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The Axiom of Choice Is Wrong (2007)

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Re: The Axiom of Choice Is Wrong (2007)

#71

Post by lurker83256 »

wbhart wrote: post #61
Tloewald wrote: post #59

Earlier quoted context omitted.

No incompleteness proves that there will be statements that are true or false that cannot be proven to be true or false. Not being able to write all the rules is a completely different and unrelated thing.

If they can neither be proved true or false within the system, that means their truth or falsity can be added as axioms to the system and a contradiction will never be reached. So I can assume them to be true or false, and develop perfectly consistent mathematics. In this way, I can always add more rules to the system (albeit a nonstandard one).

This is a misunderstanding. Statement A is actually true in the system, you just cannot prove that it is true. Adding an axiom specifying its falsity would be a contradiction (although you could not prove this).

Re: The Axiom of Choice Is Wrong (2007)

#72

Post by leereeves »

Grue3 wrote: post #55

There are many problems with this puzzle that go against the intuition. - the number of prisoners is infinite, so they will never finish answering the question. At any point in time, only a finite number of prisoners will be freed. - a single prisoner must process an infinite amount of information to reach the decision. In fact, by observing only a finite number of hats he cannot possibly choose the answer. - the num…

This puzzle seems rather hand-wavy to me.

If we actually define a specific finite number N in the equivalence relation "two such sequences [are] ‘equivalent’ if they are equal after [N] entries", no one is in a position to take advantage of the equivalence classes.

Either their position Or their position > N, in which case they cannot see the entire sequence after N entries and thus cannot determine which equivalence class the sequence belongs to.

Unless I'm missing something.

Re: The Axiom of Choice Is Wrong (2007)

#73

Post by pacala »

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.

One of the observations in programming language semantic proofs is that one doesn't need coinduction. Statements can be proven by good old fashion induction, by indexing them with the number of steps in the computation, then proving they hold for any finite number of steps.

I suspect the same technique applies to analysis. We don't need to prove a statement holds for reals with "infininte" number of digits, but merely for all rationals with finite k digits, where k is a parameter of the proof, thus can be arbitrarily large.

Re: The Axiom of Choice Is Wrong (2007)

#74

Post by waqf »

heinrichhartman wrote: post #43
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?

- Every real number can be represented as a limit of rational numbers, which are uncountable. - So for every countable subset you can find a (Cauchy) sequence of rationals that does not converge. Hence, you loose one of the most important tools in Analyisis (Cauchy criterium for convergence). You can still work with the remaining set, but formulating and proving theorems, is going to be much harder. - If integrals ov…

I"m not sure what you're saying. There are only countably many computable sequences of rational numbers, so why shouldn't they all converge?

Re: The Axiom of Choice Is Wrong (2007)

#75

Post by daxfohl »

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?

Interesting observation! Yes, I think reduced to its essence, it boils down to: the expected value of "a finite number" is infinity. Which is strange in itself.

So you end up with a sequence that converges to 100%, but that convergence never starts, but it certainly happens eventually.

I guess a similar, less verbose thing would be "pick a random rational in (0, 1)". In decimal representation it'll repeat after "a finite number" of random digits, but "a finite number" again is expected to be infinity. So is a random rational really rational? Someone more educated on set theory will have to comment.

Re: The Axiom of Choice Is Wrong (2007)

#76

Post by waqf »

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?

If you take any finite subset of N prisoners, it's easy to see that they won't do better than chance, so the expectation is for N/2 of them to guess wrongly. Since N can be arbitrarily large, your intuition is correct.

Re: The Axiom of Choice Is Wrong (2007)

#77

Post by stablemap »

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…

Maybe I'm missing your point, but for δ I think the hack is using the word "function"; as a measure or distribution it's a pedestrian object.

Re: The Axiom of Choice Is Wrong (2007)

#78

Post by HelloNurse »

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…

You need to draw a line between mathematical entities that have "a basis in physical reality" or not. For most people, performing an infinite computation is what puts the axiom of choice firmly on the fiction side, as an impossible procedure, regardless of the application and the outcome.

Re: The Axiom of Choice Is Wrong (2007)

#79

Post by baddox »

wbhart wrote: post #61
Tloewald wrote: post #59

Earlier quoted context omitted.

No incompleteness proves that there will be statements that are true or false that cannot be proven to be true or false. Not being able to write all the rules is a completely different and unrelated thing.

If they can neither be proved true or false within the system, that means their truth or falsity can be added as axioms to the system and a contradiction will never be reached. So I can assume them to be true or false, and develop perfectly consistent mathematics. In this way, I can always add more rules to the system (albeit a nonstandard one).

A Godel sentence claims that itself cannot be proven in its axiomatic system. If you add another axiom, you have a new axiomatic system that can prove the Godel sentence in the original axiomatic system, but the new axiomatic system will have its own new Godel sentence.

Re: The Axiom of Choice Is Wrong (2007)

#80

Post by baddox »

lurker83256 wrote: post #71
wbhart wrote: post #61

Earlier quoted context omitted.

If they can neither be proved true or false within the system, that means their truth or falsity can be added as axioms to the system and a contradiction will never be reached. So I can assume them to be true or false, and develop perfectly consistent mathematics. In this way, I can always add more rules to the system (albeit a nonstandard one).

This is a misunderstanding. Statement A is actually true in the system, you just cannot prove that it is true. Adding an axiom specifying its falsity would be a contradiction (although you could not prove this).

Adding either the Godel sentence or its complement would ruin the consistency of the axiomatic system, because the whole point of the Godel sentence is that it claims that itself cannot be proven to be true in its axiomatic system. But you don't get to add axioms to an axiomatic system anyway, because doing so yields a new axiomatic system to which the original Godel sentence does not refer.
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