# Tagged Questions

forcing, large cardinals, descriptive set theory, infinite combinatorics, cardinal characteristics, forcing axioms, ultrapowers, measures, reflection, pcf theory, models of set theory, axioms of set theory, independence, axiom of choice, continuum hypothesis, determinacy, Borel equivalence relations,...

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### A button for individual reals

Hamkins introduced the notion of a "button" in forcing. This is a set-theoretic statement that can be forced, and can never be made false by further forcing. An example is $V \not= L$. Another ...
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### References for higher descriptive set theory surveys

A student of Adi Jarden and mine attempts at generalizing results on selection principles from the Baire space $\omega^\omega$ to the higher Baire space $\kappa^\kappa$ ($\kappa$ uncountable), and ...
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### When does the generalized Cantor space embed in a $\kappa$-compact space

The generalized Cantor space is the space $2^\kappa$, with basic open sets $$[\sigma] := \{f\in 2^\kappa : \sigma\subseteq f\},$$ for $\sigma\in 2^{<\kappa}$. A space is $\kappa$-compact if ...
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### Example to $2^\kappa\nrightarrow (3)^2_\kappa$, plus closed walks of odd length?

Let $\kappa$ be an infinite cardinal. Consider the following example to $2^\kappa\nrightarrow (3)^2_\kappa$. $V$ is a set of vertices, each of which is an element of $2^\kappa$. Color the edge ...
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### About the limitation by size

This could be a big post, so I'll try to summarize my thoughts and divide them into several questions. When working in category theory, I used to choose the following definition. A category $C$ is ...
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### Absoluteness and Tree Representations

Suppose $T$ is a tree on $\omega \times \omega \times \delta$ for some ordinal $\delta$ is a homogeneous tree (with some coherent set of measures witnessing homoegeneity). ($T$ can have additional ...
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### Direct axiomatization of ordinal and cardinal numbers

Again, this question is related (**) to a previous one: in standard books on basic set theory, after stating the axioms of ZFC, ordinal numbers are introduced early on. Afterwards cardinals appear: ...
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### Some very weak statements on choice

This is a follow-up question to Does $|(X\times\{0\}) \cup (X\times\{1\})| \leq |X|$ for $X$ infinite imply ${\sf AC}$? Consider the statements $(\text{S}1)$ For any infinite set $X$ there ...
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### Frankl's union-closed sets conjecture for infinite families

This question is motivated by Frankl's union-closet sets conjecture. Let $X$ be a non-empty set. We say that a family ${\cal A} \subseteq {\cal P}(X)$ is union-closed if $\emptyset\notin{\cal A}$ and ...
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### Does $|(X\times\{0\}) \cup (X\times\{1\})| \leq |X|$ for $X$ infinite imply ${\sf AC}$?

Consider the statement For any infinite set $X$ there is an injection $\varphi$ from $(X\times\{0\}) \cup (X\times\{1\})$ into $X$. Does this imply the ${\sf AC}$?
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### A nice subcategory of the category of measurable spaces

Is there some notion of "nice" measurable spaces and "nice" maps between them which satisfies the following properties? The real line equipped with the Lebesgue $\sigma$-algebra is nice. Any ...
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### Axiom of countable choice need for the cantor-bernstein theorem [closed]

Is the axiom of countable choice need for proving the cantor-bernstein theorem?
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### Set theory and forcing from the point of view of a formal system $G^+$ of Gentzen type

There are four papers by Vladimir Alfeevich Kuznetsov, which discuss the above titled topic: (1) Some problems in set theory from the standpoint of a formal system G+ of Gentzen type. (Russian) Akad. ...