Your Axiom of Choice is SILLY

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Re: Your Axiom of Choice is SILLY

Postby Swap » Fri Jun 13, 2008 7:42 pm UTC

antonfire wrote:You can't describe every subset of the natural numbers in full detail either. Is the power set axiom also untenable?


I suppose I'm talking from the point of view of an applied mathematician (i.e. someone who actually solves PDEs approximately). The axiom of choice is mostly useless for applied mathematics, and it will forever be so, because none of the sets it applies to can be described by any computer (I'm pretty sure not even quantum computers can hold a reasonable representation of a non-measurable set, but correct me if I'm wrong).

Yeah, completeness of the reals also gets you a bunch of nonsense (does it require choice?), but that's ok, a real number is whatever the IEEE says it is. ;-)

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Re: Your Axiom of Choice is SILLY

Postby Yakk » Fri Jun 13, 2008 7:58 pm UTC

Swap wrote:Yeah, completeness of the reals also gets you a bunch of nonsense (does it require choice?), but that's ok, a real number is whatever the IEEE says it is. ;-)


IEEE doubles suck ass. :p Nevermind analytical issues, create an area that is a few billion units in size, you are screwed!

R = { f:Q+->Q ST |f(a)-f(b)| < b for all a < b } isn't bad. :)

Heck, even interval mathematics on doubles generates better real-like results than just using those horrid things! (at least if your interval mathematics results in a ridiculously huge interval, you know your result is garbage...)
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Re: Your Axiom of Choice is SILLY

Postby antonfire » Fri Jun 13, 2008 8:29 pm UTC

Swap wrote:Yeah, completeness of the reals also gets you a bunch of nonsense (does it require choice?), but that's ok, a real number is whatever the IEEE says it is. ;-)
It does not require choice. The most dubious thing it requires is the power set axiom.

The point is that you should have just as much apprehension about the power set axiom as you have about the axiom of choice. It also gives a bunch of stuff that you can't describe. To me, Banach-Tarski (which needs the axiom of choice) is a lot less unsettling than the idea that we can't even talk about "most" real numbers (which requires only ZF).

But, yes, the applied mathematician doesn't care. The stuff I'm currently working on also doesn't run into any of thees philosophical issues, which is good as far as I'm concerned.
Jerry Bona wrote:The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?

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Re: Your Axiom of Choice is SILLY

Postby Mathmagic » Fri Jun 13, 2008 11:00 pm UTC

kevbrown wrote:
fortyseventeen wrote:That is merely a representation of the number, but I digress.


mathmagic wrote:No no no. It's because e4πi/3 is an imaginary number. It doesn't actually exist, silly!


It's not a matter of existence versus nonexistence. The numbers we call 'complex numbers' meet certain criteria, and those are useful, and that's perfectly good reason for existence.

I'm not sure if you caught it or not, but my comment was entirely sarcastic.
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Re: Your Axiom of Choice is SILLY

Postby kevbrown » Sun Jun 15, 2008 8:51 am UTC

my bad. I'm not so great at internet sarcasm.
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Re: Your Axiom of Choice is SILLY

Postby shinjak » Tue Jun 24, 2008 11:26 pm UTC

i don't understand this claim everyone's making that the well-ordering principle is so counterintuitive. i find it no harder to visualise than the fact that there are larger cardinals than the cardinality of the reals. just imagine a sufficiently long line and place the elements of your set at the integer points.

meanwhile, the equivalent formulation of the axiom of choice that i've used the most often by far is zorn's lemma. which is not visually intuitive. at all.

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Re: Your Axiom of Choice is SILLY

Postby antonfire » Tue Jun 24, 2008 11:36 pm UTC

People aren't saying that they can't visualize the well-ordering principle, people are saying that the truth of the well-ordering principle is not obvious and even that it's counterintuitive.

In particular, it's not obvious that there is a "sufficiently long line". Any such line would have some pretty weird properties. For larger cardinals, it won't even be path-connected.
Jerry Bona wrote:The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?

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Re: Your Axiom of Choice is SILLY

Postby ThomasS » Fri Jun 27, 2008 2:23 am UTC

Swap wrote:
antonfire wrote:I suppose I'm talking from the point of view of an applied mathematician (i.e. someone who actually solves PDEs approximately). The axiom of choice is mostly useless for applied mathematics, and it will forever be so, because none of the sets it applies to can be described by any computer (I'm pretty sure not even quantum computers can hold a reasonable representation of a non-measurable set, but correct me if I'm wrong).


You are correct that non-measurable sets are by their nature hard to define uniquely with finite memory.

However, computer "solutions" of differential equations are called such because they converge to actual solutions. Proving that this is the case can involve Sobelov spaces and other wonders of functional analysis. You quite possibly can get what you need without the axiom of choice, but things like Tychonoff's theorem appear in the darnedest places.

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Re: Your Axiom of Choice is SILLY

Postby Porges » Sun Jun 29, 2008 1:15 am UTC

What’s with all this ‘The AoC is intuitive’ crap :P I find it to be completely the opposite. :/

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Re: Your Axiom of Choice is SILLY

Postby Owehn » Mon Jun 30, 2008 2:13 pm UTC

Another way to state Choice is that products of nonempty sets are nonempty. That seems pretty intuitive to me.
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Re: Your Axiom of Choice is SILLY

Postby Yakk » Mon Jun 30, 2008 2:49 pm UTC

Ie, for sets A and B, A x B is defined as { (a,b) such that a is an element of A, and b is an element of B }.

The axiom of choice can be written as: If A and B are non-empty sets, then A x B is non-empty.
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Re: Your Axiom of Choice is SILLY

Postby antonfire » Mon Jun 30, 2008 2:56 pm UTC

No, it can't. What you said is that a finite product of nonempty sets are nonempty, which is provable in ZF. The axiom of choice is equivalent to saying that any product of nonempty sets is nonempty, which is not provable in ZF, which is why you need another axiom.

It's still pretty intuitive in this formulation though. It's hard to imagine that adding more nonempty sets will make the product bigger.
Jerry Bona wrote:The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?

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Re: Your Axiom of Choice is SILLY

Postby Woxor » Mon Jun 30, 2008 3:05 pm UTC

It's essentially "If you have a bunch of sets, and none of them are empty, then you can pick an element out of each set." As long as you accept things like "infinity" and such, I believe it's very intuitive. Zorn's Lemma seems like magic sometimes, and Banach-Tarski is pretty trippy, but the Axiom of Choice itself seems obvious.

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Re: Your Axiom of Choice is SILLY

Postby ConMan » Tue Jul 01, 2008 5:19 am UTC

Correct me if I'm wrong (and I almost certainly am), but isn't "the product of a countable number of non-empty sets is non-empty" also provable in ZF? It's only when you get to "the product of an uncountable number of non-empty sets is non-empty" that you need choice ... or so my vague recollections of a brief explanation of set theory in a measure theory course tell me.
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Re: Your Axiom of Choice is SILLY

Postby Yakk » Tue Jul 01, 2008 3:27 pm UTC

Gah, ya, I'm dumb. :)
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Re: Your Axiom of Choice is SILLY

Postby antonfire » Tue Jul 01, 2008 5:23 pm UTC

ConMan wrote:Correct me if I'm wrong (and I almost certainly am), but isn't "the product of a countable number of non-empty sets is non-empty" also provable in ZF? It's only when you get to "the product of an uncountable number of non-empty sets is non-empty" that you need choice ... or so my vague recollections of a brief explanation of set theory in a measure theory course tell me.
I'm pretty sure that that statement is equivalent to the axiom of countable choice, which is not provable in ZF, but is weaker than the axiom of choice.
Jerry Bona wrote:The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?

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Re: Your Axiom of Choice is SILLY

Postby Token » Tue Jul 01, 2008 9:01 pm UTC

antonfire wrote:
ConMan wrote:Correct me if I'm wrong (and I almost certainly am), but isn't "the product of a countable number of non-empty sets is non-empty" also provable in ZF? It's only when you get to "the product of an uncountable number of non-empty sets is non-empty" that you need choice ... or so my vague recollections of a brief explanation of set theory in a measure theory course tell me.
I'm pretty sure that that statement is equivalent to the axiom of countable choice, which is not provable in ZF, but is weaker than the axiom of choice.

Yep - as soon as you get infinite, the generalised cartesian product kicks in and you're dealing with choice functions.
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Re: Your Axiom of Choice is SILLY

Postby mattisbusiness » Sun Jul 06, 2008 4:45 am UTC

The axiom of choice is a crutch. It sticks around, because it makes proofs easier. In that sense, it's overly silly. Some people look down upon people that employ it, but they're all applied mathematicians.

The reality is that we don't really deal with infinite things in the real world, so its consequences don't really matter. I mean, can you actually decompose a tennis ball into infinite point sets with no volume?

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Re: Your Axiom of Choice is SILLY

Postby NathanielJ » Mon Jul 07, 2008 5:16 pm UTC

mattisbusiness wrote:The reality is that we don't really deal with infinite things in the real world, so its consequences don't really matter.


I've probably never disagreed more with something that I've read on these boards. All of calculus is based on the idea of infinities, so do the consequences of calculus not really matter?

I work in quantum computing, and here are two things that the Axiom of Choice does just in that one particular field that I can think of off the top of my head:
- AoC implies the Hahn-Banach theorem, which in turn implies Arveson's Extension theorem, which says that any completely positive map defined on an operator system can be extended to a completely positive map on the entire overlying space. What this does in quantum computing is ensures that if you have a quantum channel defined on an algebra (relatively nice small subsection of you whole space), you can twerk it to make sure that it also behaves nicely elsewhere.
- AoC implies that every vector space has a basis, which is quite necessary for a lot of the theory of C*-algebras to hold water, and the mathematical formulation of quantum mechanics is in terms of C*-algebras.

I'm sure there are many, many other applications of the AoC in the real world in other fields as well.
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Re: Your Axiom of Choice is SILLY

Postby Yakk » Mon Jul 07, 2008 7:23 pm UTC

NathanielJ wrote:
mattisbusiness wrote:The reality is that we don't really deal with infinite things in the real world, so its consequences don't really matter.

I've probably never disagreed more with something that I've read on these boards. All of calculus is based on the idea of infinities, so do the consequences of calculus not really matter?

You can do calculus without infinities.

Using infinities can make it easier and less messy. :-)

I work in quantum computing, and here are two things that the Axiom of Choice does just in that one particular field that I can think of off the top of my head:
- AoC implies the Hahn-Banach theorem, which in turn implies Arveson's Extension theorem, which says that any completely positive map defined on an operator system can be extended to a completely positive map on the entire overlying space. What this does in quantum computing is ensures that if you have a quantum channel defined on an algebra (relatively nice small subsection of you whole space), you can twerk it to make sure that it also behaves nicely elsewhere.

You read it wrong. If you can actually twerk it so that it behaves nicely elsewhere, then you didn't need to AoC to prove it for that case. The cases for which the AoC would be needed to prove that it "could be twerked" are those that you cannot actually find the twerking for.

Basically, you can do all of your work in ZF~C instead of ZFC, and anything that applies to concrete reality still holds -- because the AoC is about infinite cross products of sets, not about finite cross products of sets.

Now, AoC helps -- you can presume the basis without finding it, or presume there exists an exact solution that you are finding successive approximations to, instead of having to prove that there is a means of producing successive answers that, in some sense, approximate the actual answer. Lots of wording gets better.

The vector spaces for which you can find an explicit basis for - they have a basis regardless of AoC. The vector spaces that you cannot find the basis for, those are the ones that AoC claims that there is a basis for them.

Having the AoC around helps lots. But don't give it too much credit -- much of mathematics can be done without it, it is just much messier.
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Re: Your Axiom of Choice is SILLY

Postby NathanielJ » Mon Jul 07, 2008 7:38 pm UTC

You read it wrong. If you can actually twerk it so that it behaves nicely elsewhere, then you didn't need to AoC to prove it for that case. The cases for which the AoC would be needed to prove that it "could be twerked" are those that you cannot actually find the twerking for.


Yes, of course if you actually have the twerk sitting in front of you then you don't need an existence theorem to tell you that it exists, but the extension theorem is nice nonetheless because it assures you that certain things work. Whether or not you can actually find the twerking is not important in my particular area of research, as existence is enough to make things work.
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