbers to me and it has helped me envision infinity as less abstract, at least as no more abstract than the concept of zero.

[aegis]

I didn't read your whole post, but my understanding is that the integers and fractions are countably infinite (i.e. when you 1, 2, 3, 4 etc. if you do that forever you will get all of the integers, there's a more sophisticated way of listing all the fractions). However you cannot list all the numbers between 1 and 2, so you have an uncountable infinity of those.

So why is uncontable infinity > countable infinity? I can't give a matheamtical justification, maybe someone else can

[LetUsRatiocinate]

A couple examples of differently sized infinite sets:

The numbers in the Fibonacci sequence. {1,1,2,3,5,8,13,21,34...}
Prime numbers {1,2,3,5,7,11,13,17,19...}
All positive rational numbers {1,2,3,4,5,6,7,8...}

Why do you think one infinity should = all other infinities?

[premjan]

This is taken up in Non-standard analysis which contains both infinite and infinitesimal numbers. The reason there are different types of infinity is because infinity is not just a number on the number line, but actually the cardinality of various types of sets, which can all be infinite in membership without having the same size. The number line doesn't actually contain infinity anyway, as it is at the "end" of the line, which you can never get to.

[Blackbeard]

I didn't read your whole post, but my understanding is that the integers and fractions are countably infinite (i.e. when you 1, 2, 3, 4 etc. if you do that forever you will get all of the integers, there's a more sophisticated way of listing all the fractions). However you cannot list all the numbers between 1 and 2, so you have an uncountable infinity of those.

So why is uncontable infinity > countable infinity? I can't give a matheamtical justification, maybe someone else can

That is pretty much what I wrote in my OP, second paragraph. The rest of my post, if you care to read it, partly explains my problem with that. As a further explanation, it seems to me that any set is countable to infinity, the fact that there are "infinities inside infinities" does not seem to preclude the ability to count that set to infinity, of which you will see that I believe there can only be one infinity in my last paragraph and I also suggest a different way of looking at numbers that may be more "realistic".

[Corona688]

That is pretty much what I wrote in my OP, second paragraph. Not exactly. You leaped from different sizes of infinities to quantifying them as infinity^2, infinity^3. They aren't quantified. We just know that we can't count all the values in the set of reals with all the values in the set of integers.

Think of it this way. f(x)=x approaches infinity as x approaches infinity. f(x)=x^2 also approaches infinity as x approaches infinity. Does this mean that x^2 = x as x approaches infinity? Of course not. One grows much faster than the other, and will always be larger than the other no matter how high you count. Similarly, there will always be more reals to count than integers no matter how high you count.

[Blackbeard]

A couple examples of differently sized infinite sets:

The numbers in the Fibonacci sequence. {1,1,2,3,5,8,13,21,34...}
Prime numbers {1,2,3,5,7,11,13,17,19...}
All positive rational numbers {1,2,3,4,5,6,7,8...}

Why do you think one infinity should = all other infinities?


From those brief examples, I see wat you are getting at, but just because an integer is repeated or skipped in a given set, does not change the fact that that set can still be counted to infinity, IMHO. In the Fibonacci sequence, the first 1 I would count as "1", the second 1 I would count as "2", etc. In the prime number sequence, I would count the 5 as "4" and the 7 as "5", etc. I think aegis' (and my own) explanation of "infinities inside infinities" better illustrates "different infinities", but IMHO still falls short. As for why I think all infinities are the same, I cover that in my OP.

[Blackbeard]

This is taken up in Non-standard analysis which contains both infinite and infinitesimal numbers. The reason there are different types of infinity is because infinity is not just a number on the number line, but actually the cardinality of various types of sets, which can all be infinite in membership without having the same size. The number line doesn't actually contain infinity anyway, as it is at the "end" of the line, which you can never get to.

Thanks for the link, I will check it out when I have more time. But if you read the last paragraph of my OP, I propose that the number line is an inadequate representation of numbers, and try to give rationale and an alternate view.

[Corona688]

An idea I had a long time ago when pondering infinity is that the whole "number line" concept is inherently wrong and not fitting to the natural order of things. A number circle to me would be more realistic, a circle with 0 on one pole, infinity on the other 180 degrees opposite, positive numbers on one hemisphere, negative on the other. So as you go up or down from zero numerically, either way you will eventually arrive at the same infinity, which like zero has neither a negative or a positive. There is such a thing as negative infinity. You can't get rid of that without getting rid of negative numbers altogether.now I know this has no impact on how mathematics is performed, but it is a much neater way to envision numbers to me and it has helped me envision infinity as less abstract, at least as no more abstract than the concept of zero. Problem is, it doesn't work. There is such a thing as negative infinity. Would you say that x = -x as x grows arbitrarily large? I certainly hope not.

[Blackbeard]

Not exactly. You leaped from different sizes of infinities to quantifying them as infinity^2, infinity^3. They aren't quantified. We just know that we can't count all the values in the set of reals with all the values in the set of integers.

Think of it this way. f(x)=x approaches infinity as x approaches infinity. f(x)=x^2 also approaches infinity as x approaches infinity. Does this mean that x^2 = x as x approaches infinity? Of course not. One grows much faster than the other, and will always be larger than the other no matter how high you count. Similarly, there will always be more reals to count than integers no matter how high you count.

well, that's a more elegant way of saying the same thing. And I do understand that concept, really I do. But one can make a mathematical formula that "proves" 0=1 (yes, using a shallowly hidden division by 0 .... I am assuming you are familiar with this trick of math). Somehow, as I stated in my OP, I see infinity as nothing more than the reciprocal of 0 and tried to logically state why. If that is the case, then mathematics using infinity would logically create the same kind of conundrums that division by zero can. IOW, if using "zero" incorrectly can make 1=0, then why can't using infinity "incorrectly" result in x=x^2? :huh:

So far I have just seen variations of what I already stated that I understood in my OP, just using different terms. At least I know my thinking was correct as to why it is said that there are more irrational than rational numbers, etc. Any comment on my suggestion that infinity is merely 0's reciprocal? That makes logical sense to me, am I crazy? :huh:How about my suggestion that a number circle would be more appropriate than a number line, with infinty being on one side and 0 being 180 degrees opposite, positive numbers occupying one semicircle and negatives the other? It seems no one got to that last paragraph.

[mac_philo]

A couple examples of differently sized infinite sets:
[...]
Prime numbers {1,2,3,5,7,11,13,17,19...}
All positive rational numbers {1,2,3,4,5,6,7,8...}


No, those sets are exactly the same size. They are both aleph zero.

So why is uncontable infinity > countable infinity? I can't give a matheamtical justification, maybe someone else can

Because "same size" means "can be put into one-to-one correspondence". Here is an example of one-to-one correspondence:

http://upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Bijection.svg/200px-Bijection.svg.png

You cannot do this between a uncountably infinite set and the natural numbers. In this picture above you have a function that takes you from the set {1,2,3,4} to the set {A,B,C,D}. The function is a bijection, which means here that each number points to one letter, and each letter gets pointed at by some number.

When you try to make a similar function from the natural numbers to an uncountable set, there will be members in the uncountable set that are not paired with any member of the countable set. So the uncountable set is bigger.

Read the wiki entry on cardinality and follow the links about bijections and injections.

[premjan]

Thanks for the link, I will check it out when I have more time. But if you read the last paragraph of my OP, I propose that the number line is an inadequate representation of numbers, and try to give rationale and an alternate view.I think your circle representation is actually used in complex analysis and is known as the Riemann sphere. It does give a geometric meaning to infinity.

[Blackbeard]

There is such a thing as negative infinity. You can't get rid of that without getting rid of negative numbers altogether. Problem is, it doesn't work. There is such a thing as negative infinity. Would you say that x = -x as x grows arbitrarily large? I certainly hope not.

No, I would never say that x= -x. But just as there is no -0, if infinity were 0's reciprocal ( 1/0 = infinity , 1/infinity = 0) then it would be reasonable to assume that there is no negative infinity. I have never heard of a negative infinity, and yes on the number line, which as I suggested may not be a fair representation of real numbers, eliminating negative infinity would wrongly eliminate all negative numbers, this is how the idea of the number circle was born in my mind several years ago:

http://www.johnnygold.com/numbercircle.jpg

You can call me crazy to go against the accepted number line, but I thought this was a site for freethinkers, ones who don't necessarily subscribe to traditional views and thought, but rather try to find things that make sense for them. This makes sense to me, a number line does not and never did, just like the concept of god never made sense to me. The number line presents conundrums like the ones noted, the number circle to me seems to avoid those same conundrums, just as atheism avoids all the paradoxes and conundrums associated with the existence of god.

[Blackbeard]

I think your circle representation is actually used in complex analysis and is known as the Riemann sphere. It does give a geometric meaning to infinity.

Thank you for that. I knew I couldn't have been the first to conceive of it, it just seemed too logical and with all the geniuses floating around through time/space, me not nearly being one of them, I couldn't imagine that it was a unique thought, though I had never seen nor heard of it before.

I do really have to go now and will examine both of your links, thanks again, later.

[Jet Black]

A couple examples of differently sized infinite sets:

The numbers in the Fibonacci sequence. {1,1,2,3,5,8,13,21,34...}
Prime numbers {1,2,3,5,7,11,13,17,19...}
All positive rational numbers {1,2,3,4,5,6,7,8...}

Why do you think one infinity should = all other infinities?

actually those are all the same iirc, though I am not entirely sure about the prime numbers. The issue here is mapping. If you can have a one to one mapping between two sets, they are the same size. So for example all the positive rational numbers and all the even positive rational numbers:

1,2
2,4
3,6
4,8
5,10

can be mapped by r=2*l

but this is not the case for irrational numbers. There is actually a mathematical proof that the number of irrational numbers is bigger than the number of rational.

[TNorthover]

It would probably be a good idea to firmly distinguish between numbers for counting things and numbers for arithmetic (and possibly even geometric ones).

The idea of counting numbers can be extended to categorise infinite sets, just as it does finite ones. mac_philo seems to have given the best description so far (even a pretty picture!), but the intuitive idea is that two sets are the same size if we can pair off elements of each and have none left over. This is what we naturally do for finite sets, and it's made the definition for infinite ones -- hence uncountable is a larger infinity than countable. It turns out that you can't do things like division properly for this concept of infinity, and you do get various identities like inf+inf = inf (provided it's the same infinity in each case).

Arithmetic numbers become the nonstandard analysis premjan mentioned when you start trying to add something like infinity to them. The result has no direct bearing on counting the elements of sets, but they behave more like "normal" numbers under addition etc. inf+inf is not equal to inf -- it's a different infinity; and you can divide them (but still not divide by 0).

Finally you could add an infinity geometrically. This is what your circle was attempting to do, and as premjan said it's similar to the Riemann sphere. This arises from thinking about how sequences approach infinity and so on. It actually turns out that for the number line it is more natural to have a separate +infinity and -infinity for things to approach, largely because an important part of the number line is being able to compare the size of two numbers. When we move over to the complex numbers though, we've already lost that ability, and the Riemann sphere single infinity is almost exclusively used. Anyway, there's basically no arithmetic on this version of infinity, not even addition (and it has no relation to the size of sets).

[Blackbeard]

I just finished reading the article on the Riemann sphere (and on non-standard analysis) . Very interesting. Even though the mathematics are well beyond my abilities, I think I do get the concepts. And it would seem that either viewpoint about infinity (only one or more than one) could be equally well argued, or perhaps more precisely, could be equally well argued when used in the context most fitting their use. One thing I found interesting is the applications that the Riemann sphere seems to be fitted for:

The Riemann sphere has many uses in physics. In quantum mechanics, points on the complex projective line are natural values for photon polarization states, spin states of massive particles of spin 1/2, and 2-state particles in general. The Riemann sphere has been suggested as a relativistic model for the celestial sphere. In string theory, the worldsheets of strings are Riemann surfaces, and the Riemann sphere, being the simplest Riemann surface, plays a significant role. It is also important in twistor theory.


I find it interesting that it does seem to find a "home" in theories attempting to describe matter/energy in it's most fundamental forms, ie quantum mechanics and string theory.

It also allows for 1/0 = infinity and 1/infinity = 0. It puts infinity "on the map", so to speak. I feel somewhat vindicated. :)

[LetUsRatiocinate]

No, those sets are exactly the same size. They are both aleph zero.

See, that's why I love this place! Because I learn such cool stuff. Since I had to look up "aleph-0" I learned "aleph-null bottles of beer on the wall, aleph-null bottles of beer, take one down, pass it around, aleph-null bottles of beer on the wall." :eek:

The issue here is mapping. If you can have a one to one mapping between two sets, they are the same size. So for example all the positive rational numbers and all the even positive rational numbers:

1,2
2,4
3,6
4,8
5,10

can be mapped by r=2*l

but this is not the case for irrational numbers. There is actually a mathematical proof that the number of irrational numbers is bigger than the number of rational.

I stand corrected. Thanks! :wave:

[jeffevnz]

If anyone's still a little hazy on how one infinitely-sized set can be bigger than another, see mac philo's explanation. The point is that infinite sized sets don't have a "size" in the usual sense. What matters is whether the elements of the two sets can paired up, without leaving any elements out on either side. If you can do this, the sets are considered the same size, in the sense of cardinality (and that may be counterintuitive, as with integers and rationals). If you can show that this cannot be done, without leaving elements out on one side, then one set is "larger".
I find it interesting that it does seem to find a "home" in theories attempting to describe matter/energy in it's most fundamental forms, ie quantum mechanics and string theory.
Whenever someone mentions string theory I have to work in a warning that it is one of the most colossally oversold theories of all time. Not that there's any evidence against it. In fact, there can never be any evidence against it. That's the problem! It's untestable. And it's starting to look like it always will be. Furthermore, it makes so many new assumptions about the laws of nature, that it's almost certainly wrong... I feel obligated to warn people about it, because string theorists have an amazing PR campaign. Don't get me wrong, marketing your ideas is an important part of science. But it's bad for everyone when such good PR is paired with such a shaky theory.

[atonal chaotic]

You can call me crazy to go against the accepted number line, but I thought this was a site for freethinkers, ones who don't necessarily subscribe to traditional views and thought, but rather try to find things that make sense for them.That doesn't mean you can just think up whatever and people will just sit there and say "woah, cool man." It's usually productive to try to understand the traditional view as thoroughly as you can before simply inventing an "alternative." This is needed to actually argue for one or the other, and to develop an alternative that has a chance of actually being better than the established view.

[Corona688]

well, that's a more elegant way of saying the same thing. And I do understand that concept, really I do. But one can make a mathematical formula that "proves" 0=1 (yes, using a shallowly hidden division by 0 .... I am assuming you are familiar with this trick of math). A) One can do stupid math tricks.
B) I don't know where, but I'm sure you've done it somewhere.
C) ???
D) Profit!

You don't need a better argument, you need a argument, period...Somehow, as I stated in my OP, I see infinity as nothing more than the reciprocal of 0 and tried to logically state why. If that is the case, then mathematics using infinity would logically create the same kind of conundrums that division by zero can. IOW, if using "zero" incorrectly can make 1=0, then why can't using infinity "incorrectly" result in x=x^2? :huh: I hate to break it to you, but: 1 != 0
x != x^2 when x>1
1+1 != 3 When you get results that say otherwise, they're wrong. It's not some kind of special new math you've discovered. It's wrong.

So wrong, in fact, that contradictions are one of the most powerful forms of proof in mathematics. If you assume your premise and derive a contradiction from it, then you have proven that premise to be false.So far I have just seen variations of what I already stated that I understood in my OP, just using different terms. At least I know my thinking was correct as to why it is said that there are more irrational than rational numbers, etc. Any comment on my suggestion that infinity is merely 0's reciprocal? Crazy in that you've denied the existence of a negative infinity, and ignorant in not knowing that 1/0 is not infinity. 1/0 is undefined. Equations that approach 1/0 can be evaluated to see what they approach, but a flat literal 1/0 is simply undefined.That makes logical sense to me, am I crazy? :huh:How about my suggestion that a number circle would be more appropriate than a number line, with infinty being on one side and 0 being 180 degrees opposite, positive numbers occupying one semicircle and negatives the other? It seems no one got to that last paragraph. There didn't seem much point in discussing it when the basic assumptions underpinning it don't work. But since you ask, I'll explain the fundamental problem with it: It is discontinuous. Somewhere, your number line flips from an arbitrarily large number to an arbitrarily small number for absolutely no reason. Discontinuities cause a lot of problems in mathematics. Arbitrarily redefining the number line as something discontinous would break an awful lot of things that we know actually work. Things like...calculus. And negative numbers.

[Corona688]

No, I would never say that x= -x. But just as there is no -0, if infinity were 0's reciprocal ( 1/0 = infinity , 1/infinity = 0) then it would be reasonable to assume that there is no negative infinity. And yet, x=-x is a logical extrapolation when there's no negative infinity. Allow me to present a proof by contradiction of your number line.

A) f(x) = -x
B) infinity = -infinity
C) |f(x)| approaches infinity as x approaches infinity
D) From B, |f(x)| approaches f(x) as x approaches infinity.
E) Contradiction. f(x) never approaches |f(x)|.
F) Therefore infinity cannot be the same as negative infinity.You can call me crazy to go against the accepted number line, but I thought this was a site for freethinkers, ones who don't necessarily subscribe to traditional views and thought, but rather try to find things that make sense for them. If you don't want to sound like a crackpot, it's best not to spout the whole martyr line. They laughed at galileo. They also laughed at bozo the clown.

I'm not bashing it because I don't like it. I'm bashing it because it doesn't work. Without a negative infinity on the real number line, you get contradictory results.

[d0t]

And yet, x=-x is a logical extrapolation when there's no negative infinity. Allow me to present a proof by contradiction of your number line.

A) f(x) = -x
B) infinity = -infinity
C) |f(x)| approaches infinity as x approaches infinity
D) From B, |f(x)| approaches f(x) as x approaches infinity.
E) Contradiction. f(x) never approaches |f(x)|.
F) Therefore infinity cannot be the same as negative infinity.

Negative infinity is needed when talking about limits on the reals. I think he read something about Stereographic projection (projection of the circle to the line), which has only one infinity. x and -x then both approach this "infinity" for increasing x, and 1/0 is by definition the point "infinity" on the circle.

Of course, using this you abandon the real number line and need to (re)define a few concepts like metric used, arithmetic with infinity, etc.

Regardless of the wrongness of "infinity=-infinity" on the reals, your proof contains a few mistakes:
- different notions of "approaches" mixed up (try putting everything in epsilon/delta notation)
- the conclusion in D (this would imply x -> x^2 because both -> infinity).

[Corona688]

Regardless of the wrongness of "infinity=-infinity" on the reals, your proof contains a few mistakes:
- different notions of "approaches" mixed up (try putting everything in epsilon/delta notation) I didn't feel like playing with iidb's tex plugin. ;)- the conclusion in D (this would imply x -> x^2 because both -> infinity). Ah. I should have been more specific, yes. Allow me to add:
C1) Since f(x)<=0 for all x>=0, and given g(x) = |f(x)|, then g'(x) = -f'(x) for all x>=0 (aka they approach their respective infinities at precisely the same absolute rate)
C2) f(0)=g(0) (they both start in the same place)
C3) f''(x) = 0, g''(x) = 0 (they're both lines)

[Blackbeard]

That doesn't mean you can just think up whatever and people will just sit there and say "woah, cool man." It's usually productive to try to understand the traditional view as thoroughly as you can before simply inventing an "alternative." This is needed to actually argue for one or the other, and to develop an alternative that has a chance of actually being better than the established view.

And that is exactly what I am trying to do, hence this thread. And as I said, in reality I didn't not make anything up, there is precedent and application for my idea in the Riemann sphere, something I was unaware of previously.

[Blackbeard]

A) One can do stupid math tricks.
B) I don't know where, but I'm sure you've done it somewhere.
C) ???
D) Profit!

You don't need a better argument, you need a argument, period... I hate to break it to you, but: 1 != 0
x != x^2 when x>1
1+1 != 3 When you get results that say otherwise, they're wrong. It's not some kind of special new math you've discovered. It's wrong.

So wrong, in fact, that contradictions are one of the most powerful forms of proof in mathematics. If you assume your premise and derive a contradiction from it, then you have proven that premise to be false. Crazy in that you've denied the existence of a negative infinity, and ignorant in not knowing that 1/0 is not infinity. 1/0 is undefined. Equations that approach 1/0 can be evaluated to see what they approach, but a flat literal 1/0 is simply undefined. There didn't seem much point in discussing it when the basic assumptions underpinning it don't work. But since you ask, I'll explain the fundamental problem with it: It is discontinuous. Somewhere, your number line flips from an arbitrarily large number to an arbitrarily small number for absolutely no reason. Discontinuities cause a lot of problems in mathematics. Arbitrarily redefining the number line as something discontinous would break an awful lot of things that we know actually work. Things like...calculus. And negative numbers.

Read the whole thread. There is precedent set for 1/0 = infinity and 1/infinity = 0. And never in my "number circle" does a number jump from incredibly large to incredibly small, if you look at it logically (perhaps I did not illustrate this but I thought it was self-evident) that as you go up in positive numbers you eventually get to infinity and then flip to an incredibly large (just shy of infinite) negative number from an incredibly large (just shy of infinite) positive number. Just like as you move on the number line from positive to negative, from an infinitesimal positive number to an infinitesimal negative number.

Your condescending and insulting manner is not appreciated, BTW. I may not have studied math to the extent you have, but I am certainly no idiot. And the fact that you come across as so dogmatic indicates a closed mind. Any further responses from you will be duly ignored, there are people on here who seem to have a much better grasp on the mathematics than you anyway and present their knowledge without the belittling comments.

[Blackbeard]

And yet, x=-x is a logical extrapolation when there's no negative infinity. Allow me to present a proof by contradiction of your number line.

A) f(x) = -x
B) infinity = -infinity
C) |f(x)| approaches infinity as x approaches infinity
D) From B, |f(x)| approaches f(x) as x approaches infinity.
E) Contradiction. f(x) never approaches |f(x)|.
F) Therefore infinity cannot be the same as negative infinity. If you don't want to sound like a crackpot, it's best not to spout the whole martyr line. They laughed at galileo. They also laughed at bozo the clown.

I'm not bashing it because I don't like it. I'm bashing it because it doesn't work. Without a negative infinity on the real number line, you get contradictory results.


Then why, as in the Riemann sphere, is there only one infinity and this form of mathematics has found very much use in physics, quantum mechanics and string theory? Perhaps it is you who needs to read up more before making blanket statements. I at least admitted I see and understand both sides of the argument. Read this article before spouting further Reimann Sphere. (http://en.wikipedia.org/wiki/Riemann_sphere)

[Tenek]

And as I said, in reality I didn't not make anything up

something I was unaware of previously.

Unless that double negative was intentional, you made stuff up

Also the reason we don't have 1/0 = infinity is because it's asking what number, when multiplied by zero, equals 1. There aren't any. Same with 1/infinity - you multiply 0 by infinity, you get 0; anything else and it's ±infinity. Once again, there isn't anything that fits the bill.

edit:

And I'm going to remain somewhat skeptical of your in-depth knowledge of the issues considering that 16 hours ago you didn't understand why there are more reals than integers.

[Blackbeard]

Unless that double negative was intentional, you made stuff up

The double negative was unintentional.

Also the reason we don't have 1/0 = infinity is because it's asking what number, when multiplied by zero, equals 1. There aren't any. Same with 1/infinity - you multiply 0 by infinity, you get 0; anything else and it's ±infinity. Once again, there isn't anything that fits the bill.

From a Wiki article (http://en.wikipedia.org/wiki/Riemann_sphere) that was linked to twice on this thread, now three times, about the Riemann sphere (http://en.wikipedia.org/wiki/Riemann_sphere): "In mathematics, the Riemann sphere is a way of extending the plane of complex numbers with one additional point at infinity, in a way that makes expressions such as 1/0=infinity well-behaved and useful, at least in certain contexts. It is named after 19th century mathematician Bernhard Riemann. It is also called the complex projective line."

In addition, is it true that 1/2 = .5? Yes. Then the reciprocal is true, 1/.5 = 2. Break out your calculator. I think the reciprocal rule states that if 1/x = y then 1/y = x. True? So 1/0 = infinity leads to 1/infinity = 0. Yes, I see how that equation does lead to 1 = infinity x 0. Hence the resultant mathematical illegality. hence the work of Bernhard Riemann which allows for 1/0=infinity and that work, as I have said before, has found use in physics, quantum physics and string theory. Not my words. read the article

edit:

And I'm going to remain somewhat skeptical of your in-depth knowledge of the issues considering that 16 hours ago you didn't understand why there are more reals than integers.

I never claimed I was an expert nor claimed that I have in-depth knowledge of this issue. If you are skeptical of the work of Bernhard Riemann, fine. I find vindication that I thought of a similar, though overly simplistic as I already stated, concept years ago. Discovering that there is a more formalized and accepted version makes this thread worth it for me despite close minded folks like you who feel the need to belittle others for no good reason. No, I may not be a math expert. Honestly I never even went to college, but I sport a very high IQ and possess excellent logical abilities. I'll take my logical abilities over your knowledge any day. Knowledge, if you can't figure out how to use it, is virtually useless.

[Arctish]

<MD to P per request of the OP>

[TNorthover]

I never claimed I was an expert nor claimed that I have in-depth knowledge of this issue. If you are skeptical of the work of Bernhard Riemann, fine. I find vindication that I thought of a similar, though overly simplistic as I already stated, concept years ago.

Before I lay into Corona, you got lucky. You were mixing concepts of infinity like there's no tomorrow, and happened upon applying the idea that there's a single point at infinity to the number line; you were then fortunate enough not to pursue it far enough to run into a contradiction (well, until later posts). There's far more to the Riemann sphere than the number line, or even that it has a single point at infinity. The essence of the Riemann sphere is that it's right that a single point at infinity is added to the complexes: you didn't establish that (not a criticism -- there's no way I could have done so on my own either). In fact it's usually not the right thing to do with the number line.

Having said that:
Equations that approach 1/0 can be evaluated to see what they approach, but a flat literal 1/0 is simply undefined. There didn't seem much point in discussing it when the basic assumptions underpinning it don't work. But since you ask, I'll explain the fundamental problem with it: It is discontinuous.
That's not true. When you add a (or some) points at infinity, you define continuity. You'll get different continuous functions from one infinity and from two, and both will be different again from the continuous functions on \mathbb{R} itself. Both are valid, but what you lose with one point at infinity is an ordering that behaves well with continuity. In general, arithmetic takes a dive, but for analysis points at infinity are very useful.

E) Contradiction. f(x) never approaches |f(x)|.

This is explicitly wrong: f(x) approaches |f(x)| as x approaches 0 (well, actually |f(x)-|f(x)|| approaches 0 but I'm sure that's what you meant). It's also wrong as x approaches infinity under the one-point compactification of the reals -- as you showed with D.

[untermensche]

How many elements are there in each infinite set?

Are there not infinite, by definition?

If every infinite set has infinite elements, where is the difference?

[Blackbeard]

Before I lay into Corona, you got lucky. You were mixing concepts of infinity like there's no tomorrow, and happened upon applying the idea that there's a single point at infinity to the number line; you were then fortunate enough not to pursue it far enough to run into a contradiction (well, until later posts). There's far more to the Riemann sphere than the number line, or even that it has a single point at infinity. The essence of the Riemann sphere is that it's right that a single point at infinity is added to the complexes: you didn't establish that (not a criticism -- there's no way I could have done so on my own either). In fact it's usually not the right thing to do with the number line.

Having said that:

That's not true. When you add a (or some) points at infinity, you define continuity. You'll get different continuous functions from one infinity and from two, and both will be different again from the continuous functions on \mathbb{R} itself. Both are valid, but what you lose with one point at infinity is an ordering that behaves well with continuity. In general, arithmetic takes a dive, but for analysis points at infinity are very useful.



This is explicitly wrong: f(x) approaches |f(x)| as x approaches 0 (well, actually |f(x)-|f(x)|| approaches 0 but I'm sure that's what you meant). It's also wrong as x approaches infinity under the one-point compactification of the reals -- as you showed with D.

Thank you. And you may criticize away, I make no claims to be an expert mathemetician. And I do understand that the Riemann sphere deals with complex numbers (a+bi where a and b are real numbers and i is the square root of -1). I actually wouldn't mind a primer in complex numbers to better understand how they are explained and applied as the square root of -1 is obviously a difficult concept to grasp.

[Phil Scott]

There is precedent set for 1/0 = infinity and 1/infinity = 0.Both of these expressions are still undefined even when extending the complex plane to create the Riemann sphere. "1/0" means "that unique number which when multiplied by 0 makes 1". Even with a point at infinity, there is still no such number. However, when extending certain functions to the complex plane, it is a useful rule to remember "1/0 goes to infinity" and "1/infinity goes to 0". For instance, an important class of functions on the extended plane are the Möbius transformations, which have the form f(z) = \frac{az + b}{cz + d}. In analytically extending them to the extended plane, we can take as a rule that if the denominator is 0, the result is infinity, and if the denominator is infinity while the numerator is finite, the result is 0 (if both numerator and denominator are infinity, the result is a/c). For the Möbius transformation f(z) = \frac{1}{z}, we have \bar{f}(0) = \infty and \bar{f}(\infty) = 0.


For some more illustration, some circles drawn on the Riemann sphere will pass through the point at infinity, and when they are mapped back to the plane, will appear as unending straight lines. That means you can think of a straight line on the extended plane as an infinitely large circle, made by joining both "ends" up at the point at infinity. You can even use a Möbius transformation to map the "generalised circle" onto an ordinary circle in the plane and map back again. You could then have two points, one running around the ordinary circle in a single direction, and a corresponding point running along the line. As the point on the circle makes a revolution, the other will move along the line more and more quickly towards infinity, arrive at the point at infinity, and then suddenly appear from the other direction to get back to its start.

[JohannGoodflag]

A remark about division by zero:

You can certainly write "1/0" and define it to be "something", in the sense that division is a function which you can extend in any way you please. However, multiplication and addition may not have the properties which were guaranteed for them originally if you do so.

More precisely: most number systems that we like talking about --- the integers, rationals, reals, complexes, and even the quaternions --- are examples of a ring, which is just a number system with a fixed collection of properties we find reasonable/useful to talk about.
http://en.wikipedia.org/wiki/Ring_%28mathematics%29
It is easy to prove that division by zero cannot be defined in a ring; that is, if you extend your definition of multiplication so that division by zero is possible, you don't have a ring any more --- and as a result, arithmetic has been fouled up.

In the case of adding infinity to the complex numbers and stipulating that 1/0 = infinity, you cannot cancel infinities. That is, the equality x + infinity - infinity = x does not hold for all x, nor does x * infinity / infinity = x. If you can sleep at night with this, that's fine; but it is not an essential feature of the complex numbers to have a single point at infinity. And 0/0 still can't be defined in a nice way.

[OneWayTraffic]

How many elements are there in each infinite set?

Are there not infinite, by definition?

If every infinite set has infinite elements, where is the difference?

Some infinities are bigger than others.

The Reals are bigger than the Natural numbers, which are the same size as the Rationals. All easily provable in university mathematics.

Basically you can count the Rationals. You order them up by the denominators in a long list. It's impossible to do so with the Real numbers. Give me any arbitarily long list of real numbers and I can define a number that isn't on the list. Moreover, I can show you that this method would work for even a list that went on forever without end.

[Huon]

The important mathematician on these subjects is Georg Cantor :
http://en.wikipedia.org/wiki/Georg_Cantor

[Blackbeard]

OK, I get it. I pondered and read and do see the problems with 1/0 = infinity, and I do see why the irrational and real numbers are uncountable (I am figuring that the reals are uncountable because they include the irrationals). I even read articles on the infinitesimal and how the infinitesimal presents similar problems as infinity, but like infinity many brillian people have figured out ways to deal with those problems if not ultimately solve them.

That said, please allow me to muse for a moment though without flaming me for it. I couldn't help but think that perhaps our whole "digital" number system is inherently flawed in describing the real world. I began to think of how it is impossible to accurately describe a waveform "digitally". I am an ex-sound engineer and started doing that before CD's were invented (showing my age), when practical digital audio was in it's infancy and not yet used in professional recording studios (Well, early in my career I did get to witness one of the first uses of a digital audio multitrack recorder in the studio, a prototype 32 track machine made by 3M used to record Bob Dylan's record Infidels, ironically enough, in 1983). As such, when digital audio came out, I had to learn all about it. Most sound engineers were very resistant to it for the reason that it was realized that digital audio could not capture an analog waveform accurately. I am sure you all know why. No matter how many bits you use, no matter how high a sample rate you use, you are missing some of the waveform. Sure, digital audio as it is today is good enough so as to be indiscernible from even the most hi-fi analog recording, in fact it will sound better due to a much lower noise floor, but I digress. In reality, if an audio waveform is viewed on an oscilliscope of sufficient resolution both pre and post digitization (but before the "smoothing" output filters are applied), you will be able to see the difference in the waveform. The undigitized wave form will be smooth while the digitized waveform will be stepped. Again I say it doesn't matter how high your sample rate is or what bit depth you use, if the oscilliscope is of sufficient resolution to see the quantization of the digital waveform you will be able to see the "steps". So I began to think that perhaps our number system is somewhat analogous in it's inability to perfectly accurately describe the real world, the natural universe. Our precision using our current number system of real numbers (I am not sure another is possible) is certainly good enough for the very large majority of applications but perhaps not precise enough, just like no digitization of a waveform can ever be perfectly accurate ...... unless you can manage an infinite sample rate and infinite bit depth, which would obviously be impossible, especially since logically you would have to use a set of infinity for real numbers. So what's my point? Is it possible that our number system, as it is inherently quantified and "digital", not good enough to desribe the real world which it seems to me, logically, is not digital and perhaps not quantifiable? If you can't describe a wave form accurately with numbers (regardless of the base used, the base is irrelvant), how can we expect to describe the most fundamental natural laws with numbers, natural laws that seem to be based on waves and the interaction of perhaps infinitessimaly small particles? Our number system seems inadequate to accurately define the infinitessimal, not to mention the infinite, when it seems to me that both "values" must surely "exist" in nature. IOW, how can we quantify an infinitessimally small particle, or an infinitely large universe, with a number system that does deal naturally with either of those values? It is for this reason that the Riemann Sphere remains so compelling to me, however based on what I have read I think it is not the answer yet, but may be a step towards a better solution.

Again, please don't flame, I can almost see the responses already. I am just musing.

[premjan]

I think the Riemann sphere concept is present in the area of projective geometry, which is commonly used in computer graphics actually. The projective geometry concept for the one-dimensional Riemann Sphere is the Projective line.

[kennethamy]

The important mathematician on these subjects is Georg Cantor :
http://en.wikipedia.org/wiki/Georg_Cantor

Sure is. He distinguished among degrees of infinite sets. A simple example is that the set of all even numbers is infinite; and the set of all odd and even numbers is infinite. But the latter set is greater than the former set since it includes it. An elementary, but good book is by George Gamow. One, Two, Three, Infinity

[Huon]

jgold6 : your post #36.

From a mathematical point of view, waves are treated with Fourier series, and Fourier transforms. This does not imply any "digital" numeration. It is exactly the same problem as the numeric approximation of sin (x) in the old logarithms tables of my youth. How many decimals ? We use units which are adapted to the problem, meters, kilometers, microns, angströms, etc...

[Blackbeard]

OK, I'll read up on those. But again, even using the smallest units possible, you still can not describe accurately the point in a waveform that is infinitessimally above the 0 crossing point, or that is infinitessimally less than the peak value, using any number system, as far as I can see. Yes, I know that given a frequency and amplitude, you can easily draw and describe a wave accurately using math. This, it seems to me, is far different from being able to quantify every specific point on that wave, because there is at least one that is infinitessimally small, and there are, it would seem, an infinite number of points that are infinitessimally smaller or larger than any given point ... not to mention the infinite number of irrationals the wave must pass through .... you can only ever approximate those using our number system, true? So, if you want to find out exactly where a wave is at a given point in time it seems to me that mostly you can only ever get an approximation, sure a very accurate approximation, but most often not a perfectly accurate number.

Which brings me to another thought .... LOL, you are all going to love this! Maybe if we used a number system in base infinity? :Cheeky:

[Phil Scott]

Sure is. He distinguished among degrees of infinite sets. A simple example is that the set of all even numbers is infinite; and the set of all odd and even numbers is infinite. But the latter set is greater than the former set since it includes it.Neither is greater than the other in the senses Cantor considered. The subset relation cannot be used to order all the infinite sets.

[kennethamy]

Neither is greater than the other in the senses Cantor considered. The subset relation cannot be used to order all the infinite sets.

Thank you. When I wrote that, I had a feeling that was wrong, but did not remember why. In fact, I think I made that mistake before. Can you explain, though?

[untermensche]

Some infinities are bigger than others.

The Reals are bigger than the Natural numbers, which are the same size as the Rationals. All easily provable in university mathematics.

Basically you can count the Rationals. You order them up by the denominators in a long list. It's impossible to do so with the Real numbers. Give me any arbitarily long list of real numbers and I can define a number that isn't on the list. Moreover, I can show you that this method would work for even a list that went on forever without end.
How is it possible to have more than infinite elements in a set?

[kennethamy]

How is it possible to have more than infinite elements in a set?

As I was corrected, this isn't what Cantor means by degrees of infinity, but this give you the idea:

"In the early 1600's Galileo began to show signs of a modern attitude toward the infinite, when he proposed that "infinity should obey a different arithmetic than finite numbers." But it was not until the late 19th century that Georg Cantor (1845-1918), a German mathematician, finally put infinity on a firm logical foundation and described a way to do arithmetic with infinite quantities useful to mathematics. His basic definition was simple: a collection is infinite, if some of its parts are as big as the whole. For example, even though from one point of view the entire list of numbers we count with {1,2,3,4,5,.......} is twice as large as the list of even numbers {2,4,6,8,10,.......}, the two lists can be matched-up in a one-to-one fashion.


So the two lists are exactly the same size, infinite. (This idea has been amusingly elaborated in the story of "The Hotel Ad Infinitum" as told by David Stacy.)

Cantor was able to demonstrate that there are different sizes of infinity. The infinity of decimal numbers that are bigger than zero but smaller than one is greater than the infinity of counting numbers. (Click to see Cantor's "diagonalization proof.") "

[untermensche]

As I was corrected, this isn't what Cantor means by degrees of infinity, but this give you the idea:

"In the early 1600's Galileo began to show signs of a modern attitude toward the infinite, when he proposed that "infinity should obey a different arithmetic than finite numbers." But it was not until the late 19th century that Georg Cantor (1845-1918), a German mathematician, finally put infinity on a firm logical foundation and described a way to do arithmetic with infinite quantities useful to mathematics. His basic definition was simple: a collection is infinite, if some of its parts are as big as the whole. For example, even though from one point of view the entire list of numbers we count with {1,2,3,4,5,.......} is twice as large as the list of even numbers {2,4,6,8,10,.......}, the two lists can be matched-up in a one-to-one fashion.


So the two lists are exactly the same size, infinite. (This idea has been amusingly elaborated in the story of "The Hotel Ad Infinitum" as told by David Stacy.)

Cantor was able to demonstrate that there are different sizes of infinity. The infinity of decimal numbers that are bigger than zero but smaller than one is greater than the infinity of counting numbers. (Click to see Cantor's "diagonalization proof.") "
I asked how is it possible to have more than infinite elements in a set.

What you say is interesting.

But, do you have an answer to my question?

[Phil Scott]

Thank you. When I wrote that, I had a feeling that was wrong, but did not remember why. In fact, I think I made that mistake before. Can you explain, though?The idea is that the way to determine whether two sets of objects contain the same number of elements is by seeing whether they can be paired so that each pair contains an element from each set. For instance, the sets {1,2,3,4} and {a,b,c,d} must contain the same number of elements because they can be paired: (a,1), (b,2), (c,3), (d,4). Other pairings are possible (23 others), but only one is needed to show that the two sets are the same size.

This definition allows us to determine whether infinite sets contain the same number of elements. You can, for instance, show that the set of even numbers is the same size as the set of even and odd numbers. We just form the pairing (0,0), (1,2), (-1,-2), (2,4), (-2,-4),...,(k,2k),.... This pairing matches every integer with every even number.

Cantor showed that every integer can also be paired with every rational number, but that it is impossible to pair all the integers with the real numbers. At best, we can match up the set of integers with a subset of the reals, meaning that, by Cantor's definition of size, there are more reals than integers. Cantor also showed that no set can be the same size as its power set (the power set of a set A being the set of all subsets of A.) That means we immediately have an infinitely large hierarchy of ever "larger" sets. The power set of the naturals is larger than the naturals. The power set of the power set of the naturals is larger than the power set of the naturals. And so on.... Of course, Cantor had much much more to say about infinite sets than just this, but that's the very basic definition.

Cantor also considered well-ordered sets. A well-ordered set A is a set together with an order relation such that any subset of A has a minimum element. One well-ordered set A can be considered smaller than another well-ordered set B if A is structurally identical to an initial-segment of B. For instance, one well-ordered set is the natural numbers with the relation <. Another well-ordered set consists of two copies of the natural numbers side by side: 0,1,2,3,...,0,1,2,3,... where the numbers in the second copy are somehow distinguished from those in the first. In this case, the natural numbers, being identical to just the first half of the concatenation are considered smaller than the concatenation.

By this account, the set of integers with their natural ordering is structurally identical to the even integers with their natural ordering, meaning again, neither is greater or less than the other.

[Phil Scott]

How is it possible to have more than infinite elements in a set?The point is that some infinite sets are, in a sense, larger than others, so we need a more precise way to talk about a set having an infinite number of elements. Assuming a certain axiom of set theory, a precise hierarchy of numbers emerge to discuss sets with an infinity of members, known as the cardinal numbers. Each is denoted by the Hebrew letter \aleph with a so-called "ordinal number" for a subscript(ordinals are associated with the well-ordered sets I mention above). Since ordinary whole numbers form the smallest of the ordinals, our first infinite number is \aleph_0. Then we have \aleph_1, \aleph_2, \aleph_3 and so on until we get to \aleph_\omega and then we keep going to \aleph_{\omega + 1}.

We don't, therefore, ask how it is possible to have more than an infinite number of elements. A set such as the natural numbers has \aleph_0 elements. How is it possible to have more than \aleph_0 elements? Easy. Just have \aleph_1 elements.

[kennethamy]

I asked how is it possible to have more than infinite elements in a set.

What you say is interesting.

But, do you have an answer to my question?

I thought I gave you an example. Aren't there more odd and even numbers than there are just odd numbers? But but the set of (just) odd numbers has an infinite number of elements, and the set of (just) odd numbers has an infinite number of elements. So the set of both odd numbers and even numbers has an infinite number of elements, and is larger than just the set of either odd or the set of even numbers, each of which also has an infinite number of elements.

[JohannGoodflag]

I asked how is it possible to have more than infinite elements in a set.

It is not possible to have "more than infinite" number of elements. Infinite describes a type of quantity. Just as "plural" means "more than one", "infinite" means "larger than any finite* quantity". Just as there is more than one kind of "plural" quantity (2, 3, 4, etc are all different quantities), there also more than one kind of infinite quantity (the number of real numbers is a larger infinite quantitiy than the number of integers).

[untermensche]

It is not possible to have "more than infinite" number of elements. Infinite describes a type of quantity. Just as "plural" means "more than one", "infinite" means "larger than any finite* quantity". Just as there is more than one kind of "plural" quantity (2, 3, 4, etc are all different quantities), there also more than one kind of infinite quantity (the number of real numbers is a larger infinite quantitiy than the number of integers).
That does not make sense.

If there can be no more elements than infinite elements, then all things with infinite elements have an amount of elements that cannot be exceeded. None can have more elements than any other.

[perfessor]

That does not make sense.
This is a good attitude. The first step in solving a problem is to recognize that you have one.

If there can be no more elements than infinite elements, then all things with infinite elements have an amount of elements that cannot be exceeded. None can have more elements than any other.
First of all, "infinity" is not a number or even a quantity. It is a concept. You cannot say that the set of integers has more elements than the set of primes - both are infinite. The most you can say is that one is a subset of the other. You seem to understand this part.

There are, however, properties of infinite sets that are worth understanding, such as cardinality. The integers are countable, as are rationals. The reals are not. So in a sense there are "more" reals than rationals, but "more" tends to make us think of counting, which we cannot do with reals. Think of them as a "denser" set.

[untermensche]

This is a good attitude. The first step in solving a problem is to recognize that you have one.


First of all, "infinity" is not a number or even a quantity. It is a concept. You cannot say that the set of integers has more elements than the set of primes - both are infinite. The most you can say is that one is a subset of the other. You seem to understand this part.

There are, however, properties of infinite sets that are worth understanding, such as cardinality. The integers are countable, as are rationals. The reals are not. So in a sense there are "more" reals than rationals, but "more" tends to make us think of counting, which we cannot do with reals. Think of them as a "denser" set.
You have not explained anything.

How is it possible to have more than infinite elements?

Just conceptually explain that without a bunch of extraneous mathematical jargon.

[perfessor]

You have not explained anything.

How is it possible to have more than infinite elements?

Just conceptually explain that without a bunch of extraneous mathematical jargon.
Ha ha you're funny. We are talking about mathematical concepts, you know. That's what the jargon is about. Try a dictionary.

ETA: The concept of countable and uncountable sets (cardinality) was explained early in the thread. Perhaps I presumed too much, to think you might have read those early posts.

[untermensche]

Ha ha you're funny. We are talking about mathematical concepts, you know. That's what the jargon is about. Try a dictionary.

ETA: The concept of countable and uncountable sets (cardinality) was explained early in the thread. Perhaps I presumed too much, to think you might have read those early posts.
Explaining the differences in those sets does not answer my question in any way.

If there is some logic which says one infinite set can contain more elements than another infinite set I would like to see it.

[Corona688]

Read the whole thread. There is precedent set for 1/0 = infinity No. 1/x approaches positive or negative infinity as x approaches 0, but that is not the same thing as 1/0 = infinity.and 1/infinity = 0. This postulate, on the other hand, is completely true since both an arbitrarily large positive number and an arbitrarily large negative number produce the same answer.And never in my "number circle" does a number jump from incredibly large to incredibly small, if you look at it logically (perhaps I did not illustrate this but I thought it was self-evident) that as you go up in positive numbers you eventually get to infinity and then flip to an incredibly large (just shy of infinite) negative number from an incredibly large (just shy of infinite) positive number. Your argument is semantic. The problem remains the same. Your number set is discontinous.Just like as you move on the number line from positive to negative, from an infinitesimal positive number to an infinitesimal negative number. They are nothing alike. The space between -1 and 1 is continuous: for all x, f(x+h) approaches f(x) as h approaches zero. aka for all points on the line, no matter how small you subdivide it, you will always find a number between any two numbers. There is no discontinuity. You might say it is "smooth" with no jumps.

Now take, for example, f(x)=1/(x-3). It is discontinuous at 3.

f(x)=1/(x-3)

x h f(x+h) f(x-h)
3 1 1 -1
3 .1 10 -10
3 .01 100 -100
3 .001 1000 -1000
... There is a "jump" at f(3), where it goes from an arbitrarily large negative number to an arbitrarily large positive number. Take the limit from the left and you get something negative, take the limit from the right and you get something positive, approach it from both sides and you get infinity minus infinity, which would seem to be zero (if you can even do that with infinity!).

Three seperate answers depending on how you approach the problem. Which one is right?

It can't be a positive number: From the left, it becomes an arbitrarily large negative number.

It can't be a negative number: From the right, it approaches an arbitrarily large positive number.

Lastly, it can't be zero: The only two values at which f(x) approaches zero are positive and negative infinity.

Hence we say, f(3) = 1/(3-3) = 1/0 is undefined. It is not something that can be calculated.

Incidentally, the reinmann sphere does not help you. "complex numbers have a sphere therefore real numbers have a circle" is an argument from analogy, you need something much stronger.Your condescending and insulting manner is not appreciated, BTW. Neither is yours. Yes, I have read the thread, thank you very much. That I disagree with you does not immediately indicate I am wrong. Nor have I seen you present any evidence whatsoever of your position: just a feeling that it must be right, and a pretty but meaningless diagram plotted from no data at all.I may not have studied math to the extent you have, but I am certainly no idiot. I wouldn't call you an idiot. Stubborn? Yes.And the fact that you come across as so dogmatic indicates a closed mind. Yawn. I've never been told this by anyone who had any evidence for their position. If they had evidence, they'd post it instead of insulting people.Any further responses from you will be duly ignored Hardly suprising, if the only response you have to my disproofs are insults and non-sequitors.there are people on here who seem to have a much better grasp on the mathematics than you anyway and present their knowledge without the belittling comments. I'm sure they'll either leave or become similarly acid once you tell them they're "closed minded" for not acceding to the inevitability of your correctness.

[JohannGoodflag]

If there can be no more elements than infinite elements, then all things with infinite elements have an amount of elements that cannot be exceeded. None can have more elements than any other.

"Infinite" does not mean "a number which cannot be exceeded": this is the source of your confusion.

There are a number of equivalent definitions of "infinite", one of which is simply "more than any finite number" --- where "more" is carefully defined in terms of one-to-one mappings. This definition does not imply that there cannot be more than one size of infinite quantity.

If you are interested in understanding what we're talking about, I suggest you abandon your prior notions of what "infinite" entails, and start over.

[perfessor]

Explaining the differences in those sets does not answer my question in any way.

If there is some logic which says one infinite set can contain more elements than another infinite set I would like to see it.
OK, I'll try again.

Rational numbers i.e. numbers of the form a/b, where a and b are integers. This is an infinite set. It is countable - it can be put in a one-to-one correspondence with the set of Natural Numbers (1,2,3,4,...).

Integers are a subset of Rationals. Obviously, Integers are also countable.

The cardinality of these sets is called Aleph-Null. I don't care if you don't like the jargon. It is what it is.

The set of Reals - you can think of them as points on the number line. These are not countable. They are dense. There are "too many" of them to count. There's a proof, but for now I don't want to derail the explanation.

The cardinality of the set of Reals is Aleph-One.

I think you want to assign some sort of quantity, but think that you can't (you are right about that), therefore they have the same number of elements (you are wrong about that). But while quantity is meaningless, cardinality is not. The cardinality of these two sets, Reals and Rationals, is different.

[Kronocide]

That does not make sense.

If there can be no more elements than infinite elements, then all things with infinite elements have an amount of elements that cannot be exceeded. None can have more elements than any other.

It only doesn't make sense if you think of "infinite" as a quantity. No set can have more elements than >1 (that is, there is no set that is too big to be >1) but sets that are >1 can be of different size, right? In the same way, "infinite" means greater than a finite number. There can be such numbers of different sizes.

I think the best way to understand cardinality is to see a concrete example, such as the diagonal proof that all the infinite subsets of the natural numbers can not be matched, one to one, with the natural numbers themselves. That is, there are more infinite subsets of natural numbers than there are natural numbers, even though the latter is an infinite set.

Let's describe a set of the natural numbers as a sequence of 1's and 0's, where a 1 means that the natural number on this position is included in this set and a 0 means this number is not included. So each infinite subset of natural numbers is defined by an infinite sequence of 1's and 0's. Then we list these sequences, one after another in some random order:

1 0 0 0 1 1 0 1 0 0 1 1 1 1 0 ...
1 1 0 1 0 0 0 0 1 1 0 0 0 0 1 ...
0 1 0 0 0 0 1 1 1 1 1 1 0 1 0 ...
1 1 0 1 0 1 1 1 1 0 0 1 0 1 1 ...
...

The matrix goes on to infinity to the right as well as downward, since each set is of infinite size and there is an infinite number of them. Now to prove that this infinite list can not be the list of all subsets of the natural numbers. Construct a new set by starting at row 1 column 1 and switch whatever value you have there, then proceed diagonally and proceed to switch each 1 to a 0 and vice versa. This will be a new set of natural numbers, and it will not be in the list, because it will differ from every set in the list at at least one position.

Hence, an infinite list can not contain all infinite subsets of the natural numbers. These sets can not be counted, they are not denumerable. There are more of them than of natural numbers.

That rational numbers are denumerable can be proved by another matrix. Put all the single parts on the first row, the two-parts on the second, etc:

1/1 1/2 1/3 1/4 ...
2/1 2/2 2/3 2/4 ...
3/1 3/2 3/3 3/4 ...
...

Then sort them into a single row by picking the first one on the first row, then the first on the second row, then the second on the first row, then the third on the first row, then the second on the second row, the first on the third row, etc. describing diagonals through the matrix:

1/1, 2/1, 1/2, 1/3, 2/2, 3/1, ...

It's obvious that you can work your way through the matrix this way and create one countable series of the rational numbers. The useful thing is that you can do the same with sets of any finite number of coordinates, by first doing this with the first pair, turning them into a countable series, then adding the third coordinate to that series, creating a new matrix, doing the procedure on the new matrix, again arriving at a denumerable series, adding the next coordinate, etc. If the number of coordinates is infinite, we have the situation described above, and the sets are not countable.

This is all just about the first two cardinalities of transfinite numbers. But there is an infinite number of cardinalities as well! :)

[untermensche]

OK, I'll try again.

Rational numbers i.e. numbers of the form a/b, where a and b are integers. This is an infinite set. It is countable - it can be put in a one-to-one correspondence with the set of Natural Numbers (1,2,3,4,...).

Integers are a subset of Rationals. Obviously, Integers are also countable.

The cardinality of these sets is called Aleph-Null. I don't care if you don't like the jargon. It is what it is.

The set of Reals - you can think of them as points on the number line. These are not countable. The are dense. There are "too many" of them to count. There's a proof, but for now I don't want to derail the explanation.

The cardinality of the set of Reals is Aleph-One.

I think you want to assign some sort of quantity, but think that you can't (you are right about that), therefore they have the same number of elements (you are wrong about that). But while quantity is meaningless, cardinality is not. The cardinality of these two sets, Reals and Rationals, is different.
I say these are simply two different conceptions of infinity that cannot be compared as if they are the same conception.

One a set of imaginary objects. A set of stationary elements. While the objects can be counted, the set can not be counted.

One a set of a line, in other words a set as the answer to some equation. A set as the product of some function. A set as the product of a process.

And while these sets are different conceptually, the elements in each set is infinite. And as sets containing infinite elements, one set cannot have more elements than the other set.

And I do not think infinity is a quantity. That is why I say the number of elements cannot be exceeded, which is a concept not a quantity.

[Kronocide]

I say these are simply two different conceptions of infinity that cannot be compared as if they are the same conception.

There aren't just two cardinalities, there is an infinite number.

[JohannGoodflag]

And while these sets are different conceptually, the elements in each set is infinite. And as sets containing infinite elements, one set cannot have more elements than the other set.

So, you don't think it is at all interesting that the elements of these two sets cannot be matched off, one by one, even in principle?

[Corona688]

Explaining the differences in those sets does not answer my question in any way.

If there is some logic which says one infinite set can contain more elements than another infinite set I would like to see it. Well, the problem is that infinity is not a number. Name any integer number and I can count one higher, but counting higher than infinity still leaves you at infinity.

You can map positive integers to positive and negative integers easily: map zero to zero, map even numbers map to negative numbers with f(n)=-n/2, map odd numbers to positive numbers with g(n)=((n-1)/2) + 1. Mapping positive and negative integers to positive integers alone is even easier. Zero maps to zero, positive integers map to 2n, negative integers map to 2n-1.

No matter what number you pick from which set, it can be freely converted back and forth between the two. You have to count twice as high in the positive integers but that's immaterial since "forever" never had a stopping point anyway. In this sense, positive integers alone are the same "size" as positive and negative integers together.

But try and map integers to real numbers; you can't. Try and count up to pi in infinite decimal places and you'll never finish saying the first number in the list; the integers get "stuck" counting the first number while the rationals go on. In this sense, the real numbers are "larger" than the integers.

[untermensche]

So, you don't think it is at all interesting that the elements of these two sets cannot be matched off, one by one, even in principle?
But the matching of elements would never end so it never really gets interesting.

[untermensche]

Well, the problem is that infinity is not a number. Name any integer number and I can count one higher, but counting higher than infinity still leaves you at infinity.

You can map positive integers to positive and negative integers easily: map zero to zero, map even numbers map to negative numbers with f(n)=-n/2, map odd numbers to positive numbers with g(n)=((n-1)/2) + 1. Mapping positive and negative integers to positive integers alone is even easier. Zero maps to zero, positive integers map to 2n, negative integers map to 2n-1.

No matter what number you pick from which set, it can be freely converted back and forth between the two. You have to count twice as high in the positive integers but that's immaterial since "forever" never had a stopping point anyway. In this sense, positive integers alone are the same "size" as positive and negative integers together.

But try and map integers to real numbers; you can't. Try and count up to pi in infinite decimal places and you'll never finish saying the first number in the list; the integers get "stuck" counting the first number while the rationals go on. In this sense, the real numbers are "larger" than the integers.
Even if larger can never mean, "the set has more elements than"?

[Corona688]

But the matching of elements would never end so it never really gets interesting. How can the matching never end when they cannot be matched?

[JohannGoodflag]

But the matching of elements would never end so it never really gets interesting.

Well, we can match the set of all the integers to the even numbers just by saying that we match n to 2*n. We don't match them literally "one at a time": we give a rule which matches all of the integers to all of the even numbers, in a way that there's no overlap where two integers get mapped to the same even number, or vice versa.

So, the concept of matching up the elements fo two infinite sets is well-defined, and doesn't take any time: we just define a rule which describes how it is done for all of the elements.

This can be done for many pairs of infinite sets; but it cannot for the integers and the reals.

[untermensche]

There aren't just two cardinalities, there is an infinite number.
Does that change anything?

[Corona688]

Even if larger can never mean, "the set has more elements than"? Correct. It's not the number of elements that's being counted, it's the ability of one set to map to another.

You can map the integers to the digits of one irrational number. For every integer number, there is a corresponding digit of pi. Given a concrete input value, a concrete output value can be calculated. But when asked to enumerate any concrete real value, the count is infinity. f(n) = 1, n is infinity. f(n) = 2, n is infinity. f(n)=2^0.5, n is infinity. This is not a valid mapping.

[untermensche]

Well, we can match the set of all the integers to the even numbers just by saying that we match n to 2*n. We don't match them literally "one at a time": we give a rule which matches all of the integers to all of the even numbers, in a way that there's no overlap where two integers get mapped to the same even number, or vice versa.

So, the concept of matching up the elements fo two infinite sets is well-defined, and doesn't take any time: we just define a rule which describes how it is done for all of the elements.

This can be done for many pairs of infinite sets; but it cannot for the integers and the reals.
You haven't really matched anything.

You've simply defined things differently.

And you wonder why you end up with different things?

[JohannGoodflag]

What have I defined differently? I've just got the two sets, and I've told you how to determine, for each integer, which even number it gets matched up with. I'm not doing any replacements or anything: I'm just describing a matching-rule.

[untermensche]

Correct. It's not the number of elements that's being counted, it's the ability of one set to map to another.
But number of elements is what defines denseness.

If something is more dense than something else then it has more elements per area.

[untermensche]

What have I defined differently? I've just got the two sets, and I've told you how to determine, for each integer, which even number it gets matched up with. I'm not doing any replacements or anything: I'm just describing a matching-rule.
You've defined the sets differently. You have not described the number of elements. You've described the realtionship between the concepts and made believe this changes the number of elements in some way.

[JohannGoodflag]

You've defined the sets differently. You have not described the number of elements. You've described the realtionship between the concepts and made believe this changes the number of elements in some way.

You mean that your complaint is that the set of real numbers and the set of integers don't have the same definition, and so you're guessing that it's for this reason that we cannot match them up against one another?

[untermensche]

You mean that your complaint is that the set of real numbers and the set of integers don't have the same definition, and so you're guessing that it's for this reason that we cannot match them up against one another?
No.

Not that they don't have the same defintion. They have a relationship.

But you act as if this relationship can effect the number of elements as if it were a quantity.

[JohannGoodflag]

They have a relationship; but it's not just an arbitrary relationship.

We basically just use three definitions, which do seem quite reasonable.

There are at least as many elements in a set B as in a set A if we can match up the elements of A with some subset of B.
There are exactly as many elements in a set B as in a set A if we can match up the elements of A with the elements of B itself. (Note that B is "a subset" of itself, in the usual parlance, and so if B has exactly as many elements as A, we can weaken this to say that B also has at least as many elements as A.)
There are more elements in a set B than in a set A if B has at least as many elements as A (in the above sense), but not exactly as many (also in the above sense).


These are the relationships we care about. Is there any clear reason why this is a ridiculous way of trying to guage the relative sizes of two sets?

[Corona688]

You haven't really matched anything.

You've simply defined things differently.

And you wonder why you end up with different things? Sure you can match things. Watch this!

1 becomes 2.
2 becomes 4.
3 becomes 6.
4 becomes 8.
5 becomes 10.

You can even go backwards!

100 becomes 50.
4 becomes 2.
65536 becomes 32768.

No matter what number you pick from either set we always know exactly what it's match in the other set is. The ability to do this is what mapping a set means.

You can't do this when mapping the integers to the reals. You can't tell me which integer pi maps to.

[untermensche]

Sure you can match things. Watch this!

1 becomes 2.
2 becomes 4.
3 becomes 6.
4 becomes 8.
5 becomes 10.

You can even go backwards!

100 becomes 50.
4 becomes 2.
65536 becomes 32768.

No matter what number you pick from either set we always know exactly what it's match in the other set is. The ability to do this is what mapping a set means.

You can't do this when mapping the integers to the reals. You can't tell me which integer pi maps to.
All that says is pi has no relationship to the integers. It doesn't say anything about the number of elements in a set that has no quantity.

[Corona688]

All that says is pi has no relationship to the integers. Actually, no. Name which part of pi you want -- say, 0.04 -- and I'll give you a corresponding integer n, 2. Given 2, I can give you the corresponding part of pi, 0.04.

But you cannot do this with the reals.

[Blackbeard]

Actually, no. Name which part of pi you want -- say, 0.04 -- and I'll give you a corresponding integer n, 2. Given 2, I can give you the corresponding part of pi, 0.04.

But you cannot do this with the reals.

That is one of the best explanations to a layman that I have seen to describe why the set of irrationals can not be counted.

What I get is that there is no function whereby one can "convert" (perhaps not a good word, but it helps me to think this way) a particular integer to a particular real. No function like 2n to "convert" the set of integers to the set of even numbers. No matter how complex you make the function, you will never be able to make one that will pair a unique integer to a unique irrational number, or real number for that matter as irrational numbers are a subset of reals. Am I getting it?

[perfessor]

What I get is that there is no function whereby one can "convert" (perhaps not a good word, but it helps me to think this way) a particular integer to a particular real.
The common word is "mapping". Such a mapping, if it existed, would have to set up a correspondence to every real.

No function like 2n to "convert" the set of integers to the set of even numbers. No matter how complex you make the function, you will never be able to make one that will pair a unique integer to a unique irrational number, or real number for that matter as irrational numbers are a subset of reals. Am I getting it?
Yes, yes you are! :wave:

[Blackbeard]

The common word is "mapping". Such a mapping, if it existed, would have to set up a correspondence to every real.


Yes, yes you are! :wave:

Cool, OK, I am done doing this >:banghead:

Now I would like to try to clarify in my mind another concept I read about thanks to referrences to it on this thread, and that is the "Axiom of Choice". Let me see if I got this correctly. The Axiom of Choice tries to get around this "problem" of mapping the reals by allowing one to essentially arbitrarily map every real number to an integer. IE, if you have a set A that consists of every possible set of reals (which would be infinite), you can make a new set B by choosing exactly one element from each subset of A. If you carefully choose a different real number from each subset of A, then you can map that new set B to the set of integers and count them. Do I have this right?

[JohannGoodflag]

Let me see if I got this correctly. The Axiom of Choice tries to get around this "problem" of mapping the reals by allowing one to essentially arbitrarily map every real number to an integer. IE, if you have a set A that consists of every possible set of reals (which would be infinite), you can make a new set B by choosing exactly one element from each subset of A. If you carefully choose a different real number from each subset of A, then you can map that new set B to the set of integers and count them. Do I have this right?

Er, no.

The Axiom of Choice is much simpler --- deceptively simple, even. It just says that if you have some sequence of non-empty sets (e.g., subsets of the reals --- but really, just any sequence of sets), that there is a way of picking one element from each set.

That is: suppose you have a function S, which for each positive integer gives you some non-empty set. Then, there exists a "choice function" c, which is a function such that for all n, c(n) is an element of S(n). For each n, c(n) just "picks one element" from S(n).

The fact that such functions exist can be proven for infinite sequences of finite sets, and finite sequences of infinite sets: you don't need to assume it. But it cannot be proven for infinite sequences of infinite sets. Because it is considered important to make possible a number of theorems in real analysis, many mathematicians consider it a reasonable axiom. But some disagree. (Personally, I think it's just fine.)

The Axiom of Choice doesn't "get around" anything to do with the fact that the set of real numbers is larger than the set of the natural numbers. It does, however, imply that given any two sets, either they will be the same size, or one will be larger than the other. It may seem that this should obviously hold, but the Axiom of Choice is neccessary to guarantee the existence of the various functions one needs between two otherwise unspecified sets in order to prove this. (Indeed, this can be taken as another motivation for allowing the Axiom of Choice, given that one is interested in studying cardinality the way that Cantor introduced.)

[Corona688]

But number of elements is what defines denseness. Number of elements over a range. In the integers, there are 10 numbers in [1,10]. In the reals, there are infinite numbers in [1,10].

[david bowditch]

i just don't understand how we can talk about infinity having size

isn't that like talking about the size of a point? if a point has any size it is multiple points. if infinity has a size it is finite

when we talk about matching an infinite set of odd numbers with an infinite set of both odd and even numbers why would any numbers not have a match?
1-1
3-2
5-3
7-4
ad infinitum

if this continues on forever im not seeing why any numbers wouldn't be paired up

can anyone explain what numbers from these sets couldn't be paired up? or why these numbers would not be able to be paired up?

[johno]

A particularly elegant discussion of infinity in its various guises is given by Rudy Rucker in a very accessible book entitled Infinity. Rucker is particularly skilful at presenting ideas in concrete imagery and my particular favourite is the book with infinitely many pages. The first page is half an inch thick, the second page a quarter, and so on. The book is an inch thick, but it contains infinitely many pages. It would be curious to turn the book over and have a look at the back because the book has no last page. Rucker goes on to suggest that you take the first page, and slice it into more pages, so the first subpage is a quarter of an inch thick, the next, an eightth of an inch thick, and so on. You do the corresponding slicing on each of the original pages. You end up with a book containing an infinity of pages, each one subdivided into an infinity of pages. Evidently there are more pages in the book after the process of subdivision than there were before, and of course, there is nothing to stop you sub-dividing the pages a third time, a fourth time, an infinite number of times. You always have a book with an infinite number of pages, but some infinities are bigger than others.

Does this help?

(You can tell that I am not a mathematician . . .)

johno

[premjan]

i just don't understand how we can talk about infinity having size

isn't that like talking about the size of a point? if a point has any size it is multiple points. if infinity has a size it is finite

when we talk about matching an infinite set of odd numbers with an infinite set of both odd and even numbers why would any numbers not have a match?
1-1
3-2
5-3
7-4
ad infinitum

if this continues on forever im not seeing why any numbers wouldn't be paired up

can anyone explain what numbers from these sets couldn't be paired up? or why these numbers would not be able to be paired up?

In fact, the even numbers and the integers are considered to be sets of equal size. The criterion of equal size is that a one-to-one correspondence can be created between elements of the two sets. But not all correspondences will show this. For instance the even numbers look a priori like a smaller set than the integers, as every even number is an integer but not every integer is an even number. We could say that the two sets have equal cardinality, but the integers are a proper superset of the even numbers.

[Kronocide]

One very familiar fact that this thread illustrates well is that our intuitions about transfinite numbers and infinite sets are not very useful or trustworthy. There is no use trying to intuitively reason about infinity, since your conclusions will most likely be wrong. To understand cardinality or countability you have to read and understand the actual proofs and definitions. Only then does it make sense.

[SophistiCat]

No matter what number you pick from either set we always know exactly what it's match in the other set is. The ability to do this is what mapping a set means.

You can't do this when mapping the integers to the reals. You can't tell me which integer pi maps to.

Sure I can. There is nothing special about pi as such. I can, for example, match the set {pi, 0, 1, 2, 3, ... } with the set of integers. What I can't do is match the entire set of reals with integers.

i just don't understand how we can talk about infinity having size

Strictly speaking, we can't. "Size" is just an informal alias for "cardinality" when we are talking about infinite sets. Cardinality of infinite sets is intuitively analogous to size of finite sets, but it is not the same thing. You can only take the analogy so far.

[david bowditch]

One very familiar fact that this thread illustrates well is that our intuitions about transfinite numbers and infinite sets are not very useful or trustworthy. There is no use trying to intuitively reason about infinity, since your conclusions will most likely be wrong. To understand cardinality or countability you have to read and understand the actual proofs and definitions. Only then does it make sense.

i don't doubt that

[Blackbeard]

Er, no.

The Axiom of Choice is much simpler --- deceptively simple, even. It just says that if you have some sequence of non-empty sets (e.g., subsets of the reals --- but really, just any sequence of sets), that there is a way of picking one element from each set.

That is: suppose you have a function S, which for each positive integer gives you some non-empty set. Then, there exists a "choice function" c, which is a function such that for all n, c(n) is an element of S(n). For each n, c(n) just "picks one element" from S(n).

The fact that such functions exist can be proven for infinite sequences of finite sets, and finite sequences of infinite sets: you don't need to assume it. But it cannot be proven for infinite sequences of infinite sets. Because it is considered important to make possible a number of theorems in real analysis, many mathematicians consider it a reasonable axiom. But some disagree. (Personally, I think it's just fine.)

The Axiom of Choice doesn't "get around" anything to do with the fact that the set of real numbers is larger than the set of the natural numbers. It does, however, imply that given any two sets, either they will be the same size, or one will be larger than the other. It may seem that this should obviously hold, but the Axiom of Choice is neccessary to guarantee the existence of the various functions one needs between two otherwise unspecified sets in order to prove this. (Indeed, this can be taken as another motivation for allowing the Axiom of Choice, given that one is interested in studying cardinality the way that Cantor introduced.)

from Wiki "Intuitively speaking, the Axiom of Choice says that given a collection of bins each containing at least one object, exactly one object from each bin can be picked and gathered in another bin—even if there are infinitely many bins and there is no "rule" for which object to pick from each. AC is not required if the number of bins is finite or if such a selection "rule" is available."

and:

"Not every situation requires the axiom of choice. For finite sets X, the axiom of choice follows from the other axioms of set theory. In that case it is equivalent to saying that if we have several (a finite number of) boxes, each containing at least one item, then we can choose exactly one item from each box. Clearly we can do this: We start at the first box, choose an item; go to the second box, choose an item; and so on. There are only finitely many boxes, so eventually our choice procedure comes to an end. The result is an explicit choice function: a function that takes the first box to the first element we chose, the second box to the second element we chose, and so on. (A formal proof for all finite sets would use the principle of mathematical induction.)

For certain infinite sets X, it is also possible to avoid the axiom of choice. For example, suppose that the elements of X are sets of natural numbers. Every nonempty set of natural numbers has a smallest element, so to specify our choice function we can simply say that it takes each set to the least element of that set. This gives us a definite choice of an element from each set and we can write down an explicit expression that tells us what value our choice function takes. Any time it is possible to specify such an explicit choice, the axiom of choice is unnecessary.

The difficulty appears when there is no natural choice of elements from each set. If we cannot make explicit choices, how do we know that our set exists? For example, suppose that X is the set of all non-empty subsets of the real numbers. First we might try to proceed as if X were finite. If we try to choose an element from each set, then, because X is infinite, our choice procedure will never come to an end, and consequently, we will never be able to produce a choice function for all of X. So that won't work. Next we might try the trick of specifying the least element from each set. But some subsets of the real numbers don't have least elements. For example, the open interval (0,1) does not have a least element: If x is in (0,1), then so is x/2, and x/2 is always strictly smaller than x. So taking least elements doesn't work, either.

The reason that we are able to choose least elements from subsets of the natural numbers is the fact that the natural numbers come pre-equipped with a well-ordering: Every subset of the natural numbers has a unique least element under the natural ordering. Perhaps if we were clever we might say, "Even though the usual ordering of the real numbers does not work, it may be possible to find a different ordering of the real numbers which is a well-ordering. Then our choice function can choose the least element of every set under our unusual ordering." The problem then becomes that of constructing a well-ordering, which turns out to require the axiom of choice for its existence; every set can be well-ordered if and only if the axiom of choice is true."

If you could, please explain how I misinterpreted the above because I get from that that with a finite set of infiinte sets or an infinite set of finite sets, the Axiom of Choice is not necessary. It only becomes necessary for an infinite set of infinite sets, and/or where a function can not be made to map two sets, as in the set of naturals to the set of reals.

[ex-xian]

If I may, I'm going to talk about something a few pages up. The reason we can define a single point at infinity and turn the complex plane into the sphere but can't do this with the real line and turn it into the circle is that there is no notion of order on the complex numbers.

As people have pointed out above, doing it with the real line ends up "proving" that -x = x. But with the complex numbers, there is no concept of order. That is, there is no concept of saying one complex number is greater or lesser that any other complex number.

That's why makes the real numbers so special. They are the unique ordered field with the least upper bound property. Any attempt to put an order on the complex field results in an absurdity.

[ex-xian]

I think the Riemann sphere concept is present in the area of projective geometry, which is commonly used in computer graphics actually. The projective geometry concept for the one-dimensional Riemann Sphere is the Projective line.

You have to be careful whether you're talking about real or complex spaces. You're right about if you mean the complex projective line, but wrong if you mean the real projective line. And I'm not sure what you mean by the "one-dimensional" Riemann Sphere. The Riemann sphere has Euler characteristic of 2 so it can't be one-dimensional.

[untermensche]

Number of elements over a range. In the integers, there are 10 numbers in [1,10]. In the reals, there are infinite numbers in [1,10].
Then you are not discussing two infinities. You are discussing a range of one thing, a finite set, and an infinity of another.

Yes, within a range density can differ, but within two infinities there can be no differences in density.

[ex-xian]

What do you mean when you say "dense"? The rational numbers and the irrational numbers are dense in the reals, but the rationals are uncountable. The integers form an infinite set but they are not dense in the reals.

[perfessor]

Then you are not discussing two infinities. You are discussing a range of one thing, a finite set, and an infinity of another.

Yes, within a range density can differ, but within two infinities there can be no differences in density.
Jesus dog untermensche, get yourself a math book and admit that you need to learn something new out of it.

Take the interval [0,1] The rationals in that interval can be put in a one-to-one correspondence with the natural numbers. The reals cannot - this has been pointed out to you numerous times.

The term "dense" has a mathematical meaning in this context. Again, check a dictionary for the jargon. Skip past definitions that refer to a person's ability to understand new ideas.

What do you mean when you say "dense"? The rational numbers and the irrational numbers are dense in the reals, but the rationals are uncountable. The integers form an infinite set but they are not dense in the reals.
Bold added.

You are wrong about this I'm afraid. Rationals are countable.

[ex-xian]

Jesus dog untermensche, get yourself a math book and admit that you need to learn something new out of it.

Take the interval [0,1] The rationals in that interval can be put in a one-to-one correspondence with the natural numbers. The reals cannot - this has been pointed out to you numerous times.

The term "dense" has a mathematical meaning in this context. Again, check a dictionary for the jargon. Skip past definitions that refer to a person's ability to understand new ideas.


Bold added.

You are wrong about this I'm afraid. Rationals are countable.
Whoops. typo.

[untermensche]

Jesus dog untermensche, get yourself a math book and admit that you need to learn something new out of it.
Why can't you explain it?

You seem hung up on this meaningless one to one correspondence crap which means nothing when you are talking about infinite elements.

You can not have infinite elements and then more than infinite elements.

[JohannGoodflag]

Why can't you explain it?

You seem hung up on this meaningless one to one correspondence crap which means nothing when you are talking about infinite elements.

You can not have infinite elements and then more than infinite elements.

Well, you haven't commented on my explanation in post #77.

[untermensche]

They have a relationship; but it's not just an arbitrary relationship.

We basically just use three definitions, which do seem quite reasonable.

There are at least as many elements in a set B as in a set A if we can match up the elements of A with some subset of B.
There are exactly as many elements in a set B as in a set A if we can match up the elements of A with the elements of B itself. (Note that B is "a subset" of itself, in the usual parlance, and so if B has exactly as many elements as A, we can weaken this to say that B also has at least as many elements as A.)
There are more elements in a set B than in a set A if B has at least as many elements as A (in the above sense), but not exactly as many (also in the above sense).


These are the relationships we care about. Is there any clear reason why this is a ridiculous way of trying to guage the relative sizes of two sets?
Those are reasonable if we talk about finite sets.

Infinite sets cannot have different amounts of elements because it is impossible to exceeed the number of elements in an infinite set.

[ex-xian]

Why can't you explain it?

You seem hung up on this meaningless one to one correspondence crap which means nothing when you are talking about infinite elements.

You can not have infinite elements and then more than infinite elements.
What you're calling crap is the best way we have of talking about sets, especially infinite sets. Your problem is that you seem to insist that infinity should make common sense. Us primates just didn't evolve to think that way. That's why we need to be rigorous about the way we use language.

One of the defining properties of infinite sets is that they can be put into a 1-1 correspondence with a proper subset. Far from being meaningless, the concept is essential to speaking coherently about them.

[untermensche]

What you're calling crap is the best way we have of talking about sets, especially infinite sets. Your problem is that you seem to insist that infinity should make common sense. Us primates just didn't evolve to think that way. That's why we need to be rigorous about the way we use language.

One of the defining properties of infinite sets is that they can be put into a 1-1 correspondence with a proper subset. Far from being meaningless, the concept is essential to speaking coherently about them.
If we are looking at number of elements in any two sets, all we do is take any element from one set and then take one element from another set, and then put these aside. If one set runs out of elements before the other set, then one set is larger than the other set.

If we take any two infinite sets and begin matching up elements, neither set will ever run out of elements, so one set is not larger than the other set.

[ex-xian]

If we are looking at number of elements in any two sets, all we do is take any element from one set and then take one element from another set, and then put these aside. If one set runs out of elements before the other set, then one set is larger than the other set.

If we take any two infinite sets and begin matching up elements, neither set will ever run out of elements, so one set is not larger than the other set.

Have you seen Cantor's diagonal argument that shows that it is impossible to match the reals with the natural numbers in the way you describe?

http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument

[untermensche]

Have you seen Cantor's diagonal argument that shows that it is impossible to match the reals with the natural numbers in the way you describe?

http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
What do you mean?

Take any real, and I will give you a natural number.

Go ahead. This will never end despite what Cantor says.

[ex-xian]

Okay, let me rephrase. Have you seen and understood Cantor's diagonal argument. Have about his first proof that the reals are uncountable?

http://en.wikipedia.org/wiki/Cantor%27s_first_uncountability_proof

If you disagree you have to find a flaw in the proofs. It's not enough to make empty assertions.

[perfessor]

What do you mean?

Take any real, and I will give you a natural number.

Go ahead. This will never end despite what Cantor says.
Well, I guess that settles it then. You are right, and Cantor and every other mathematician is wrong.

Let me know when you've got your book on set theory written, since I have to throw mine out.:banghead: :banghead:

[untermensche]

Okay, let me rephrase. Have you seen and understood Cantor's diagonal argument. Have about his first proof that the reals are uncountable?

http://en.wikipedia.org/wiki/Cantor%27s_first_uncountability_proof

If you disagree you have to find a flaw in the proofs. It's not enough to make empty assertions.
Don't dodge. Take my challege.

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