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The Axiom of Choice Is Wrong (2007)

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Re: The Axiom of Choice Is Wrong (2007)

#121

Post by Jweb_Guru »

smilliken wrote: post #107
Jweb_Guru wrote: post #98

Earlier quoted context omitted.

> Real space doesn't appear to be infinite Citation needed. People keep saying things like this ("real space isn't infinite" or "real spacetime must be discrete") but never show their work.

Nobody has proof, and it's probably unknowable. It could be that we're a discrete system embedded in a continuous system. Or it could be we're in a seemingly continuous system which is actually embedded in a high precision discrete system. That being said, nature has given us very strong hints at discretization. For example: https://en.m.wikipedia.org/wiki/Planck_length . Because of uncertainty principle, we can't me…

The fact that we don't know what's going on within the Planck length doesn't demonstrate that spacetime is discretized; I don't even think it strongly indicates it, unless you're already predisposed to think that way. Anyway, that seems to have no bearing on whether the universe is infinite.

Re: The Axiom of Choice Is Wrong (2007)

#122

Post by warrenpj »

Jweb_Guru wrote: post #98
candiodari wrote: post #42

Earlier quoted context omitted.

> Don't they just say that if there's such a thing as objective mathematical reality, it can't be effectively axiomatized? No they don't. Real space doesn't appear to be infinite, and Zn is not subject to Godel's incompleteness theorem. If you drop the requirement of infinite numbers and "recursive" infinites (e.g. real numbers), as reality appears to do, there is no problem.

> Real space doesn't appear to be infinite Citation needed. People keep saying things like this ("real space isn't infinite" or "real spacetime must be discrete") but never show their work.

There was a post here two weeks ago discussing this exact problem. [1] It starts by explaining Democritus' reasoning that matter must be made of discrete atoms. The main point of the article is that a similar logic applies to the future theory of quantum gravity; space and time too must be discrete.

From the article:

> Importing the atomic philosophy of Democritus into modern physics might be essential for reconciling general relativity (which assumes a continuous reality) with quantum mechanics (which very much does not).

As far as infinite vs finite space: because there is a fixed rate of information propagation, we only need to show that time is finite in extent, and it follows that space is also finite. We know that the visible universe is finite (by looking in the sky) and we can explain why: the big bang happened at a finite time in the past.

We don't know if the universe will have a finite future, but it might be the case. One possible cause of a finite future could be accelerating expansion of the universe due to dark energy. [2]

Of course there are other interpretations of "real space", but this one (causally connected) seems parsimonious to me.

[1] https://news.ycombinator.com/item?id=13466254

[2] https://en.wikipedia.org/wiki/Ultimate_fate_of_the_universe

Re: The Axiom of Choice Is Wrong (2007)

#123

Post by fmap »

cygx wrote: post #37
fmap wrote: post #24

The problem solution in the article uses the axiom of choice to construct a "nonprincipal ultrafilter" on the natural numbers. This is actually weaker than the full axiom of choice, but you can still show that no such object is computable. It's a nice exercise to show that with the same assumptions as in the article you can decide the halting problem. (hint: consider the boolean sequence where the nth element is true…

There is no objective mathematical reality, because it cannot include a statements about its own consistency I'm assuming we're talking about Gödel's incompleteness theorems? Don't they just say that if there's such a thing as objective mathematical reality, it can't be effectively axiomatized?

Yes, and indeed Gödel himself believed in an objective mathematical reality. What I meant to say is that the commonly accepted basis for mathematics (first order logic and ZFC) was first justified using arguments which later turned out to be false.

Logically, we are not finding better and better approximations to some mathematical laws of nature, but rather making a series of arbitrary choices (you can assert the truth or falsity of any independent statement). Philosophically, we can argue about the sense in which the continuum hypothesis ought to be true or false (that is what Gödel did), but practically it makes more sense to think of logic as a tool that you build to describe and solve specific problems.

Re: The Axiom of Choice Is Wrong (2007)

#124

Post by Jweb_Guru »

warrenpj wrote: post #122
Jweb_Guru wrote: post #98

Earlier quoted context omitted.

> Real space doesn't appear to be infinite Citation needed. People keep saying things like this ("real space isn't infinite" or "real spacetime must be discrete") but never show their work.

There was a post here two weeks ago discussing this exact problem. [1] It starts by explaining Democritus' reasoning that matter must be made of discrete atoms. The main point of the article is that a similar logic applies to the future theory of quantum gravity; space and time too must be discrete. From the article: > Importing the atomic philosophy of Democritus into modern physics might be essential for reconcilin…

To the best of my knowledge, mainstream quantum mechanics assumes continuous spacetime, so the argument that the primary reason it can't be reconciled with general relativity is continuity seems pretty specious to me. And it's possible to come up with a model where the universe is infinite but the visible universe is finite. In fact, I believe FLRW is one such a model, and is supported with high accuracy by the latest experimental observations. So again, your argument doesn't seem that compelling to me.

Re: The Axiom of Choice Is Wrong (2007)

#125

Post by caf »

fmap wrote: post #24

The problem solution in the article uses the axiom of choice to construct a "nonprincipal ultrafilter" on the natural numbers. This is actually weaker than the full axiom of choice, but you can still show that no such object is computable. It's a nice exercise to show that with the same assumptions as in the article you can decide the halting problem. (hint: consider the boolean sequence where the nth element is true…

This is actually weaker than the full axiom of choice, but you can still show that no such object is computable.

Presumably this is an ever-present hazard in any proof using the axiom of choice - in this case it is relatively obvious, but how easy is it to accidentally rely on such an uncomputable object in a dusty corner of your proof?

Re: The Axiom of Choice Is Wrong (2007)

#126

Post by cygx »

Jweb_Guru wrote: post #124
warrenpj wrote: post #122

Earlier quoted context omitted.

There was a post here two weeks ago discussing this exact problem. [1] It starts by explaining Democritus' reasoning that matter must be made of discrete atoms. The main point of the article is that a similar logic applies to the future theory of quantum gravity; space and time too must be discrete. From the article: > Importing the atomic philosophy of Democritus into modern physics might be essential for reconcilin…

To the best of my knowledge, mainstream quantum mechanics assumes continuous spacetime, so the argument that the primary reason it can't be reconciled with general relativity is continuity seems pretty specious to me. And it's possible to come up with a model where the universe is infinite but the visible universe is finite. In fact, I believe FLRW is one such a model, and is supported with high accuracy by the lates…

And it's possible to come up with a model where the universe is infinite but the visible universe is finite. In fact, I believe FLRW is one such a model

Just to clarify, FLRW is a class of models, which includes ones with finite as well as infinite space. We've been able to determine that our universe is close to spatially flat. However, that does not mean it's infinite: If spatial curvature is positive but small compared to the cosmic horizon (which is due to spatial expansion), we won't be able to tell.

That said, even given finite space, spacetime can of course still be infinite if the universe does not recollapse.

Re: The Axiom of Choice Is Wrong (2007)

#127

Post by cygx »

warrenpj wrote: post #122
Jweb_Guru wrote: post #98

Earlier quoted context omitted.

> Real space doesn't appear to be infinite Citation needed. People keep saying things like this ("real space isn't infinite" or "real spacetime must be discrete") but never show their work.

There was a post here two weeks ago discussing this exact problem. [1] It starts by explaining Democritus' reasoning that matter must be made of discrete atoms. The main point of the article is that a similar logic applies to the future theory of quantum gravity; space and time too must be discrete. From the article: > Importing the atomic philosophy of Democritus into modern physics might be essential for reconcilin…

we only need to show that time is finite in extent, and it follows that space is also finite [...]

That does not follow: As our universe is spatially flat (or close to it), it is consistent with the infinite-space Friedmann model. The cosmic horizon that bounds the observable universe would still exist due to the metric expansion of space.

Re: The Axiom of Choice Is Wrong (2007)

#128

Post by warrenpj »

Jweb_Guru wrote: post #124
warrenpj wrote: post #122

Earlier quoted context omitted.

There was a post here two weeks ago discussing this exact problem. [1] It starts by explaining Democritus' reasoning that matter must be made of discrete atoms. The main point of the article is that a similar logic applies to the future theory of quantum gravity; space and time too must be discrete. From the article: > Importing the atomic philosophy of Democritus into modern physics might be essential for reconcilin…

To the best of my knowledge, mainstream quantum mechanics assumes continuous spacetime, so the argument that the primary reason it can't be reconciled with general relativity is continuity seems pretty specious to me. And it's possible to come up with a model where the universe is infinite but the visible universe is finite. In fact, I believe FLRW is one such a model, and is supported with high accuracy by the lates…

> To the best of my knowledge, mainstream quantum mechanics assumes continuous spacetime, so the argument that the primary reason it can't be reconciled with general relativity is continuity seems pretty specious to me.

Your criticism is right and that is a bad argument. The real reasons they are hard to reconcile are technical (see for example [1]), and not mentioned in the source I referred to.

The article only says that a discrete theory of spacetime "might be essential" for the reconciliation; the arguments for this are philosophical - it would be beautiful and historically completing. Read the article if you're interested in the arguments, which aren't included in my comments.

> And it's possible to come up with a model where the universe is infinite but the visible universe is finite. In fact, I believe FLRW is one such a model, and is supported with high accuracy by the latest experimental observations.

I agree with you that the universe could be infinite, but that doesn't contradict my previous comment. I explained that I was talking about local (causally connected to us) space-time, only. I admit this is a bit of a trick, but the parent comment used the term "real space" which I interpreted in such a way to favour an argument for finite space.

> So again, your argument doesn't seem that compelling to me.

I've given reasons but I think it might be helpful to restate what I am actually claiming. I admit I don't know that space and time are discrete. The source I gave is a poetic and philosophical argument that it might be. I think it's established fact that energy and matter are, and that time is finite in the past, so the visible universe is finite. I think it's possible that the future is finite too.

I don't know what FLRW is. My best guess is https://en.wikipedia.org/wiki/Friedmann–Lemaître–Robertson–W...

[1] http://www.theory.caltech.edu/people/jhs/strings/str115.html

Re: The Axiom of Choice Is Wrong (2007)

#129

Post by naasking »

Jweb_Guru wrote: post #120
naasking wrote: post #105

Earlier quoted context omitted.

Google "Bekenstein Bound".

That doesn't prove anything about the continuity (or lack thereof) of spacetime, just indicates that energy is discretized. Also has no bearing on the original statement about infinity. It's also worth pointing out that the original statement is nonsensical because the theory of real closed fields doesn't suffer from incompleteness in the first place, so there's no need to "save" it from paradox the way you have to s…

The Bekenstein bound, as a starting point, addresses the more general point that all of known physics needs only computable abstractions. Continuous abstractions are merely for our convenience, and not fundamental as far as we've seen, which does address your specific question about whether space is truly continuous.

Re: The Axiom of Choice Is Wrong (2007)

#130

Post by cygx »

warrenpj wrote: post #128
Jweb_Guru wrote: post #124

Earlier quoted context omitted.

To the best of my knowledge, mainstream quantum mechanics assumes continuous spacetime, so the argument that the primary reason it can't be reconciled with general relativity is continuity seems pretty specious to me. And it's possible to come up with a model where the universe is infinite but the visible universe is finite. In fact, I believe FLRW is one such a model, and is supported with high accuracy by the lates…

> To the best of my knowledge, mainstream quantum mechanics assumes continuous spacetime, so the argument that the primary reason it can't be reconciled with general relativity is continuity seems pretty specious to me. Your criticism is right and that is a bad argument. The real reasons they are hard to reconcile are technical (see for example [1]), and not mentioned in the source I referred to. The article only say…

> I think it's established fact that energy and matter are, and that time is finite in the past, so the visible universe is finite.

None of that is established, actually:

Metric expansion of space comes with a cosmic event horizon where redshift goes to infinity. While the energy contained within that region is finite, there's no reason to assume that the universe outside the horizon looks any different than inside the horizon, and it might very well be spatially infinite.

As to time being finite in the past, it is possible to generalize Einstein's equations so that they can be extended past the big bang singularity, and what you get is a mirror universe on the other side. The issue here is that as we approach Planck scale, quantum gravity becomes relevant, and we have no predictive theory of that.

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