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The Axiom of Choice Is Wrong (2007)

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Re: The Axiom of Choice Is Wrong (2007)

#61

Post by wbhart »

Tloewald wrote: post #59
cygx wrote: post #46

Earlier quoted context omitted.

Ok. Not sure I buy that (eg right now, we have no reason to believe that the lifetime of the universe is finite), but that's not what I was getting at: My point is that at worst, the incompleteness theorems only imply that we won't be able to write down all the rules that govern the universe.

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).

Re: The Axiom of Choice Is Wrong (2007)

#62

Post by [deleted] »

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…

[deleted]

Re: The Axiom of Choice Is Wrong (2007)

#63

Post by [deleted] »

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…

[deleted]

Re: The Axiom of Choice Is Wrong (2007)

#64

Post by bananabiscuit »

This problem has exactly the wrong setup for using th the axiom of choice:

First, the axiom of choice requires that you have a countable number of sets that you are choosing elements from, but there are undoubtably many of the equivalence classes that he described [0]. So the author is using something stronger than the axiom of choice to arrive at his paradox.

Second, if you actually are in a situation where you have to choose from a countably infinite number of sets, you only need the axiom of choice if there is no selection rule for choosing an element available. In this case there is a rule you can use, namely: select the sequence in which the "finite prefix" is all zeros.

[0]: the number of equivalence classes is uncountable because there is a 1:1 relation between the equivelance class and an the infinite sequence that is common to all the sequences in the equivalence class once theirs uncommon prefixes have been truncated.

Re: The Axiom of Choice Is Wrong (2007)

#65

Post by n4r9 »

bananabiscuit wrote: post #64

This problem has exactly the wrong setup for using th the axiom of choice: First, the axiom of choice requires that you have a countable number of sets that you are choosing elements from, but there are undoubtably many of the equivalence classes that he described [0]. So the author is using something stronger than the axiom of choice to arrive at his paradox. Second, if you actually are in a situation where you have…

> the axiom of choice requires that you have a countable number of sets that you are choosing elements from

You're referring to the "axiom of countable choice", which is a different axiom. The Wikipedia entry for the axiom of choice makes it clear that the number of sets can be uncountable.

> select the sequence in which the "finite prefix" is all zeros

This doesn't really make sense as a selection rule. The size of the prefix can vary between pairs of members from the same equivalence class.

Re: The Axiom of Choice Is Wrong (2007)

#66

Post by yequalsx »

bananabiscuit wrote: post #64

This problem has exactly the wrong setup for using th the axiom of choice: First, the axiom of choice requires that you have a countable number of sets that you are choosing elements from, but there are undoubtably many of the equivalence classes that he described [0]. So the author is using something stronger than the axiom of choice to arrive at his paradox. Second, if you actually are in a situation where you have…

The axiom of choice applies to index sets of arbitrary size. Indeed there is an axiom of countable choice for those who don't like the axiom of choice.

Re: The Axiom of Choice Is Wrong (2007)

#67

Post by TimonKnigge »

bananabiscuit wrote: post #64

This problem has exactly the wrong setup for using th the axiom of choice: First, the axiom of choice requires that you have a countable number of sets that you are choosing elements from, but there are undoubtably many of the equivalence classes that he described [0]. So the author is using something stronger than the axiom of choice to arrive at his paradox. Second, if you actually are in a situation where you have…

> First, the axiom of voice requires that you have a countable number of sets that you are choosing elements from, but there are undoubtably many of the equivalence classes that he described [0].

No it does not? The axiom of countable choice [0] is a strictly weaker axiom.

[0] https://en.wikipedia.org/wiki/Axiom_of_countable_choice

Re: The Axiom of Choice Is Wrong (2007)

#68

Post by yequalsx »

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 are uncountably many polynomial over the reals. There are uncountably many functions from N to N. The premise of your question is incorrect.

Re: The Axiom of Choice Is Wrong (2007)

#70

Post by waqf »

yequalsx wrote: post #68
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 are uncountably many polynomial over the reals. There are uncountably many functions from N to N. The premise of your question is incorrect.

OP never claimed there were countably many functions from N to N, only that there are countably many Turing machines which is true.

By "polynomials" OP doesn't mean polynomials over the reals but over, for example, the rationals – the point is that there's a countable subset of R that's algebraically closed.

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