Do choice principles in all generic extensions imply AC in $V$?
It's well-known that not all choice principles are preserved under forcing, e.g. in this answer https://mathoverflow.net/a/77002/109573 Asaf shows the ordering principle can hold in $V$ and fail in a...
View ArticleAnswer by Elliot Glazer for Is every set being cardinal definable consistent...
This is consistent. Kanovei constructed a model $M$ with an infinite Dedekind finite set of reals which is lightface projectively definable. By descending to $L(R),$ we can further assume it satisfies...
View ArticleAnswer by Elliot Glazer for Extending the product measure on $2^\omega$
There is no such measure. Suppose toward contradiction $\mu$ is such a measure.Partition $\Omega$ by the mod finite equivalence relation $\sim_{\text{fin}},$ let $X \subset \Omega$ be a choice of...
View ArticleAnswer by Elliot Glazer for Strong form of $\mathtt{PSP}$ for $K_\sigma$ sets
Here's a counterexample. Let $X$ be the set of bounded sequences, and let $A$ be the set of sequences which have only finitely many nonzero terms and achieve a strict maximum at the last nonzero term....
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 ArticleComment by Elliot Glazer on Which $L$-like principles are known to be...
Thank you for the thorough answer! Can you comment on the case of extendible cardinals? I’m always nervous about those due to their high quantifier complexity.
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...
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