Intuition in Mathematics (2012) [pdf]

Re:Intuition in Mathematics (2012) [pdf] by Koshkin on Monday January 23, 2017 @10:20AM (#13462731)

Intuition in mathematics is just as important as it is in any other human endeavor. On the other hand, there are many results and methods that are deemed non-intuitive - by students and (consequently) by instructors. Some of the results are so non-intuitive that they may end up being labeled "pathological" - despite being, in fact, completely logical consequence of the assumptions, of which one of the most well-known is Axiom of Choice. Another example is that differential forms are often seen by students as very non-intuitive compared to vectors, even though they are much more useful and can be efficiently used in more general settings than vectors can.

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Re:Intuition in Mathematics (2012) [pdf] by justinpombrio on Monday January 23, 2017 @11:12AM (#13463154)

> Some of the results are so non-intuitive that they may end up being labeled "pathological" - despite being, in fact, completely logical consequence of the assumptions, of which one of the most well-known is Axiom of Choice.

That's not true. The axiom of choice is actually famous for being independent of ZFC, meaning that it cannot be proven or disproven using the usual axioms of set theory. That's why it's an axiom: you need to assume its truth without grounds.

Also, in intuitionistic logic, the axiom of choice is provably false.

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Re:Intuition in Mathematics (2012) [pdf] by waqf on Monday January 23, 2017 @12:17PM (#13463767)

Firstly, the sentence you quoted, while unfortunately ambiguous, is correct if you take "of which" as picking up "assumptions" instead of "results". That is, the assumptions, of which a famous one is AC, have counterintuitive consequences, of which the original poster does not give an example but Banach–Tarski is a commonly cited one.

Secondly, the axiom of choice may be independent of ZF (depending on whether ZF is consistent) but it is certainly not independent of ZFC :).

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Re:Intuition in Mathematics (2012) [pdf] by justinpombrio on Monday January 23, 2017 @12:23PM (#13463832)

> Firstly, the sentence you quoted, while unfortunately ambiguous, is correct if you take "of which" as picking up "assumptions" instead of "results".

Oh, that's what Koshkin was trying to say.

> Secondly, the axiom of choice is not independent of ZFC, it's independent of ZF (assuming ZF is consistent) :).

Ah, the C stands for choice...

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Re:Intuition in Mathematics (2012) [pdf] by maxiepoo on Monday January 23, 2017 @02:08PM (#13464928)

What do you mean by AC is provably false in intuitionistic logic? Intuitionistic logic + AC has as models ordinary models of classical logic since intuitionistic logic is a refinement of classical logic.

Do you mean [Diaconescu's theorem](https://en.wikipedia.org/wiki/Diaconescu's_theorem) which shows that AC implies excluded middle?

Maybe you mean there are models of intuitionistic logic where AC (and EM) is false?

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