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 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 Aug 28 asked What is the set-theoretic definition of a function? Jun 30 comment Examples of results failing in higher dimensions Chaos theory suddenly got a lot more interesting to me. Jun 22 accepted What is the relationship between the Boltzmann distribution and information theory? Jun 20 comment What kind of “symmetry” is the symmetric group about? Reminds me a little of this question on MathOverflow: mathoverflow.net/questions/993/… Jun 20 comment What is the relationship between the Boltzmann distribution and information theory? Great explanation, that's exactly what I was looking for, thanks! Jun 19 asked What is the relationship between the Boltzmann distribution and information theory? May 29 awarded Nice Question Mar 3 asked Equilibrium distributions of Markov Chains Sep 6 comment Sufficient statistics vs. Bayesian sufficient statistics Cool, thanks for the link. Nice to learn that Bayes sufficiency comes from Kolmogorov. Sep 6 comment Sufficient statistics vs. Bayesian sufficient statistics I'm not sure what you mean by your last sentence -- the definition of sufficient statistic I listed comes from Wikipedia (and the point is exactly that $f(x|\theta, t)$ should not depend on $\theta$). Sep 5 asked Sufficient statistics vs. Bayesian sufficient statistics Sep 5 awarded Supporter Aug 28 awarded Scholar Aug 28 accepted Purely combinatorial proof that$(e^x)' = e^x$ Aug 28 awarded Student Aug 28 asked Purely combinatorial proof that$(e^x)' = e^x$