# Questions tagged [set-theory]

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, Boolean-valued models, embeddings, orders, relations, transfinite recursion, set theory as a foundation of mathematics, the philosophy of set theory.

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### Representing $\forall x\in \mathbb{N}$ via ZFC

Given some arbitrary predicate $\phi(x)$ (e.g. $\phi(x)="x$ is uneven" or "x is prime"), I was wondering if one can formally make sense out of statements like \begin{equation*}{\forall ...
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### What is the “Prikry–Silver collapse” when CH fails?

We all know and love Cohen reals, and we can (and often do) define the Cohen forcing as partial functions $p\colon\omega\to 2$ with finite domain. The Prikry–Silver forcing is defined as partial ...
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### Induced subgraphs of the almost-disjointness graph

Let $[\omega]^\omega$ denote the collection of infinite subsets on $\omega$, and let $$E=\big\{\{a, b\}:a,b\in [\omega]^\omega \text{ and } |a\cap b| \text{ is finite}\big\}.$$ Is every simple, ...
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### Is this theory using defined notions of classes, sets, and membership interpretable in ZFC?

The main difference with this formal theory is that it depends in an essential manner on a defined notion of class, set, and set membership $\in$, rather than the usual appraoch of leaving them ...
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### Elementary embeddings and replacement

Let $\alpha\not= 0$ be such that for every $\beta<\alpha$ there is $\beta<\gamma<\alpha$, where $V_\gamma$ is an elementary substructure of $V_\alpha$. In other words, $V_\alpha$ is a limit ...
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### How to compare three supremums of ordinals eventually writable by Ordinal Turing Machines?

This question implies that we have fixed: (i) a particular enumeration of Ordinal Turing machines; (ii) a particular way to encode an ordinal by an infinite binary sequence. The class of $$-...
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### Self-homeomorphism of Stone-Čech boundary with an isolated fixed point

$\DeclareMathOperator\bso{\beta^*\!\omega}\DeclareMathOperator\Homeo{Homeo}$Let $\bso$ be the complement of the countable discrete space $\omega$ in its Stone-Čech compactification $\beta\omega$ (some ...
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### Can we have $\sf V=HTD$? How it relates to $\sf V=HOD$?

In my search for some type-set motivated line of thought that might prove axiom of choice, I was thinking of a concept that looks like $\sf V=HOD$, but in terms of types instead of ordinals, that is ...
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### I have 2 very stupid questions about elements and subsets in ZFC [migrated]

Question 1: if X is a set, are its elements sets? (in ZFC) I believe the answer is yes. But where is the proof? Is it an axiom? Question 2: If X is a set, Y is an entity of unknow nature which has ...
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### Can Z + Ranks + Successor cardinals + Ordinal inaccessibility be equal to ZF?

[EDIT: The axiom of successor cardinals was found by an answer by Greg Kirmayer, not to be capturing the intended meaning of it, which is simply reflected by its name, i.e. the existence of a ...
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### On a connectivity property of set systems

Let $X\neq \emptyset$ be a set and let ${\cal A}\subseteq {\cal P}(X)$ with $\bigcup {\cal A}=X$. We say that $S\subseteq X$ is indivisible if for all $T\subseteq S$ with $\emptyset \neq T \neq S$ ...
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### How many category structures are possible on two sets?

For two sets $O$ and $A$, we will call a category structure a collection of functions ${\sf dom}:A\to O,\ {\sf cod}:A\to O,\ {\sf 1}:O\to A,\ \circ:A\times_OA\to A$ satisfying the usual axioms for a ...
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### How much is known about the consistency strength of toposes and topos-like categories?

It's a well-known fact that the theory of a well-pointed topos with a natural numbers object (NNO) has the same consistency strength as MacLane set theory (also known as bounded Zermelo). There are ...
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### What does the Concordant constructible universe model?

Define a ranking function $\cal R$ as: $\mathcal{R}: V \to ON; \,\mathcal {R}(x)= \min \alpha \, \forall y \in x: \alpha > \mathcal {R}(y)$ Now the constructible rank $\mathcal R^c$ of a set $X$ ...
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### “Drinking number” of a graph

Motivation. A while ago I attended a party and I only knew some, but not all, of the attendees. There were 2 kinds of drinks: beer and soda. I noticed that amongst my acquaintances, more than half ...
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### Do saturated models require choice?

Let $T$ be a first-order theory, and suppose we want to build a saturated model $\mathbb U$ of $T$. That is, we want a model $\mathbb U$ of cardinality bigger than $|T|$, saturated in its own ...
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### Ring of global sections of a functor $\mathbf{CRing} \to \mathbf{Set}$

Let $U : \mathbf{CRing} \to \mathbf{Set}$ be the forgetful functor. For any functor $F : \mathbf{CRing} \to \mathbf{Set}$ consider the class of natural transformations \mathcal{O}(F) := \mathrm{Hom}(...
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### Can we choose a sequence of Hilbert spaces?

Let $n$ be a fixed natural number. Let $H$ be a complex Hilbert space and $H_1,\dotsc,H_n$ be closed subspaces of $H$. Set $H_0\mathrel{:=}H_1\cap H_2\cap\dotsb\cap H_n$ and let $P_i$ be the ...
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### How large is the supremum of halting times of Infinite Time Turing Machines, assuming that halting times are bounded and inputs are arbitrary?

Given a fixed enumeration of Infinite Time Turing Machines (ITTMs), let $M_i(x)$ denote a computation of an $i$-th ITTM, assuming that the input $x$ is a real (an infinite binary sequence). Then the ...
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### Is a function needed here?

This question is related to my question Can we choose an element from a class?. However, I decided to create a separate question. Let $H$ be a complex Hilbert space and $H_1,\dotsc,H_n$ be closed ...
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### Descriptive set theory for computer scientists?

It seems to me that there are scattered references of deep relationships between descriptive set theory and computability theory. For one, the relationship between the Borel hierarchy and the ...
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A while ago I asked this question on mathstackexchange: Let $\mathscr{C}\subset \mathscr{P}(\Omega)$ be a class of subsets of a nonempty set $\Omega$ containing $\Omega$ and $\varnothing$. Define $\... 0answers 85 views ### Can this theory that internalize sets from external functions be consistent? I want to know if the following theory stand a chance of being consistent?$Language:$Mono-sorted first order predicate logic + primitives of equality$="$and class membership$\in"$: Define:$...
We work in $\mathsf{ZFC+V=L}$. Define a plausible theory to be a theory $T\subseteq\mathcal{L}_{\omega_1,\omega}$ in an $\omega_1$-finite language which is $\omega_1$-c.e. and $\omega_1$-finitely ...