Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

The Axiom of Choice Is Wrong (2007)

cornellmath.wordpress.com

Post reply on HN
Print view View original

11–20 of 153 posts

Re: The Axiom of Choice Is Wrong (2007)

#11

Post by skissane »

zodiac wrote: post #8
CJefferson wrote: post #3

As time goes by, I increasingly view things like uncountable infinities and the axiom of choice as "a fun maths game", rather than having any intrinsic truth or falsity. Other's view may differ. Here is an interesting thing I've never seen anyone write down (I should do it myself) -- we don't need uncountable infinities. * How many natural numbers are there? Countable * How many rational numbers are there? Countable…

How many sets of natural numbers are there? An uncountable number. Of course, you could throw out some axioms so that you create an axiom system in which I can't define such a thing as "a subset of the natural numbers", but such a world doesn't feel "real" (in a platonic sense) to me.

> How many sets of natural numbers are there? An uncountable number.

But, at most countably many of those sets will ever be individually thought about by any human being. So there are only countably many (and maybe more strongly finitely many) subsets which any human being could ever possibly think about, and an uncountable remainder which are unthinkable. In order to do mathematics, do we really need to posit the existence of an uncountable number of objects no human mind will ever be able to individually consider?

> Of course, you could throw out some axioms so that you create an axiom system in which I can't define such a thing as "a subset of the natural numbers", but such a world doesn't feel "real" (in a platonic sense) to me.

You could introduce a set theory where from a countable set you can only infer the existence of its computable subsets, or its first order definable subsets, or something like that. Not sure why that should feel any less "real" than an uncountable infinity of mathematical objects about which no one will ever individually think.

Re: The Axiom of Choice Is Wrong (2007)

#12

Post by sn41 »

CJefferson wrote: post #3

As time goes by, I increasingly view things like uncountable infinities and the axiom of choice as "a fun maths game", rather than having any intrinsic truth or falsity. Other's view may differ. Here is an interesting thing I've never seen anyone write down (I should do it myself) -- we don't need uncountable infinities. * How many natural numbers are there? Countable * How many rational numbers are there? Countable…

One of the great theorems in logic is the Lowenheim-Skolem theorem [1], which says that if a countable first-order model has an infinite model, then there is a countably infinite model. For example, there is a countably infinite model for the theory of reals.

I have heard that Skolem [2] used this theorem as a basis for his belief that uncountable sets can be avoided, and countably infinite sets suffice.

[1] https://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_...

[2] http://www-gap.dcs.st-and.ac.uk/history/Biographies/Skolem.h...

Re: The Axiom of Choice Is Wrong (2007)

#13

Post by baddox »

skissane wrote: post #11
zodiac wrote: post #8

Earlier quoted context omitted.

How many sets of natural numbers are there? An uncountable number. Of course, you could throw out some axioms so that you create an axiom system in which I can't define such a thing as "a subset of the natural numbers", but such a world doesn't feel "real" (in a platonic sense) to me.

> How many sets of natural numbers are there? An uncountable number. But, at most countably many of those sets will ever be individually thought about by any human being. So there are only countably many (and maybe more strongly finitely many) subsets which any human being could ever possibly think about, and an uncountable remainder which are unthinkable. In order to do mathematics, do we really need to posit the ex…

> But, at most countably many of those sets will ever be individually thought about by any human being.

If your arguments are tied to such physical constraints, then there aren't even countably infinite natural numbers unless the Universe is infinite in space or in time (such that there can be infinite humans or "thinkers").

Re: The Axiom of Choice Is Wrong (2007)

#14

Post by baddox »

pherq wrote: post #9
zodiac wrote: post #8

Earlier quoted context omitted.

How many sets of natural numbers are there? An uncountable number. Of course, you could throw out some axioms so that you create an axiom system in which I can't define such a thing as "a subset of the natural numbers", but such a world doesn't feel "real" (in a platonic sense) to me.

In this case, what you "throw out" are uncomputable (or non-recursive, depending on terminology you like) sets; i.e. sets for which the membership function is not decidable. Yes, there are an uncountable number of these sets, but they can't be defined in any useful way.

Indeed, if you consider a real number to just be a decimal representation formed by concatenating all the natural numbers in a set (not at all a rigourous construction of the reals, but hopefully sufficient for this argument), the equivalence is clear.

Re: The Axiom of Choice Is Wrong (2007)

#15

Post by skissane »

baddox wrote: post #13
skissane wrote: post #11

Earlier quoted context omitted.

> How many sets of natural numbers are there? An uncountable number. But, at most countably many of those sets will ever be individually thought about by any human being. So there are only countably many (and maybe more strongly finitely many) subsets which any human being could ever possibly think about, and an uncountable remainder which are unthinkable. In order to do mathematics, do we really need to posit the ex…

> But, at most countably many of those sets will ever be individually thought about by any human being. If your arguments are tied to such physical constraints, then there aren't even countably infinite natural numbers unless the Universe is infinite in space or in time (such that there can be infinite humans or "thinkers").

Indeed, that's why I'm an ultrafinitist.

https://en.wikipedia.org/wiki/Ultrafinitism

Re: The Axiom of Choice Is Wrong (2007)

#16

Post by baddox »

im3w1l wrote: post #5

I'd assume that even if in every case the number of incorrect guesses is finite, the expected number of people that fail to guess their color is infinite. Am I right about this?

Im not sure what you mean by "expected number." Do you mean if you try to guess how many inmates the warden has managed to guarantee will fail? Per the article, the warden can guarantee that an "arbitrarily large finite number of them" will fail. But it's still always finite despite being unbounded. If you want to predict a lower bound on the number of failures the warden has guaranteed, you just have to guess a larger natural number than the warden. :)

Re: The Axiom of Choice Is Wrong (2007)

#17

Post by joatmon-snoo »

CJefferson wrote: post #3

As time goes by, I increasingly view things like uncountable infinities and the axiom of choice as "a fun maths game", rather than having any intrinsic truth or falsity. Other's view may differ. Here is an interesting thing I've never seen anyone write down (I should do it myself) -- we don't need uncountable infinities. * How many natural numbers are there? Countable * How many rational numbers are there? Countable…

    the solution to any problem any could ever have
That's a pretty tall claim to make.

Once upon a time people thought irrationals and transcendentals like pi and e didn't even exist. It wasn't until the past century that people realized abstract algebra and its ilk had practical applications. The same goes for chaos theory and fractals as well.

Admittedly it's hard for me to envision a world where notions of countability will ever have real-world uses, but I also consider calling it "a fun maths game" rather reductionist, and frankly, poor taste (especially if you're going to say something like "intrinsic truth or falsity").

    Give him three pence, since he must make gain out of what he learns.

Re: The Axiom of Choice Is Wrong (2007)

#18

Post by baddox »

skissane wrote: post #15
baddox wrote: post #13

Earlier quoted context omitted.

> But, at most countably many of those sets will ever be individually thought about by any human being. If your arguments are tied to such physical constraints, then there aren't even countably infinite natural numbers unless the Universe is infinite in space or in time (such that there can be infinite humans or "thinkers").

Indeed, that's why I'm an ultrafinitist. https://en.wikipedia.org/wiki/Ultrafinitism

Interesting. I was aware of finitism, of course, and even strict finitism, but not this particularly strict version of it.

Re: The Axiom of Choice Is Wrong (2007)

#19

Post by zodiac »

baddox wrote: post #14
pherq wrote: post #9

Earlier quoted context omitted.

In this case, what you "throw out" are uncomputable (or non-recursive, depending on terminology you like) sets; i.e. sets for which the membership function is not decidable. Yes, there are an uncountable number of these sets, but they can't be defined in any useful way.

Indeed, if you consider a real number to just be a decimal representation formed by concatenating all the natural numbers in a set (not at all a rigourous construction of the reals, but hopefully sufficient for this argument), the equivalence is clear.

I've always found dedekinds construction (where sqrt 2 is represented by the set of rationals {x in Q : x^2 < 2}) to be a nice natural place where powersets of countable infinite sets come up (in addition to being a great way to construct R)

Re: The Axiom of Choice Is Wrong (2007)

#20

Post by mrob »

Infinities aren't real, so you shouldn't be surprised if unrealistic things happen when you invoke infinities. That you can duplicate a sphere by cutting it into a finite number of pieces and reassembling it is a "fact" in the same sense as "Luke Skywalker destroyed the Death Star". It might be interesting and culturally important, but it's talking about fictional entities. Both spheres and arbitrarily detailed pieces of spheres only exist in the imaginations of mathematicians. The Axiom of Choice is obviously true, and the fact that it lets you invent weird sounding stories from weird components is no evidence against it. Don't confuse mathematical tools with reality.
Post reply on HN