Comment by Elliot Glazer on What can be the measure of a Vitali set?
Yes. I've added a note on that at the end.
View ArticleComment by Elliot Glazer on Building the real from Dedekind finite sets
You’re right. I’ve rephrased it in terms of $[A]^{<\omega}.$
View ArticleComment by Elliot Glazer on Does Well-Ordered Interval Power Set "WOIPS"...
See mathoverflow.net/a/471784/109573
View ArticleComment by Elliot Glazer on Must strange sequences wear Russellian socks?
We’ll justify $3 \rightarrow 1$ by the contrapositive. Fix $s \in \prod A_i.$ Identify $\bigsqcup A_i \setminus \{s_j: j<\omega\}$ with $\{t \in \prod A_i: \exists ! j (s_j \neq t_j)\}.$ The latter...
View ArticleAnswer by Elliot Glazer for Consistency of a strange (choice-wise) set of reals
The existence of such a set follows from $``\mathbb{R}$ is a countable union of countable sets.$"$ Let $\mathbb{R} = \bigcup_{n<\omega} S_n,$ each $S_n$ countable. Let $T_n = \{x \in \mathbb{R}:...
View ArticleAnswer by Elliot Glazer for How hard is it to get "absolutely" no amorphous...
Turning my comment into an answer, an $X$ which is the universe of any finitely axiomatized theory with an infinite model must be orderable, and there must be a bijection between between $X$ and $X^2.$...
View ArticleAnswer by Elliot Glazer for The difference between Baire 2 and 'effectively...
It's provable in ZF that every Baire-2 function is effectively Baire-2. It suffices to prove the following:(ZF) There is an explicit function which maps each Baire-1 function $f: \mathbb{R} \rightarrow...
View ArticleAnswer by Elliot Glazer for Consistency of a strong Fubini type theorem for...
ZFC refutes this principle. Let $\kappa=\text{non}(\mathcal{L}),$ i.e. the least cardinality of a set of reals of positive outer measure. Let $X \subset [0,1]$ be such that $|X|=\kappa$ and...
View ArticleAnswer by Elliot Glazer for The existence of a maximal “cross-sectional”...
No. Suppose $\mathcal{F}$ is such a filter. Clearly each $X \in \mathcal{F}$ has positive measure intersection with every positive-length interval. Then neither $[0,1/2]$ nor $[1/2, 1]$ are in the...
View ArticleAnswer by Elliot Glazer for If $A, B$ is a non-trivial partition of $S^1$, is...
This question is explored in great generality by Laczkovich inLaczkovich, Miklós, "Two constructions of Sierpiński and some cardinal invariants of ideals", Real Anal. Exch. 24(1998-99), No. 2, 663-676...
View ArticleAnswer by Elliot Glazer for Ideal-like filter on a ring not generated by ring...
This does not hold for all commutative rings, but it does hold for Noetherian rings and for valuation rings (assuming the convention that filters don't contain $\emptyset,$ or else $\mathcal{P}(R)$ is...
View ArticleAnswer by Elliot Glazer for What notable theorems cannot be automatically...
$\text{ZF}+ \text{AC}_{\omega}$ is not $\Sigma^1_4$-conservative over ZF and ZF + DC is not $\Sigma^1_4$-conservative over $\text{ZF}+ \text{AC}_{\omega}.$An example of the former: the sentence...
View ArticleAnswer by Elliot Glazer for The Parity Principle and $\mathbf{C}_2$ (choice...
Over ZFA, the Parity Principle is strictly weaker. We'll show it follows from Multiple Choice, the assertion that for any family of nonempty sets $\mathcal{F},$ there is $g: \mathcal{F} \rightarrow...
View ArticleAnswer by Elliot Glazer for A surjection from square onto power: Is limit...
In the Feferman-Levy model $M$ for $\mathbb{R}$ being a countable union of countable sets, there is $X$ with $|\mathcal{P}(X)| =^* |X^2|$and $\aleph(X)=\aleph^*(X)=\omega_1.$In this model, we can...
View ArticleAnswer by Elliot Glazer for Gently changing measure
To question 1, there is such a pair. This is a minor reworking of Ashutosh's example you linked.Start with $L,$ add an $\omega_1$-sequence of random reals $X=\langle r_{\alpha}: \alpha<\omega_1...
View ArticleAnswer by Elliot Glazer for Is "the purely probabilistic version of...
In any model of ZFC where there is no such set for $c=1$ (e.g., Friedman's model mentioned in Gro-Tsen's answer), there is no $S \subset [0,1]^2$ such that all vertical slices $S_x$ are null and all...
View ArticleAnswer by Elliot Glazer for Is Global Choice conservative over Zermelo with...
Edit: Here's my preprint filling in the details of the construction in this answer: https://arxiv.org/pdf/2312.11902.pdfGlobal Choice is not conservative over ZC. We'll build a model of ZC which...
View ArticleWhich $L$-like principles are known to be relatively consistent with large...
For which of the standard large cardinal axioms $\varphi_{LC}$ and which $L$-like principles $\psi$ (e.g. GCH, $\mathrm{V}=\mathrm{HOD},$ the ground axiom, and various diamond and square principles) is...
View ArticleAnswer by Elliot Glazer for Is it consistent with ZFC that the real line is...
Here's a ZF proof that if $S$ is a chain of sets with $\bigcup S = \mathbb{R},$ then there is $X \in S$ which contains a countable set dense in some nonempty open set.If there is $X \in S$ such that...
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 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 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 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 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 $A \subset \omega \times \omega$...
View ArticleIs there an irrational number which in every base, has only 2 digits appear...
Is there any irrational number which in every base, has only 2 digits appear infinitely often? What if we just restrict to non-Liouville irrational numbers (like $\pi$), and only consider bases 3 and...
View ArticleComment by Elliot Glazer on How much collapsing is enough to ensure that...
Yes, that’s right. I've edited this in.
View ArticleComment by Elliot Glazer on How much collapsing is enough to ensure that...
Good point. I really want to define $\lambda_{\mathbb{P}}$ to be the least $\lambda$ such that every algebra $\mathcal{A}$ which collapses $\lambda$ also forces a $(M, \mathbb{P})$-generic, and...
View ArticleComment by Elliot Glazer on Can we always force on models of ZF-Reg. to get...
If there is a proper class of Quine atoms, then $H_{\alpha}$ is a proper class for all $\alpha \ge 2.$ And this fact is preserved under any reasonable concept of forcing over foundationless models.
View ArticleComment by Elliot Glazer on What happens if we restrict Replacement to...
Well, we can canonically code $p$ by the set of all subsets of $|TC(\{p\}) \cup \omega|$ which code $p.$ One of several tricks which should allow us to build a synonymy between the two theories I...
View ArticleComment by Elliot Glazer on Can this extension of ZC evade having distinct...
@JoelDavidHamkins There are many such examples. Consider the tight theories $Z_n.$ Use the synonymy of $Z_1$ with $\mathrm{ZF} - \mathrm{Inf} + \neg \mathrm{Inf} +\mathrm{TC}$ to treat all as...
View ArticleComment by Elliot Glazer on Is this slim set theory bi-interpretable with...
$L$ adjoined by an Ord-sequence of Cohen reals satisfies your theory and is not bi-interpretable with any model of ZF.
View ArticleAnswer by Elliot Glazer for Is a nonempty proper subset of the reals that has...
Yes. Suppose $\emptyset \subsetneq A \subsetneq \mathbb{R}$ is a measurable set with countably many translates. Notice $A^c$ also has these properties. Let $X=A$ or $X=A^c.$ The equivalence relation $x...
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