Comment by Elliot Glazer on Must the number of smooth structures be countable...
Naively this equivalence relation is analytic, because it posits the existence of a diffeomorphism. I don’t think there is a simple trick to proving this conjecture. In the 4D case, it will probably...
View ArticleComment by Elliot Glazer on Turning linear ordering into well-ordering
@GabeGoldberg Assume $V=L,$ let $\kappa$ be inaccessible, let $G=\langle G_{\alpha} : \alpha < \kappa \rangle$ be Levy collapse generic. Let $R=\mathbb{R}^{L[G]}$ and define $f: R \rightarrow...
View ArticleComment by Elliot Glazer on What is the least $\alpha$ such that $L_\alpha$...
Regarding the cases mentioned in the last paragraph: $(2^{\omega})^L$ (as well as any $\omega$-model of $\mathrm{RCA}_0$) is either measure 0, maximally nonmeasurable (inner measure 0 and outer measure...
View ArticleComment by Elliot Glazer on Can there be a proper class of Dedekind-finite...
Has anyone checked if every set can be the image of a dually Dedekind-finite set?
View ArticleComment by Elliot Glazer on Sequential continuity and the Axiom of Choice
Yes, that fails for the indicator function of an infinite Dedekind finite set $S:$ globally sequentially usco, sequentially continuous off $S,$ discontinuous at the condensation points of $S$ (which is...
View ArticleComment by Elliot Glazer on Cohen's model yet again
I have a strong interest in this question. Proof strategies for several problems I've worked on are hung up on verifying this conjecture or minor variants of it.
View ArticleDefinability of isomorphisms between class well-orderings
Inspired by this question, I've been trying to figure out for myself the basic properties of definable class well-orderings in transitive models $M$ of ZFC: What is $\omega_1^{CK}(\mathsf{Ord})$?There...
View ArticleComment by Elliot Glazer on Is there an elementary proof of a better result...
I expanded the first part. Does the second part address what you were looking for regarding whether the players "almost always fail"?
View ArticleComment by Elliot Glazer on Why is inner model theory evidence for...
@n901 All reasonable formulations of "all sets of reals are Lebesgue measurable" are equivalent in ZF, and prove the $\sigma$-additivity of $\lambda$ (see mathoverflow.net/a/393162/109573). We get a...
View ArticleComment by Elliot Glazer on If $\omega_1$ is not inaccessible in $L$, how...
In $L^{\mathrm{Col}(\omega, \omega_1)}$ all OD sets of reals are measurable, so I don't think there's going to be any reasonable answer to this question.
View ArticleComment by Elliot Glazer on Do the surreal numbers enjoy the transfer...
I think for any complete theory $T$ with an infinite model, global choice is equivalent to any two proper class saturated (PCS) models of $T$ being isomorphic (or even just having a definable...
View ArticleComment by Elliot Glazer on Do the surreal numbers enjoy the transfer...
Yes, $n_{\alpha}$ is what I meant. And the redundant "standard" was just there as signposting, but I removed it since it might have just been confusing.
View ArticleComment by Elliot Glazer on What axioms are needed to show that the range of...
Is it even clear that Hahn-Banach implies there is a diffuse measure whose range is not $[0,1]$?
View ArticleAnswer by Elliot Glazer for Surreals and NSA: some foundational issues
Problem 1: There is a definable proper class saturated real-closed field $\mathbb{R}^*$, defined by a slight modification of your and Shelah's construction, such that there is an $\mathrm{OD}_p$...
View ArticleComment by Elliot Glazer on Reference request: The non-productivity of...
I'd call this fact "near-productivity of Lindenbaum numbers," while non-productivity refers to ZF models in which this inequality is sharp (see here for an example mathoverflow.net/a/456549/109573).
View ArticleComment by Elliot Glazer on Reference request: The non-productivity of...
This should also give the more general fact that $\aleph^*(X^{<\omega}) \le \aleph^*(X)^+.$
View ArticleComment by Elliot Glazer on What is the consistency strength of...
The definition Jech uses in his paper in which he proves $\omega_1$ can be weakly compact or measurable (each relative to such a large cardinal in ZFC) is the partition characterization: an uncountable...
View ArticleComment by Elliot Glazer on Global Choice bi-interpretable with Global...
Global well-ordering can be expressed as a sentence rather than a schema. If every subset of a class order has a minimum, so does every subclass.
View ArticleComment by Elliot Glazer on Global Choice bi-interpretable with Global...
There's probably some sort of meta-theorem you can get out of this but it would be pretty limited. The problem with what you have in mind is that we can only demand $\kappa$ be $\Sigma_n$-correct with...
View ArticleAnswer by Elliot Glazer for Is this version of Zorn's lemma provable in ZF?
No, this principle implies $\mathrm{DC}(\mathbb{R}).$ Suppose $T$ is a tree on $\mathbb{R}^{<\omega}$ with no leaves or branches. Let $\mathcal{A}$ consist of all $A \subset \omega \times \omega$...
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