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The Axiom of Choice Is Wrong (2007)

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Re: The Axiom of Choice Is Wrong (2007)

#101

Post by akud »

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?

Was going to say the same thing. Most of my professors were secretly platonists, though they had to pass as formalists to get respect in polite society. It's hard to make absolute claims about the nature of mathematical reality; the formalists need to explain why math is so successful in the real world, and the platonists need to give an account of the ontological nature of mathematical objects.

Re: The Axiom of Choice Is Wrong (2007)

#102

Post by kmm »

im3w1l wrote: post #7
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…

> 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. Is a hammer true or false? I don't know, but it's good for hitting nails.

Yeah, a hammer is definitely useful, but I think the point he's trying to make is that the notion of uncountable sets isn't very useful outside of things like set theory.

Re: The Axiom of Choice Is Wrong (2007)

#103

Post by twic »

hackinthebochs wrote: post #89
twic wrote: post #39

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, "th…

> You can't just take a set and arbitrarily pick something out of it! Making a choice requires information True, but how can you "have a set", i.e. reference a set in any way, without having information about that set? There seems to be a requirement of some bare minimum of information enough to specify the set, and so enough to pick out a member of the set.

Having enough information to specify the set isn't enough to pick out one particular member.

For example, if i say "the colours teal, maroon, and taupe", you have enough information to know what's in the set, but no extra information that would let you pick one element out of it.

Re: The Axiom of Choice Is Wrong (2007)

#105

Post by naasking »

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.

Google "Bekenstein Bound".

Re: The Axiom of Choice Is Wrong (2007)

#107

Post by smilliken »

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.

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 measure distances less than that. In a similar way, time also appears to be discretized, along with energy and mass.

Re: The Axiom of Choice Is Wrong (2007)

#108

Post by mcphage »

twic wrote: post #103
hackinthebochs wrote: post #89

Earlier quoted context omitted.

> You can't just take a set and arbitrarily pick something out of it! Making a choice requires information True, but how can you "have a set", i.e. reference a set in any way, without having information about that set? There seems to be a requirement of some bare minimum of information enough to specify the set, and so enough to pick out a member of the set.

Having enough information to specify the set isn't enough to pick out one particular member. For example, if i say "the colours teal, maroon, and taupe", you have enough information to know what's in the set, but no extra information that would let you pick one element out of it.

You just listed the elements, that's the easy case. You can pick one of the listed ones, like Teal.

Re: The Axiom of Choice Is Wrong (2007)

#109

Post by skissane »

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

Godel's first theorem says no formal system with recursively enumerable axioms and which is powerful enough to express elementary arithmetic can be both consistent and complete. Most people adopt the response – alas, that means no formal systems we can devise can ever be complete (except for systems too weak to express arithmetic, which are not very useful). But, paraconsistency and dialetheism give us another response: yes, we can have complete formal systems with recursively enumerable axioms and which are powerful enough to express elementary arithmetic. Okay, they'll be inconsistent, but inconsistency is not as bad as you thought it was. We can contain the inconsistency to some small area in which it isn't going to bother you, you'll scarcely notice that it is there. We can hide the contradictions in the closet and go on our merry way not thinking about their existence except on rare occasions.

If mathematics is inconsistent, does that make mathematics objectively unreal? Only if we insist objective reality must be consistent. If objective reality contains dialetheias, then inconsistent mathematics can be objectively real.

Re: The Axiom of Choice Is Wrong (2007)

#110

Post by dvt »

IshKebab wrote: post #104

How can an axiom be 'wrong'?

An axiom is simply a statement that is defined as being true. Often times on grounds of being self-evident. For example, one of Euler's axioms is "Things that are equal to the same thing are also equal to one another".

Whereas Euler's may seem pretty tame, the Axiom of Choice is not. That's why some people think it might be wrong.

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