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awarded  Good Question
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awarded  Notable Question
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awarded  Yearling
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Jan
22
accepted Descriptions of sets and the Axiom of Choice
Jan
22
asked Why are linear functions the natural analogue of exponential functions in a tropical semiring?
Oct
11
accepted Solving a maze by taking a random walk
Oct
9
asked Solving a maze by taking a random walk
Aug
31
comment Descriptions of sets and the Axiom of Choice
Another, perhaps less tautologous way I was thinking about the first question was this: if all sets (and thus all functions, I think) are specifiable by some finite formula, then assuming a countable alphabet, there exist only a countable number of functions. But it's typically said that there are an uncountable number of functions on the natural numbers, so I think that means that "most" of these functions require the Axiom of Choice?
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awarded  Commentator
Aug
31
comment Descriptions of sets and the Axiom of Choice
I studied basic formal set theory and model theory a while ago (at the level of a standard undergraduate course), and I can look up stuff I've forgotten (though more advanced things like forcing are beyond me), so details would be great -- I'm looking for an explanation on the rigorous side.
Aug
31
asked Descriptions of sets and the Axiom of Choice
Aug
30
accepted What is the set-theoretic definition of a function?
Aug
29
awarded  Yearling
Aug
28
comment What is the set-theoretic definition of a function?
@Asaf: Ha, you got me there! I guess I've seen similar explanations before and yours was better :-), and you gave me at least a partial understanding (though for all I know it's an incorrect partial understanding ;-)).
Aug
28
comment What is the set-theoretic definition of a function?
@Arturo: Cool, yeah, I'll make a separate question after thinking about this a bit more. One last thing: can we prove that AoC is required to prove that such a function exists?
Aug
28
comment What is the set-theoretic definition of a function?
@Asaf: I think your answer was excellent for a high-level understanding. The part where it breaks down for me is why we can make a choice function on a finite number of pairs of socks, but not an infinite number. (It makes sense if we have to define the choice function with a finite string, but I don't see how it follows if the function can be undefinable. Or maybe therein lies my confusion? -- I've never really understood what Russell's analogy is saying, so is there in fact an (undefinable) choice function on the infinite socks? I guess I need a rigorous explanation of the connection =).)
Aug
28
comment What is the set-theoretic definition of a function?
Hmm, I think I kind of see -- there are a countable number of definable functions on the naturals, but an uncountable number of possibly un-definable ones. However, now I don't see why AoC is required to select from an infinite pair of socks. Could you elaborate on how that follows from this definition of a function? Asaf wrote "It means that you may not be able to write a finite sentence to help you choose from infinitely many pairs without the axiom of choice" -- was the mention of finiteness more to provide a layman's answer, or can it be rigorized?
Aug
28
accepted Equilibrium distributions of Markov Chains