What is the Infinite?

[Huzington]

What is the Infinite?

Numbers and space are not infinite. Mere endlessness is not the True Infinite. It is what Hegel called the Spurious Infinite. For each number in a series is limited and finite. No matter which term (or number) we are at, we never go beyond the finite.

Mr. Stace put it succinctly:

"[We have] an endless repitition of terms each of which is finite, so that the finite is never got rid of. This is an everlasting attempt to grasp the infinite which however completely evades us. It ought to be infinite, but it never is."

Mr. Hegel also put it well:

"The result seems to superficial reflection something very grand. . . . [B]ut it is not the real infinite. . . . In the attempt to contemplate such an infinite our thought . . . must sink exhausted. It is true . . . that we must abandon the unending contemplation, not however because the task is so subline, but because it is too tedious. The same thing is constantly recurring. We lay down a limit: then we pass it: next we have a limit once more, and so forever."

Thus we never get beyond the finite. This is mere endlessness, mere limitlessness; this is not Infinity.

[DarkWraith]

In mathematics, my professors have always been careful to point out that you never -reach- infinity, you merely approach it. Infinity isn't real; it's just a mental construct useful in describing the universe.

[Rooster_Ties]

An incredible album by jazz trumpeter Dave Douglas (http://www.allmusic.com/cg/amg.dll?p=amg&uid=MISS70309032134&sql=B2kx7gjerj6ix)???


http://image.allmusic.com/00/amg/cov200/drf100/f160/f16018xby0s.jpg
"The Infinite" (http://www.allmusic.com/cg/amg.dll?p=amg&uid=MISS70309032134&sql=A9vev97bskr5t) (2002, RCA 63918)


:D

[Keith Russell]

Greetings:

I accept it as axiomatic that all 'things' are (by definition) finite...

K

[demoninho]

Read this (http://www.amazon.com/exec/obidos/tg/detail/-/156858105X/102-9992524-9881707?v=glance) book on infinity and Georg Cantor the person who first discovered the mathematics of infinity. It'll blow your mind out as it did with Cantor :confused: :banghead: :confused:

[SlateGreySky]

Thus we never get beyond the finite. This is mere endlessness, mere limitlessness; this is not Infinity.

I find Hegel's comments in Science of Logic, paragraphs 301-302, to be particularly illuminating in reference to your point.

Hegel seems ultimately to present a picture of infinity that exists not as mere negation or abstract being, but of an infinite which is essentially dialectical: the "true infinite" is a perpetual becoming, manifested at each of its moments in determinacy.

For Hegel, the true infinite is not the prize at the end of the non-terminating line; rather, the true infinite is the circle which neither begins nor ends (cf. 302). Far from being an abstraction in itself, the true infinite is totally determinate in each of its moments.

[sodium]

Originally posted by Huzington
Numbers and space are not infinite. Mere endlessness is not the True Infinite. It is what Hegel called the Spurious Infinite. For each number in a series is limited and finite. No matter which term (or number) we are at, we never go beyond the finite.


I think this is clearly the fallacy of composition.

[Bob K]

Infinity is not a number; it is a condition of having no mathematical or physical limits.

[Haleysred]

If you need to get from point A to point B, it is relatively simple, just walk, run, skip or drive there. However, if you put an infinite number of miles between point A and Point B, you can never get to your destination. Can you say that you might not even be able to get to the starting point?

[Keith Russell]

Haley, but we do get to the starting point, and we also quite often get from point A to point B.

All things (that exist) are finite.

K

[Haleysred]

Originally posted by Keith Russell

All things (that exist) are finite.

K [/B]

So is there no "actual infinity"? Is there only a potential for infinity? Meaning that if there is an "actual infinity", then all things have the ability to be infinite. Potential infinty, meaning, that infinity is just a whole bunh of fintie sequences being combined to form infinity.

[Bob K]

Infinity is not a number; it is a condition of having no mathematical or physical limits.

Within the universe are numerous infinities.

Space, having the condition of no mathematical or physical limitations, is infinite in having no limits to its dimensions; attempts to measure it will never end.

Space can be conceptualized either (A) by the expression X + i, which means for any spherical volume of X radius or diameter there is a surrounding volume described by i, which stands for infinity, or infinite dimensions, having no mathematical or physical limitations of dimensions, or (B) by the concept of the spatial matrix created by rigid rods, defined as rigid for length, width and straightness, and absolutely unaffected by changes of velocity or/and gravity, set parallel and perpendicular in three axes such that travel along one axis will never end (there will always be one more rigid rod to traverse).

Time, as measured using an invariable time-interval (a time-interval whose duration is not affected by changes of velocity/gravity, as found in (A) motion-sensing and self-adjusting clocks, or in (B) clocks which are synchronized by radio signals from a master clock, and by either A or B time-dilation is eliminated) in the following continuum, is without mathematical or physical limitations and therefore infinite into the past as well as the future:

Past Infinity <- ... <- (T-2) <- (T-1) <- (T0) -> (T+1) -> (T+2) -> ... -> Infinity Future

NOTE: T0 = Timepoint of origin.

Thus, there never was a beginning of time, nor will there ever be an ending of time.

Matter/energy, m/e, cannot be destroyed but only changed in form, therefore the m/e of the universe is infinite in duration.

The law of the conservation of matter/energy describes therefore an infinity.

In an isolated m/e system, defined as an m/e system from which m/e cannot be taken (where would it go?) and to which m/e cannot be added (where would it come from?), the m/e is a finite number, thus, the sum total of m/e in an isolated system is a constant.

The universe is an isolated m/e system, because the m/e within it cannot be taken away from it (where would it go?) and m/e cannot be added to the m/e already within it (where would it come from?).

Thus, the m/e of the universe is infinite in duration/an infinity of duration but finite in quantity/a finity for quantity.

The conservation of charge and momentum are also infinities.

[Peter Kirby]

I believe that the volume of space in our universe is finite for the simple reason that I can see the black sky at night. But I also believe that our universe has no boundaries in space.

"Infinite" is a property of uncountable sets (with a cardinality greater than or equal to that of natural numbers).

best,
Peter Kirby

[Bob K]

Peter Kirby:I believe that the volume of space in our universe is finite for the simple reason that I can see the black sky at night. But I also believe that our universe has no boundaries in space.

The volume of space in our universe is finite n= Our universe has no boundaries in space.

NOTE: n= means does not equal/is not the same as and is used to ensure that the 'does not equal'/'is not the same as' message gets through when there is a possibility that not all fonts will honor the Option = keystrokes for the traditional does not equal/is not the same symbol, != [Did your browser serve up the correct Option = symbol which should mean n=?; my browser, Internet Explorer, served up ! (exclamation mark) and = (equal sign) instead of the correct Option = symbol.]

How is it that a finite volume can have no boundaries?

If you choose to play the infinite number of points in a finite volume card, then be advised that you are not talking about physical points which have physical dimensions--material points, as A. Einstein labeled them in his book, Relativity, and if you play the geometrical points have no physical dimensions card then be advised that without physical dimensions the geometrical points have no physical relationship to reality because they cannot be used for measurements.

The fact is that for any physical dimensions assigned to physical/material points there is a finite number of physical/material points in a finite volume, and an infinite quantity of points can only exist in an infinite volume.

[Peter Kirby]

Originally posted by Bob K
Peter Kirby: "I believe that the volume of space in our universe is finite for the simple reason that I can see the black sky at night. But I also believe that our universe has no boundaries in space."

The volume of space in our universe is finite n= Our universe has no boundaries in space. Naturally. I indicated that these are separate beliefs. Finitude and boundedness are two different properties. I say that our universe is finite yet not bounded. Of course I am not the first to say it; I picked it up from a few science books I read at the public library while in high school.

How is it that a finite volume can have no boundaries? The shortest answer is: non-Euclidean geometry. A short answer, working by analogy with things known from Euclidean geometry, is this: imagine a sphere. Now imagine the surface of a sphere. Now imagine that the surface of the sphere is all that exists, or at least the known universe. The universe under consideration consists of the set of points that exist along the latitude and longitude of this two dimensional sphere-surface-world. Clearly, this universe has finite space, but one can never run up against a boundary. That's the best way that a layman such as myself knows how to explain a non-Euclidean universe that is finite but not bounded. If you need more information or a better explanation, it's probably best to seek out a scientist, such as a cosmologist or astronomer. They should be familiar with the idea, even if they don't subscribe to it (though my impression is that most do, if they buy into Big Bang cosmology or the latest inflationary model).

best,
Peter Kirby

[Bob K]

Peter Kirby:I believe that the volume of space in our universe is finite for the simple reason that I can see the black sky at night. But I also believe that our universe has no boundaries in space.

Bob K:The volume of space in our universe is finite n= Our universe has no boundaries in space.

Peter Kirby:Naturally. I indicated that these are separate beliefs. Finitude and boundedness are two different properties. I say that our universe is finite yet not bounded. Of course I am not the first to say it; I picked it up from a few science books I read at the public library while in high school.

Okay.

The universe is infinite because it is unbounded.

Bob K:How is it that a finite volume can have no boundaries?

Peter Kirby:The shortest answer is: non-Euclidean geometry. A short answer, working by analogy with things known from Euclidean geometry, is this: imagine a sphere. Now imagine the surface of a sphere. Now imagine that the surface of the sphere is all that exists, or at least the known universe. The universe under consideration consists of the set of points that exist along the latitude and longitude of this two dimensional sphere-surface-world. Clearly, this universe has finite space, but one can never run up against a boundary. That's the best way that a layman such as myself knows how to explain a non-Euclidean universe that is finite but not bounded. If you need more information or a better explanation, it's probably best to seek out a scientist, such as a cosmologist or astronomer. They should be familiar with the idea, even if they don't subscribe to it (though my impression is that most do, if they buy into Big Bang cosmology or the latest inflationary model).

'How is it that a finite volume can have no boundaries?' was a rhetorical question.

Another Rhetorical Question: What would be a boundary on or for a surface of a sphere?

In Operational Physics, to eliminate geometrical confusions of dimensionless geometrical points which cannot be used for measurement with the reality of dimensioned physical points which can be used for measurement, there is a requirement for a set of physical/material points, points which have dimensions, to be used for the measurement of a sphere's surface, and once this requirement is established, then the number of physical/material points on any surface becomes finite, therefore, for your surface of a sphere, there would be a finite quantity--not an infinite quantity--of physical/material points.

Next, in Operational Physics, there is a requirement that once a physical/material point is traversed it cannot be traversed again, and when this requirement becomes a standard, then there cannot be, as some claim, an infinite number of pathways on the surface of a sphere, therefore the surface of the sphere cannot possibly be infinite/unbounded/boundless, for the physical/material points themselves would set the boundaries for the pathways and therefore the number of pathways over which a traveler could travel could not possibly be infinite/unbounded/boundless and, instead, must be finite/bounded/unboundless. Once one great circle of physical points is traveled, then, by the rule that once traversed a physical point cannot be traversed again, the great circle would create a boundary much like a fence upon the surface of the sphere and reduce the remaining possible pathways, therefore the number of pathways a traveler could travel without traversing a previously traversed physical point becomes finite.

Again, when nondimensioned geometrical points are used, which are nonmeasurable, then chaos and confusions result; but when dimensioned physical points are used, which are measurable, then chaos and confusions are eliminated and order and clarities result.

BTW, you might enjoy checking out my website--

www.bobkwebsite.com

--for further information on Operational Physics and a theory of the Universe comprising of three realities--the spatial reality, the temporal reality, and the physical reality.

[Peter Kirby]

Originally posted by Bob K
Okay.

The universe is infinite because it is unbounded. That's a logical possibility (though the "because" is not a logically necessary implication). All four are possible for a hypothetical physical universe:

(1) Finite volume and with boundaries
(2) Infinite volume and with boundaries
(3) Finite volume and no boundaries
(4) Infinite volume and no boundaries

You are saying (4), and I am saying (3). But you do not seem to recognize the existence of (2) and (3) as geometrical possibilities. I suggest that this is due to an assumption of Euclidean geometry (or perhaps the Operational Physics you discuss further on). I don't share such an assumption.

'How is it that a finite volume can have no boundaries?' was a rhetorical question. One which actually can be answered in an intelligent way. (You even tried to cut off [misguided] possible answers with subsequent paragraphs, so I do not believe that you expected me to be silent in response to the question, which is a mirror of what I already said I believed. You asked how what I believe could be true, a finite and boundary-less universe, so naturally I answered.)

Another Rhetorical Question: What would be a boundary on or for a surface of a sphere? I can do that. Imagine a sphere. Now imagine the surface of a sphere. Now imagine a great circle from pole to pole, which would divide the surface into Western and Eastern portions. Now imagine that the known universe is this: the set of points in the "Western" (only) hemisphere. When one comes up against the Prime Meridian, there is no space beyond it; it is literally the edge of the universe. This universe would be finite in space, with a boundary, and non-Euclidean.

Personally, I have a hard time understanding how our own universe could have a boundary, and that is why I believe it is without boundary. Which is not to say that it has non-finite volume, as I've already indicated.

In Operational Physics, to eliminate geometrical confusions of dimensionless geometrical points which cannot be used for measurement with the reality of dimensioned physical points which can be used for measurement, there is a requirement for a set of physical/material points, points which have dimensions, to be used for the measurement of a sphere's surface, and once this requirement is established, then the number of physical/material points on any surface becomes finite, therefore, for your surface of a sphere, there would be a finite quantity--not an infinite quantity--of physical/material points. Perhaps you are confused about what I have said (which is not an original idea, just one with which I agree), as you are (in the above paragraph) arguing against someone who thinks that there is an infinite quantity of space, while I have said clearly from the beginning that I believe that the volume of our universe is finite. I said that the surface of a sphere has a finite quantity of space, just as I believe that this universe has a finite volume. If one defines a "material point" to be one which has some fixed volume within a three-dimensional space, then I would agree that the number of material points in our universe is finite.

Next, in Operational Physics, there is a requirement that once a physical/material point is traversed it cannot be traversed again, ... Is there any empirical data that could persuade someone that this model is accurate for our universe? What particular predictions are made by the hypothesis?

best,
Peter Kirby

[Bob K]

NOTE: When I Reply to Topic Starters and/or Repliers, out of respect for the nature/essence of reasoned discussion I prefer to be thorough and thereby create the possibility of presenting a complete and understandable explanation of the concepts/principles used for the Replies.

Thoroughness, to me, requires sufficient explanations to avoid misunderstandings.

If sufficient explanations require lengthy explanations, then, for the sake of clarity and respect for the nature/essence of reasoned discussions, I choose to present lengthy explanations.

I do not offer lengthy explanations for the purpose of annoying Readers and Potential Repliers, or to win discussions/arguments by wearing out/fatiguing the opposition.

To reduce the fatigue of reading lengthy explanations I can chop them into smaller Parts.

If Readers/Repliers are nevertheless annoyed/fatigued, please keep in mind that I could easily shorten these explanations by resorting to Xn fundie tactics of denial/evasion/obfuscation/attack.

Thus, in extremes, following the attack tactics of fundies, I could simply say "You're a @#$%^&*!!!!" or "You're wrong, therefore you're a @#$%^&*!!!"

But, then, you would not know why you're a @#$%^&*!!! or that you're wrong, would you?

Reply to Peter Kirby's Reply of 9/14/03

Part One

Bob K:The universe is infinite because it is unbounded.

Peter Kirby:That's a logical possibility (though the "because" is not a logically necessary implication). All four are possible for a hypothetical physical universe:

(1) Finite volume and with boundaries
(2) Infinite volume and with boundaries
(3) Finite volume and no boundaries
(4) Infinite volume and no boundaries

You are saying (4), and I am saying (3). But you do not seem to recognize the existence of (2) and (3) as geometrical possibilities. I suggest that this is due to an assumption of Euclidean geometry (or perhaps the Operational Physics you discuss further on). I don't share such an assumption.

When OpPhys (Operational Physics) requires all points in physics to be physical/material/dimensioned points, any kind of geometry in which dimensionless points are used for measurement is eliminated or otherwise is modified if theoreticians agree that physical points are to substituted for dimensionless points.

Space, the spatial reality of the universe, which is imaginable by X + i, wherein X = the radius or diameter of a sphere and therefore a finite volume, and i = an infinite volume (infinite = a condition of having no mathematical or physical limits) which surrounds X, we thus, at small scales, observe that for every X there is nevertheless a surrounding i, and, because extrapolation from small scales to larger scales is allowed in physics, particularly in inductive reasoning, as used in the Scientific Method, then, when we extrapolate X + i for each and every case of X at small scales we observe no exceptions to the concept/principle of X + i, and, finding no exceptions in observable scales, we are justified in concluding that for every X we would find + i surrounding it (if we were capable at a later date of observing more of the larger scales, but we are presently capable of observing Xs of X dimensions measured in light-years, so the question becomes when have we observed enough cases of X + i to conclude that until someone observes n(X + i) that X + i is indeed the description of the spatial reality of the universe?

As per the inductive reasoning method, not all cases have to be observed to verify that the data observed support the hypothesis and that, therefore, until a contravening case is found, the hypothesis is accepted as a fact.

Premise #1: For every observed volume of X radius/diameter at observable scales there is a volume of + i which surrounds it, thus every volume of X is describable by the expression X + i. [Verifiable experimentally]

Premise #2: Extrapolation from small scales to larger scales is acceptable in the inductive reasoning process of the Scientific Method, and X + i is extrapolable from observed small scale cases to larger scale cases.

Conclusion #1: X + i describes the spatial reality of the universe--space, and the spatial reality/space is without mathematical or physical limitations and is therefore infinite in volume.

Conclusion#2: Of the four possible descriptions of the spatial reality of the universe--space,

(1) Finite volume and with boundaries,
(2) Infinite volume and with boundaries,
(3) Finite volume and no boundaries,
(4) Infinite volume and no boundaries,

the only logical and physical possibility is #4: (4) Infinite volume and no boundaries.

To challenge the conclusion(s), challenge the premises, and one excellent challenge is to find an exception to the observable/observed data, therefore, find us a case of X in which X + i does not describe the volume of the universe--the spatial reality/space.

[Bob K]

Reply to Peter Kirby's Reply of 9/14/03

Part Two

Bob K:'How is it that a finite volume can have no boundaries?' was a rhetorical question.

Peter Kirby:One which actually can be answered in an intelligent way. (You even tried to cut off [misguided] possible answers with subsequent paragraphs, so I do not believe that you expected me to be silent in response to the question, which is a mirror of what I already said I believed. You asked how what I believe could be true, a finite and boundary-less universe, so naturally I answered.)

Okay.

But the point of the rhetorical question and its OpPhys answer is that the expression X + i accurately describes the spatial reality of the universe--space--and, therefore, space is infinite--without mathematical or physical limitations--and therefore unbounded.

Prediction: For all cases of X, there will be observed the condition of X + i, wherein an infinite volume, i, will be found surrounding every volume of X finite dimensions, hence the universe will be confirmed to infinite in volume, unbounded, without boundaries.

Bob K:Another Rhetorical Question: What would be a boundary on or for a surface of a sphere?

Peter Kirby:I can do that. Imagine a sphere. Now imagine the surface of a sphere. Now imagine a great circle from pole to pole, which would divide the surface into Western and Eastern portions. Now imagine that the known universe is this: the set of points in the "Western" (only) hemisphere. When one comes up against the Prime Meridian, there is no space beyond it; it is literally the edge of the universe. This universe would be finite in space, with a boundary, and non-Euclidean.

Personally, I have a hard time understanding how our own universe could have a boundary, and that is why I believe it is without boundary. Which is not to say that it has non-finite volume, as I've already indicated.

When you think/say that you have "a hard time understanding how our own universe could have a boundary" then you are on track with developing an accurate concept of the spatial reality of the universe--space, and its relevant/related principle of being describable by the expression X + i and therefore its infinite/unlimited/unbounded/boundless volume, which is its nature/essence, and your conclusion, "that is why I believe it is without boundary," is therefore accurate and in agreement with (4) Infinite volume and no boundaries.

But when you start thinking/saying "[which] is not to say that it has non-finite volume" you lapse into definitional confusions because you are trying to describe the spatial reality of the universe--space--by both (3) Finite volume and no boundaries AND (4) Infinite volume and no boundaries, and, in concord with the axiom of logic that says a thing can only itself be, A = A, the spatial reality of the universe--space--logically can only be either (3) or (4) but not both, and from the expression X + i, the description of the spatial reality at all observable 'small' scales (including scales of X dimensions of diameters/radii measurable in light-years) and extrapolable to all larger scales, the only logical possibility, regardless of the claims of geometricians using dimensionless/non-physical/non-material points, is (4).

[Bob K]

Reply to Peter Kirby's Reply of 9/14/03

Part Three

Bob K:In Operational Physics, to eliminate geometrical confusions of dimensionless geometrical points which cannot be used for measurement with the reality of dimensioned physical points which can be used for measurement, there is a requirement for a set of physical/material points, points which have dimensions, to be used for the measurement of a sphere's surface, and once this requirement is established, then the number of physical/material points on any surface becomes finite, therefore, for your surface of a sphere, there would be a finite quantity--not an infinite quantity--of physical/material points.

Peter Kirby:Perhaps you are confused about what I have said (which is not an original idea, just one with which I agree), as you are (in the above paragraph) arguing against someone who thinks that there is an infinite quantity of space, while I have said clearly from the beginning that I believe that the volume of our universe is finite. I said that the surface of a sphere has a finite quantity of space, just as I believe that this universe has a finite volume. If one defines a "material point" to be one which has some fixed volume within a three-dimensional space, then I would agree that the number of material points in our universe is finite.

OpPhys argues that the volume of space is infinite--without mathematical or physical limits, and is therefore describable by the expression X + i. [It can also be described by a three-dimensional matrix/grid of nondeformable rigid rods (which cannot be deformed by spacetime/gravity/etc. regardless of the claims of SR/GR/QM fans) set in three axes parallel and perpendicular to each other whereupon travel in one direction along one axis would produce travel over an infinite number of rigid rods and result in the continuous observation of no dimensional boundaries to the volume of space/the spatial reality.]

You have said thus: "[You] are ... arguing against someone who thinks that there is an infinite quantity of space."

If we agree that quantity inre: space/the spatial reality = the volume of space/the spatial reality, then by X + i the fact resulting from the conclusion of the inductive reasoning from the Scientific Method using the premises described above is that the volume/quantity of space/the spatial reality is infinite--without mathematical/physical limits/limitations/boundaries.

The opening question of this Topic is What is the infinite?

OpPhys gives the specification of the infinite to be not a number but instead a condition of having no mathematical or physical limits/limitations/boundaries/etc.

When this definition/specification of infinite/infinity is accepted and used, then the volume of the universe can easily be shown by X + i to be infinite--having the condition of having no mathematical or physical limitations.

You have said thus: "If one defines a 'material point' to be one which has some fixed volume within a three-dimensional space, then I would agree that the number of material points in our universe is finite."

The 'point' of--the reason for--requiring all points in discussions of space/the spatial reality to be physical/material/dimensioned/bounded/limited/etc. points instead of non-physical/non-material/non-dimensioned/etc. points is precisely that only by physical points can we measure reality.

How can anyone measure reality--real volumes, real quantities--by non/physical points?

OpPhys says--and proves--that it is impossible.

When physical points are used for the measurement of volumes, then, by X + i, the volume of space/the spatial reality, is without mathematical or physical limits and is therefore infinite.

Thus, 'the number of material points in our universe' is infinite--having no mathematical or physical limitations, no boundaries.

Infinity is not a number; it is the condition of having no mathematical or physical limitations.

Therefore, 'the number of material points in our universe' is not a finite number.

[Bob K]

Reply to Peter Kirby's Reply of 9/14/03

Part Four

Bob K:Next, in Operational Physics, there is a requirement that once a physical/material point is traversed it cannot be traversed again, ...

Peter Kirby:Is there any empirical data that could persuade someone that this model is accurate for our universe? What particular predictions are made by the hypothesis?

Yes.

By actual observations of small scale X-dimensioned finite volumes we will find that X + i describes the condition wherein for every X there is a surrounding volume, i, which has no mathematical or physical limits/limitations/etc. Some of the Xs in astronomical observations will be diameters/radii using measurements by light-years, including millions of light-years, rather large scales for most of us, but not necessarily for those of us who are known as critics, so the usage of the principle of extrapolation from observable/small scales produces no exceptions to X + i.

By gedankenexperiments (German: thought experiments--Einstein used them for his insights and his explanations, and he was not alone is his usage of them), you can, I can, everyone else can, we all can together imagine large scales beyond millions/billions of light-years for Xs in which X + i still holds without exception, thus the real as well as the imagined data will confirm that X + i is an accurate description of reality.

Prediction: For any and all scales of X the expression X + i will describe without exceptions the spatial reality of the universe--space--to be having no mathematical or physical volumetric/spatial limits and therefore to be infinite in volume.

You challenge a theory's conclusion(s) by challenging its premises.

Premise #1: For every observed volume of X radius/diameter at observable scales there is a volume of + i which surrounds it, thus every volume of X is describable by the expression X + i. [Verifiable experimentally]

Premise #2: Extrapolation from small scales to larger scales is acceptable in the inductive reasoning process of the Scientific Method, and X + i is extrapolable from observed small scale cases to larger scale cases.

Conclusion: X + i describes the spatial reality of the universe--space, and the spatial reality/space is without mathematical or physical limitations and is therefore infinite in volume.

If you challenge Premise #2, then be reminded that the observable scales of X include millions/billions/etc. of light-years, certainly large enough scales for most normal people to accept the extrapolation of X+ i as being valid for the entire universe/the spatial reality/space.

[Peter Kirby]

I have no interest in making four posts in reply to your four, especially when you haven't acknowledged that you lied about posing a "rhetorical question" (which was actually a question of how what I think is true could be true) and when you've given little indication that you understand what I am saying (continually attacking strawmen like the idea that I am proposing an infinite number of "material points" within a finite space). So I am going to address the one new point you've brought out in your last salvo, and I reiterate the advice to seek out a practicing scientist for more information.

Your prediction is that we will find that space increases without running up to any boundaries; when we have a volume X within the universe, we can always add some volume i (up to infinity) and that volume will seem to exist. This is actually an argument against the idea of boundedness, not the idea of finitude as I've explained the model with which I agree. A similar experiment could be conducted by the inhabitants of the sphere-surface-world, and they would find that they never run up to a boundary when discovering wider areas of space--as there are no boundaries to this world--and draw the incorrect conclusion that the space of their sphere-surface-world is not finite. And if their surface-world is large enough, they wouldn't be able to do the experiment of going far enough in one direction to come back around to the starting point (especially if, as in our universe, it is expanding).

There is one particular prediction, however, that is made by the standard Big Bang model of a finite and unbounded universe. That prediction is that one should expect an even radiation throughout all of space as a residue of the initial Big Bang events, what is called the cosmic background radiation. Within expected statistical limits, this even amount of energy spread throughout the universe is exactly what we find. However, on the model of a universe that has infinite space (and, as your web site says, existed with an infinite, if perhaps hohum, past before the Big Bang)--there should be parts of the universe near the infinite nothingness where the heat has dissipated more than usual (and, of course, far out enough we should find almost no heat). We do not find such in reality, and this falsifies the idea that the universe has infinite volume.

best,
Peter Kirby