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The Axiom of Choice Is Wrong (2007)

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Re: The Axiom of Choice Is Wrong (2007)

#31

Post by mrob »

llamaz wrote: post #27
mrob wrote: post #20

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 piece…

Mathematics is formalised is to avoid this sort of philosophizing. I used to think, for example, that the dirac delta function was mathematical fiction - a mathematical "hack". But then in an engineering control systems class, we did an experiment where we used a step function to approximate a dirac delta function. I could see the results both on the computer screen and in physical reality through a mass-spring-dampe…

The Dirac delta function is a mathematical fiction. It doesn't look weird in your case because you didn't mix it with any other weird mathematical fictions. But maybe somebody could invent a system that violated conservation of energy by using Dirac deltas. That wouldn't mean the Dirac delta is a bad fiction, because if you actually tried to build it using your step function approximation it wouldn't work.

Re: The Axiom of Choice Is Wrong (2007)

#32

Post by pherq »

zodiac wrote: post #22
mrob wrote: post #20

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 piece…

There are a small number of people who think it's obviously false! (I'm not one of them). And a larger number who think it's not obviously true nor obviously false.

There's the old joke that the axiom of choice is obviously true, the well-ordering theorem is obviously false, and Zorn's lemma is too complicated to say.

Re: The Axiom of Choice Is Wrong (2007)

#33

Post by CJefferson »

heinrichhartman wrote: post #28
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…

Good luck developing analysis with only countable infinities. Limits will take you out of the realm of countable spaces. Most of your derivatives and integrals won't exists, if you force them to take values in countable sets.

Why not, can you give an example.

Most of the "common" derivatives and integrals I can think of would work just fine, unless I'm missing something obvious?

Re: The Axiom of Choice Is Wrong (2007)

#34

Post by kutkloon7 »

CJefferson wrote: post #33
heinrichhartman wrote: post #28

Earlier quoted context omitted.

Good luck developing analysis with only countable infinities. Limits will take you out of the realm of countable spaces. Most of your derivatives and integrals won't exists, if you force them to take values in countable sets.

Why not, can you give an example. Most of the "common" derivatives and integrals I can think of would work just fine, unless I'm missing something obvious?

No, for example the integral sqrt(1 - x^2) from x = 0 to x = 1. Or any other integral over a circle/sphere for that matter.

Re: The Axiom of Choice Is Wrong (2007)

#35

Post by CJefferson »

kutkloon7 wrote: post #34
CJefferson wrote: post #33

Earlier quoted context omitted.

Why not, can you give an example. Most of the "common" derivatives and integrals I can think of would work just fine, unless I'm missing something obvious?

No, for example the integral sqrt(1 - x^2) from x = 0 to x = 1. Or any other integral over a circle/sphere for that matter.

Nothing wrong with that, it's still a computable number -- I'm not suggesting doing away with all real numbers, just the non-computable ones.

I don't claim to have thought through every detail, but having now gone and done some reading, this seems fine in the world of constructivism, which only requires computable numbers (and therefore only countable infinities of numbers)

Re: The Axiom of Choice Is Wrong (2007)

#36

Post by IsaacL »

baddox wrote: post #6
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…

This is discussed heavily in topics around constructing the reals and constructivism. Wikipedia has a short section about using the computables instead of the reals: https://en.m.wikipedia.org/wiki/Computable_number It is pretty fascinating to think about: that almost all real numbers are not computable ("almost all" of course meaning "all but a countable set"). And yet you almost certainly will never run into a nonc…

Are there any good books on this topic?

Re: The Axiom of Choice Is Wrong (2007)

#37

Post by cygx »

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?

Re: The Axiom of Choice Is Wrong (2007)

#38

Post by heinrichhartman »

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…

Are you aware of the concept of periods ? This is a quite fascinating, countable, ring of numbers, that captures all of the above:

Kontsevich and Zagier introduce and develop this concept quite far:

http://www.maths.ed.ac.uk/~aar/papers/kontzagi.pdf

Re: The Axiom of Choice Is Wrong (2007)

#39

Post by twic »

The axiom of choice always seemed intuitively wrong to me. You can't just take a set and arbitrarily pick something out of it! Making a choice requires information, and you can't pluck information out of thin air at whim; applying the axiom amounts to creating information out of nothing.

I suppose this is because i'm not a mathematician, but have a natural sciences background. In the physical universe, memorably, "the law that entropy always increases holds, I think, the supreme position among the laws of Nature" [1], and so we do not accept the mathematicians' fake information.

More specifically related to choice from a set, Curie's principle that "when certain causes produce certain effects, it is the elements of symmetry of the causes that may be found in the effects produced" [2] forbids something uniform from becoming arbitrarily non-uniform; whenever that appears to happen, there must be some hidden cause which already carries that non-uniformity.

[1] https://en.wikipedia.org/wiki/Second_law_of_thermodynamics

[2] https://hal.archives-ouvertes.fr/jpa-00239814 - "Enfin, lorsque certaines causes produisent certains effets, les éléments de symétrie des causes doivent se retrouver dans les effets produits."; please excuse the not-so-literal translation

Re: The Axiom of Choice Is Wrong (2007)

#40

Post by twic »

CJefferson wrote: post #33
heinrichhartman wrote: post #28

Earlier quoted context omitted.

Good luck developing analysis with only countable infinities. Limits will take you out of the realm of countable spaces. Most of your derivatives and integrals won't exists, if you force them to take values in countable sets.

Why not, can you give an example. Most of the "common" derivatives and integrals I can think of would work just fine, unless I'm missing something obvious?

Are the computable numbers continuous? If not, is that a problem for analysis? If so, can you construct some computable analogue of continuity which is sufficient?
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