Asaf Karagila
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 Oct 2 comment Does Borel's proof for existence of normal numbers make an essential use of axiom of choice? I'm not sure which point you're alluding to. Oct 2 comment Does Borel's proof for existence of normal numbers make an essential use of axiom of choice? @Wojowu: It is now known that the early school of measure theory (Borel, Lebesgue, Baire) who were very opposed to the axiom of choice did in fact use the axiom of dependent choice implicitly. But what Nate said is true. The lack-of-constructiveness is not due to the axiom of choice, as much as it is due to the law of excluded middle. Oct 2 awarded Announcer Oct 1 comment $\kappa$-support iterations of $<\kappa$-strategically closed forcing Sounds like a job for a grad student... (not me, though... I have other stuff on my plate. :-P) Sep 30 comment Why should we believe in the axiom of regularity? @David: You can't see the forest for the trees if you look at it this way! Sep 29 revised Ascertain properties of a new kind of rectilinear-convex set edited tags Sep 28 comment Cardinality of definable sets of reals @Carlo: That depends on how you formulate CH, though. :-) Sep 26 comment Who needs RCS iterations? Yes, and Kunen points out that for finite support it's the same; but for larger supports it isn't the same (Kunen 1980 edition, the chapter about iterations, exercises K2-K3), at least in terms of Boolean completions. And if the Boolean completion is not the same, then the forcings are not the same. Or am I missing something? (I'm probably missing something.) Sep 26 comment Who needs RCS iterations? Correct me if I'm wrong, but for non-finite support iteration the definition with $p\restriction\beta\Vdash_\beta p(\beta) \in Q_\beta$ is not equivalent to $1_\beta\Vdash_\beta p(\beta) \in Q_\beta$. And the latter is "the usual" definition for iteration anyway. Sep 25 revised Do semialgebraic sets depend outer semicontinuously on their defining polynomials? edited tags Sep 24 revised Ordering of large cardinals by cardinality - to _ Sep 24 comment Ordering of large cardinals by cardinality The use of $\sim$ for undecidability seems particularly strange. How about $\bot$ instead? Sep 23 comment Is the axiom schema of replacement used in algebraic number theory (or more generally outside logic) @eric: While some choice is needed for functional analysis; for statements about integers like FLT you can avoid the axiom of choice. This had been discussed before on this site, and elsewhere. Sep 22 comment Some “axiom of choice” and “dependent choice” issues @David: I guess proof by authority is not enough here, eh? :-P Alright. The Fefeman-Levy model (where the reals are a countable union if countable sets) and the Truss models for the perfect set property (where the countable union of countable sets if reals is countable, but every set is Borel). In both $\omega_1$ is singular (and it has to be to avoid the inaccessible) Sep 22 comment Some “axiom of choice” and “dependent choice” issues @David: We have very different notions of fun, in this case! :-P Sep 22 comment Some “axiom of choice” and “dependent choice” issues @David: I know. But the two models to witness that are weirddddddd. Sep 22 comment Some “axiom of choice” and “dependent choice” issues @David: No, that works fine as it is. I think you can probably squeeze it from absoluteness as well. If $f$ is a polynomial, look at $L[f]$, there choice holds, there are roots for $f$, and this is upwards absolute. Sep 22 comment Some “axiom of choice” and “dependent choice” issues @Qiaochu: If the reals are a countable union of countable sets measure theory as we know it goes out the window. Then you have to resort to working directly Borel codes and that is terrifying. Sep 22 comment Some “axiom of choice” and “dependent choice” issues @David, then no inaccessible is needed. Plain ZF. Sep 21 comment Some “axiom of choice” and “dependent choice” issues Yes. Maybe it's a hint that I should go to bed. I felt like this answer wasn't... ideally formulated, at least in terms of clarity. Thank you for the helpful comment!