Clarifying Infinity

[TheVat]

Crimony, I went back to page one, and see that Someguy and Lomax* were explaining and clarifying beautifully back in April 2017. Newbies, please read the start of a thread before posting. Grrr.


*miss you, dude

[someguy1]

Seems a bit like saying there can't be any odd numbers because of the even numbers crowding them out in their infinitude. Brent, an infinite space is a continuum and has abundant room for infinite sets of anything - atoms, beach balls, short bald men named Balthazar, you name it. Have you read the thread here prior to posting? If nothing else, look up Hilbert's Hotel, and see how there's always room for more guests.

Well Hilbert's hotel refers to a countable set, in fact a countable set of the same order type as the natural numbers in their usual order. A continuum is like the real numbers. The naturals aren't a continuum, nor are the rationals. And there are no beach balls in the real numbers. Can you clarify? I'm afraid I didn't follow your examples.

[someguy1]

The known universe means the observable universe -- our Hubble volume. WMAP indicates the universe is probably flat, hence spatially infinite. If so, there are an infinite humber of atoms.

I read your WMAP link and I do not believe it supports what you say.

If the universe is expanding from the big bang, suppose we say it's flat, hence it will expand "forever." Doesn't that still mean that at any given moment, it's finite? It's just unbounded, but not actually infinite. Is that correct? Or do you see this differently?

Secondly, even if the universe is spatially infinite, why does that imply there are infinitely many atoms? How does that follow? At the moment of the big bang there is a finite amount of stuff. It may travel outward forever, but even if the space become actually infinite (dubious, but just for sake of argument), there would still only be a finite amount of stuff.

In both cases you seem to be equating an unbounded but finite space with an actually infinite one. This distinction is critical since we're discussing the possibility of an actual infinity in the real world.

[TheVat]

Sorry, SG, I used continuum in the casual physics sense of having spatio-temporally separable events and elements. If unbounded, then any sort of discrete object can exist in infinite quantity, potentially. I shouldn't have strayed from the math definition of continuum without defining it. To get into the sort of space we exist in, would I need to get into topology quite a bit? Pseudo Riemannian manifolds? Hausdorff spaces? I'd get lost. :-0

[davidm]

The known universe means the observable universe -- our Hubble volume. WMAP indicates the universe is probably flat, hence spatially infinite. If so, there are an infinite humber of atoms.

I read your WMAP link and I do not believe it supports what you say.

If the universe is expanding from the big bang, suppose we say it's flat, hence it will expand "forever." Doesn't that still mean that at any given moment, it's finite?

No, it does not mean that. The Big Bang model posits that if the universe is infinite, it was always infinite, including at the bang, which happened everywhere.

It's just unbounded, but not actually infinite. Is that correct? Or do you see this differently?

Yes, unbounded means finite but unbounded, such that if you travel far enough, you will return to where you started. An infinite universe does not have this property.

Secondly, even if the universe is spatially infinite, why does that imply there are infinitely many atoms? How does that follow? At the moment of the big bang there is a finite amount of stuff.

No, there was an infinite amount of stuff, even at the bang, if the universe is spatially infinite.

In both cases you seem to be equating an unbounded but finite space with an actually infinite one.

I am not equating the two. A finite but unbounded space has positive or negative global curvature. Flat space is infinite and always was and will always be.

[TheVat]

I was thinking in terms of a multiverse, with infinite Big Bang bubbles spreading out forever. Tegmark stuff. If there's no starting point of such a multiverse, temporally, then it is infinite at any point in time one could select.

[davidm]

Discussion

[davidm]

Another good discussion, at Physics Forum

[someguy1]

Discussion

Interesting link but inconclusive. People are raising the same objections I am, and the responses are handwavy at best. "Consider a big expanding balloon. Now imagine it's infinite." Puhleeeze.

The thread refers to the LCDM model: https://en.wikipedia.org/wiki/Lambda-CDM_model

But that Wiki page does not contain the word "infinite" nor does it say anything about the big bang creating or containing or involving (whatever verb you like) an infinite amount of stuff.

I don't find your argument compelling. And I don't take my notions of infinity from astronomers.

Your second link from Physicsforum doesn't help either. Someone says, "Imagine an infinite sheet that doubles in size." DUHHHHH then it would still be the same size because it was infinite to start with.

The links you gave have the exact same flaw as many of the posts in this thread and also in the literature. People who haven't given the subject of infinity much critical thought, wave their hands around.

I do appreciate your links but they didn't make your point or frankly even support it.

So tell me. If there are infinitely many hydrogen atoms in the world, is that a countable or uncountable infinity? Do the axioms of ZFC set theory apply? Which ones? Choice? Powerset? Replacement? Is the Continuum hypothesis true or false? Why haven't any clever physics postdocs applied for grants to study these questions? Because nobody takes these questions seriously. The claim that there are infinitely many atoms falls apart the moment you think critically about it.

Likewise, by the way, the claim that there are infinitely many universes in the multiverse. I'd ask the exact same questions. Nobody asks these questions because they immediately refute the idea of an actual infinity of anything.

[TheVat]

DM, that helps. And no multiverse is needed. Grasping this will take a while, I suspect. Though without an infinite expanse of separate Big Bang events, I'm still struggling with infinite matter as possible. If there is a singularity, just one, then the Lambda CDM model does not lead to infinite matter,, does it? Unless you somehow treat the vacuum energy in an infinite space as....no, still not there.

SG: Just to clarify again, I wasn't saying the cardinality of all beachballs was the same as the real numbers, i. e. the cardinality of the continuum. But the cardinality of beachballs and bald guys is the same in an infinite multiverse, right? Potentially at least. Well no, actually the same. We can hand bald guys beachballs forever....

[someguy1]

SG: Just to clarify again, I wasn't saying the cardinality of all beachballs was the same as the real numbers, i. e. the cardinality of the continuum.

BIV, You said there's a beachball in the real numbers. There is not. Here's the quote: "Brent, an infinite space is a continuum and has abundant room for infinite sets of anything - atoms, beach balls, short bald men named Balthazar, you name it." That's false. There are no beachballs in the real numbers. Infinite sets don't necessarily have "room" in them. I hate to be picking on this but I really don't know what you mean.

But the cardinality of beachballs and bald guys is the same in an infinite multiverse, right?

No, why should it be? Maybe there are 5 bald guys and 7 beachballs. The notion that there must be infinitely many of everything in an infinite universe is false. Also, the infinite universe idea is statistical. There could be all sorts of measure zero anomalies. It's simply not true that "everything must happen infinitely many times." In the infinite sequence 010101010101... there is never a 2. In the infinite sequence 01111111... there are indeed infinitely many 1's but 0 never recurs. In an infinite universe where matter can only take a bounded number of states in a given region of spacetime, at best you can say is that SOMETHING recurs infinitely often. But not everything.

[davidm]

Discussion

Interesting link but inconclusive. People are raising the same objections I am, and the responses are handwavy at best. "Consider a big expanding balloon. Now imagine it's infinite." Puhleeeze..

Maybe this will help

[someguy1]

Maybe this will help

Have you entirely given up conversing and regressed to Google-mining?

Jeez man that link does not explain anything. Are you serious? "Woo woo in an infinite universe a finite part of it is INFINITESIMAL wooo hoo!" Dude that is childish. It's insulting frankly. At no point did they show the universe is infinite or even make a plausibility argument. Honestly, did you read the link?

" If we go to smaller and smaller times since the Big Bang, the green circle shrinks to a point ..."

This is the exact same confusion between unboundedness and actual infinity that permeates this discussion. As the universe gets larger, the observable universe is a smaller percentage. But the universe never becomes infinite. Their own argument shows this.

[davidm]

Maybe this will help

Have you entirely given up conversing and regressed to Google-mining?

Have I what? Where have I given up conversing? Anyone who has read my posts, even if they disagree with them, must know that I write thoughtful and often quite long posts, including in this thread.

I am trying as best I can to help you overcome your total confusion about this subject. I thought this link might help you visualize where you are going wrong. Instead, you resort to trollish behaviior. Suit yourself, but don't expect me to indulge you.

[ronjanec]

“Imagine an infinite sheet that doubles in size”? An “infinite” sheet that could double in size was never infinite to begin with(!), this was always finite. After it doubles in size, it will still always and forever be be finite and never ever infinite(An object or thing cannot (ever) grow to an infinite state...it will just get larger and larger and larger finite). Double DUHHHHH.

(It’s not my fault someguy: these other guys forced me to comment about this in a philosophical way) :)

[someguy1]

I am trying as best I can to help you overcome your total confusion about this subject. I thought this link might help you visualize where you are going wrong. Instead, you resort to trollish behaviior. Suit yourself, but don't expect me to indulge you.

The argument in that link supports my point, not yours. They have two finite boxes. You're seeing what's not there. I am not confused in the least about this subject.

[davidm]

Maybe this will help

Have you entirely given up conversing and regressed to Google-mining?

Jeez man that link does not explain anything. Are you serious? "Woo woo in an infinite universe a finite part of it is INFINITESIMAL wooo hoo!" Dude that is childish. It's insulting frankly. At no point did they show the universe is infinite or even make a plausibility argument. Honestly, did you read the link?

" If we go to smaller and smaller times since the Big Bang, the green circle shrinks to a point ..."

This is the exact same confusion between unboundedness and actual infinity that permeates this discussion. As the universe gets larger, the observable universe is a smaller percentage. But the universe never becomes infinite. Their own argument shows this.

You have no clue what you are talking about. You don't even know what "unboundedness" means.

Good day.

[davidm]

“Imagine an infinite sheet that doubles in size”? An “infinite” sheet that could double in size was never infinite to begin with(!), this was always finite.)

Wrong.

[ronjanec]

“Imagine an infinite sheet that doubles in size”? An “infinite” sheet that could double in size was never infinite to begin with(!), this was always finite.)

Wrong.

Explain. :)

[davidm]

Imagine an infinite sheet of black paper. On it are painted white dots. Since the paper is infinite in length and breadth, then there are an infinite number of white dots on it. Let those white dots stand for galaxies, and the black paper stand for the universe.

At time 1 (t1) each dot is separated by one inch from all its neighboring dots. At t2 each dot is separated by two inches from all its neighboring dots. An infinite universe has just doubled in size, which means: The distance between each dot and its neighbors is, at t2, twice as great as it was at t1.

[someguy1]

Imagine an infinite sheet of black paper. On it are painted white dots. Since the paper is infinite in length and breadth, then there are an infinite number of white dots on it. Let those white dots stand for galaxies, and the black paper stand for the universe.

At time 1 (t1) each dot is separated by one inch from all its neighboring dots. At t2 each dot is separated by two inches from all its neighboring dots. An infinite universe has just doubled in size, which means: The distance between each dot and its neighbors is, at t2, twice as great as it was at t1.

Let's do the experiment in two dimensions, on the Euclidean plane. It's infinite in all directions.

For galaxies, take the integer lattice. That's the set of points having integer coordinates. The horizontal or vertical distance between each neighboring pair of points is 1.

Now suppose we rescale the coordinate system so that the distance between neighboring hor/vert points is 2. It's perfectly clear that there is still an integer lattice with hor/vert distance 1, composed of the points that were at hor/vert distance 1/2 before.

The plane is obviously the exact same size as it was before, namely infinite. All we've done is introduce a new coordinate system based on a different metric. We haven't altered the universe in the least.

How can you say the size of the plane is larger after a change of coordinates? It's the exact same infinite plane. The points of the before-plane and after-plane are still in bijection. The points of the integer lattice are still in bijection. All you did was apply a linear scaling factor to the coordinates.

The argument that the plane would "double in size" only works if it was finite to begin with. This becomes even more obvious in one dimension. If we take a line segment of length 1, doubling it in size makes it twice as long. If we "double in size" the entire real line, it's still the same real line. It doesn't change size at all. It doesn't matter if you re-label the points. The length of the line is infinite both before and after. It's the same exact line.

[ronjanec]

Wrong. An infinite sized sheet of black paper/or universe with an infinite number of white dots/galaxies still cannot double in size davidm no matter what the dots/galaxies do.

Again, if an “infinite” universe could ever double in size for any reason(or was previously limited in size before it doubled), it was never infinite to begin with. You cannot double an infinite sized anything no matter what happens anywhere else.

[davidm]

The plane is obviously the exact same size as it was before, namely infinite.

Right. It is. That is why I said, "doubling in size" means, specifically, that at t2, the distance between the white dots is twice as great as it was at t1. Are you denying that this is possible?

[davidm]

Wrong. An infinite sized sheet of black paper/or universe with an infinite number of white dots/galaxies still cannot double in size davidm no matter what the dots/galaxies do.

Again, if an “infinite” universe could ever double in size for any reason(or was previously limited in size before it doubled), it was never infinite to begin with. You cannot double an infinite sized anything no matter what happens anywhere else.

See my reply to someguy1. I ask you the same question: are you denying that what I just described is possible?

In an infinite universe, "doubling in size" just means, "doubling the distance between objects." And this is certainly possible in infinite space.

[davidm]

This is just a variant of Hilbert's hotel, btw.

[ronjanec]

Wrong. An infinite sized sheet of black paper/or universe with an infinite number of white dots/galaxies still cannot double in size davidm no matter what the dots/galaxies do.

Again, if an “infinite” universe could ever double in size for any reason(or was previously limited in size before it doubled), it was never infinite to begin with. You cannot double an infinite sized anything no matter what happens anywhere else.

See my reply to someguy1. I ask you the same question: are you denying that what I just described is possible?

In an infinite universe, "doubling in size" just means, "doubling the distance between objects." And this is certainly possible in infinite space.

I am telling you, that it is impossible for a universe that exists in an infinite size to grow in size for any reason...period. And yes, the rest of this with the dots is impossible and illogical for the aforementioned reason.

This scenario actually sounds kind of like the question of what would happen if an irresistible force meets an immovable object?. Another impossible, and illogical scenario just like the one you are talking about here(Mr. Hilbert should stick to the hotel business)

[davidm]

Wrong. An infinite sized sheet of black paper/or universe with an infinite number of white dots/galaxies still cannot double in size davidm no matter what the dots/galaxies do.

Again, if an “infinite” universe could ever double in size for any reason(or was previously limited in size before it doubled), it was never infinite to begin with. You cannot double an infinite sized anything no matter what happens anywhere else.

See my reply to someguy1. I ask you the same question: are you denying that what I just described is possible?

In an infinite universe, "doubling in size" just means, "doubling the distance between objects." And this is certainly possible in infinite space.

I am telling you, that it is impossible for a universe that exists in an infinite size to grow in size for any reason...period.

What you are telling me, is wrong.

Please explain why, in an infinite universe, the distance between objects cannot double, or treble, or whatever, in size? Why not? In fact, just the opposite is the case: In a finite and bounded universe, such a doubling or trebling can only continue for a limited time, until the objects crash against a boundary.

In a finite but unbounded universe, such as one with positive curvature, such a doubling or trebling can only continue until the objects wrap around the positively curved universe and crash into one another. In this case, though, if the universal expansion is greater than light speed (which is perfectly possible without any violation of relativity) then such collisions will never happen.

In an infinite universe, the distance between objects can increase a thousand fold, and the size of the universe is the same as before: Infinite. You should actually study the Hilbert hotel.

[davidm]

Once again, let us be clear: "doubling the size of the universe" means: the distance between objects doubles. It does not mean that space is an object that doubles in size.

[ronjanec]

You’re still missing the most important point that I am trying to make here. With 100% no doubt metaphysical certainty, it is impossible for an infinite sized universe to increase in size, because if it could ever increase in size it was never infinite to begin with. You/Hilbert are describing an impossible what if situation here.

Or again in other words: What is a known absolute certainty here, must make (or negate) the other opposing what if scenario (always) impossible.

[Brent696]

Seems a bit like saying there can't be any odd numbers because of the even numbers crowding them out in their infinitude. Brent, an infinite space is a continuum and has abundant room for infinite sets of anything - atoms, beach balls, short bald men named Balthazar, you name it. Have you read the thread here prior to posting? If nothing else, look up Hilbert's Hotel, and see how there's always room for more guests.

in·fi·nite
ˈinfənət/Submit
adjective
adjective: infinite
1. limitless or endless in space, extent, or size; impossible to measure or calculate.
"the infinite mercy of God"

synonyms:boundless, unbounded, unlimited, limitless, never-ending, interminable;

MATHEMATICS
greater than any assignable quantity or countable number.

MATHEMATICS
(of a series) able to be continued indefinitely.

You will notice that in Mathematics, they use potential "infinities" within a relative context.

For an actual Infinity, you need not other word but "boundless". For anything to be truly boundless, then nothing can exist beyond it. In an absolute set of odd numbers, then no even numbers would exist, hence the SET. But all these work as clarifying tools, and if you had read the thread you would have seen this definition provided also as regards mathematical sets and how they function.

But just for fun lets look at counting again, where does an infinite set for odd numbers begin????
We can pretend that there is no ending to the set as we might assumes time will never cease, unfortunately you have a beginning though, that would be a boundary.

"Sets" play a game with Time, not actual "things" or "object" in existence, we might say they exist potentially, but just not actually. As for hotels rooms, since nothing could exist beyond the infinite number of rooms, there would be nothing to add. Since Hilbert disregarded the definitive definition of boundlessness, He never even used an actually infinity as an example.

To clarify: 0 and 1(only), 0 is boundless in its nothingness, 1(only) is boundless in it's somethingness. Multiplicity cannot exist with Infinity.