Questions tagged [set-theory]

This tag is for set theory topics typically studied at the advanced undergraduate or graduate level. These include cofinality, axioms of ZFC, axiom of choice, forcing, set-theoretic independence, large cardinals, models of set theory, ultrafilters, ultrapowers, constructible universe, inner model theory, definability, infinite combinatorics, transfinite hierarchies; etc. More elementary questions should use the "elementary-set-theory" tag instead.

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Why does forcing not obey some simple rules of Propositional Calculus?

In Cohen "Set Theory and the Continuum Hypothesis" Cohen states on page 118: "Also forcing does not obey some simple rules of the propositional calculus. Thus p may force $\neg \neg A$ ...
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$H_\kappa\models AC$

Let $\kappa$ be a regular uncountable cardinal and $H_\kappa=\{x : |trcl(x)|<\kappa\}$. I'm trying to show that $H_\kappa\models AC$. I already proved $H_\kappa\models ZFC$ minus Power Set Axiom ...
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Jech "Set Theory" exercise 8.10

Can someone check my unfinished solution of exercise 8.10 from Jech and give me a hint? Exercise: If $\kappa$ is singular, then there is no normal ideal on $\kappa$ that contains all bounded subsets ...
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Class functions in constructive set theory

In Aczel’s book (draft) on constructive set theory (https://www1.maths.leeds.ac.uk/~rathjen/book.pdf) there is a proposition (4.2.4 on page 34) regarding when a class function exists. I’m not ...
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Does $ x \notin (A\ominus B)\ominus C $ imply $ x \notin A\ominus B \ominus C $

I read a proof for the following statement: $ x \in (A\cap B)\setminus C \Rightarrow x \notin A \ominus B \ominus C $ $ \ominus = $ symmetric difference Here is the proof: $ x \in (A \cap B)\setminus ...
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Translating independence to the meta-theory. [duplicate]

The proof of the independence of $\mathsf{CH}$, goes like this: one proves in $\mathsf{ZFC}$ that assuming $\mathsf{CON(ZFC)}$ there is a model of $\mathsf{ZFC}$ in which $\mathsf{CH}$ holds and ...
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Does Z+worldly cardinals fulfill Muller's criteria to found Mathematics?

Recall Muller's criteria in Sets, Classes and Categories: page 14 for a theory that founds Mathematics. Arn't those captured simply by $$\sf Zermelo + \text {worldly cardinals exist} $$ define sets ...
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How can one refer to a "class of all classes (of a certain type)"?

Context: Self-study, working through Smullyan and Fitting's Set Theory and the Continuum Problem, chapter 5, section $\S1$. Please excuse the woolly language in the title. I am proving the statement ...
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How do we know that mathematics is independent of the definition of the cartesian product?

In my first analysis lecture, I learnt what might be the most common way to rigorously deal with functions: Firstly, ordered pairs were defined and then a function was defined as a triple $(X,Y,G)$ ...
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For an extension of $\bigcap$ that outputs a set even for the empty set, how can we formulate and prove that the output exists uniquely for any input?

When we construct math on set theory, we have two different ways to introduce the symbol $\bigcap$ for the intersection. Namely, Introduce the symbol $\bigcap$ as a partial function symbol; do not ...
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Does Z₂ Prove the iteration theorem?

iteration theorem: Let (ℕ,0,S) be the structure of natural numbers. And let W be an arbitrary set and c be an arbitrary member of W and g be a function from W into W. Then, there is a unique function ...
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Can this choice-collection principle prove the well ordering theorem without power set axiom?

Choice Collection: if $\phi(x,y)$ is a formula in which "$B$" doesn't occur free, and "$x,y$" are among its free variables; then: $$\forall A \exists B \forall x \in A \, (\exists ...
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Extendible cardinals

In Kanamori's book on large cardinals, he defines a cardinal $\kappa$ to be extendible if for any $\eta$ there is some $j:V_{\kappa+\eta} \prec V_\zeta$ with crit($j$)=$\kappa$ an $j(\kappa)$> $\...
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What 'type' of reflection principle is used in ill-founed set theories, and how does it work?

Recently I have been studying the reflection principle and non well founded set theories. As of lately I have wondered what is there relation? and how does a reflection principle fit in a non-well ...
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Questions about proof of consistency of GCH

In Jech's Set Theory, Theorem 13.20 states the following: If $V=L$ then $2^{\aleph_\alpha}=\aleph_{\alpha+1}$ for every $\alpha$. In the proof, he writes the following: Let $X\subseteq \omega_\...
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4 answers
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Is the dual of Cantor's theorem provable without choice? without excluded middle?

For the sake of concreteness let's say we're talking about ZF, though I imagine this question can be asked for any 'typical' set theory without a choice axiom (and would prefer an answer that doesn't ...
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Definition of interior point with quantifiers

Suppose that $x\in \mathbb{R}^m$ and $E\subset \mathbb{R}^m$. Definition. We say that $x$ is an interior point of $E$ iff there is an open set $G$ containing $x$ such that $G\subset E$. I was ...
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Construct the embedding of a subset of a well-order to the full set

I'm trying to build the theory of well-orders without mentioning ordinals. Define a map $f : X \to Y$ between well-ordered sets to be a simulation, if it is an order equivalence (i.e. for all $x_0, ...
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1 answer
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Can a model of $\sf ZFC$ be a subset of $V_\omega$?

Can a model of $\sf ZFC$ be a subset of $V_\omega$? If so, in which theory this is provable?
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Transfinite Recursion Theorem - Particular case - Enderton

I have the following theorem for any formula $\gamma(x,y)$: Theorem of Transfinite Recursion: Given a well-ordered set $A$ such that for any $f$ there is a unique $y$ such that $\gamma(f,y)$ holds, ...
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I have some questions about the Ross-Littlewood Paradox

TLDR at the end. Hi, I recently saw this comment given by "completely-ineffable" on the r/badmathematics subreddit. And I just wanted to make sure if I understand it correctly and wanted to ...
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2 votes
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Strength of Krein-Milman vs Dependent Choice

I am wondering about the relationship between the Krein-Milman theorem (KM) and some other weak forms of the axiom of choice (AC). I currently basically know the following: KM + BPI (Boolean prime ...
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Infinite Lexicographic Order on Bijections is Well-Order

Problem. Prove, without using $\mathsf{AC}$ if possible , that if $\alpha$ and $\beta$ are ordinals such that $\alpha$ is countable and $\beta>1$, then $\alpha^\beta$ is countable. The induction ...
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Creating an ultrapower of V by an ultrafilter over V

I've been reading about ultrapowers, and one of the most interesting things about them is when you build them using various large cardinals. For instance, taking the ultrapower of $V$ by a measurable ...
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4 votes
1 answer
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Are sets in ZFC a primitive notion?

I read everywhere (including here on math.stackexchange) that the notion of set in ZFC is primitive. To my (probably mis-)understanding, though, a primitive notion is a concept that is not defined in ...
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Is there an uncountable (infinite) set for which all of its elements can be defined with finite information?

By finite information I mean exclusion of non computable real numbers, random real numbers, etc, or any other mathematical "object" or "element" which would require in general ...
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1 answer
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Principle of induction in ZF

I'm going through Enderton's Elements of Set Theory and he's just defined the natural numbers and proved that it is an inductive set. Then he says the following: Induction principle for $\mathbf{w}$: ...
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Definition of natural numbers in ZF

I'm going through Enderton's Elements of Set Theory and he's just defined a natural number as a set that belongs to any inductive set. The problem is that this statement seems to need the notion of ...
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Defining the indexed union/intersection formally in $\mathsf{ZFC}$

An indexed set is often informally defined as $\{A_i: i \in \mathcal{I}\}$. But this is informal in $\mathsf{ZFC}$, since it doesn't know indices. Likewise, this problem carries over to the definition ...
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Can a transitive model of $\sf ZC$ satisfy false statements about modeling within it?

This comes in connection with this posting. My question here is what happens if we only make the sole change of having the mother model $M$ satisfying $\sf ZC$ instead of $\sf ZFC$. To clarify, form ...
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2 votes
1 answer
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How can a function be defined without specifying codomain in set theory?

Sets and functions, for example, did not form a category under the set theorist's definition of a function. Most often the set theorist's definition requires a function to have a set as domain of ...
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What is the cardinality of the set of all intervals of real numbers?

I am currently reading the following in "Probabilistic Machine Learning", in section 2.2.2 Continuous random variables: ... If X ∈ R is a real-valued quantity, it is called a continuous ...
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Proof that the Borel hierarchy on $\mathbb{R}$ collapses after $\omega_1$ steps.

I want to proof that the Borel hierarchy on $\mathbb{R}$ collapses after $\omega_1$ steps. However, I'm stuck and don't know where to begin, I suspect I need to use the fact that $\aleph_1$ is a ...
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$\aleph_\alpha=\sum_{\xi<\alpha}\aleph_\xi^+$, $\alpha\neq 0$ ordinal

If we suppose that $\xi\le\alpha$, since $\aleph_\xi\le\aleph_\alpha$ and $|\alpha|\le\aleph_\alpha$, then $$\sum_{\xi\le\alpha}\aleph_\xi\le\aleph_\alpha\aleph_\alpha=\aleph_\alpha.$$ It is simple to ...
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Is there an accessible $\omega$-model of this theory with $\sf ZFC$ inside $\sf ZC$?

I've once suggested the following theory as a theory that meets Muller's criteria: page 14 for a founding theory of Mathematics. I'll re-exposit it here: Define an upward membership chain as a ...
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Why doesn't $\omega ^ {\omega}$ have cardinality equal to real numbers?

One set of order type $2\omega$ is the set {1,2} $\times N$ One set of order type $3\omega $ is {1,2,3} $\times N$ Similarly, order type $\omega ^2$ would be the set $N \times N$ $\omega ^3$ would be ...
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Is this theory with nested sequence of universes of ZFC, equi-consistent with ZFC?

Based on answers to the following question. Is the following theory consistent? Add a primitive constant $V_i$ for each natural $i$, to the language of $\sf ZFC$. Axiomatize $\sf ZFC^{V_i}$ for each ...
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Sets Having Special Properties

Let $\mathbb{F}^{n}_{2},$ where $n>3$ be the vector space of all $n$-tuples over binary field $\mathbb{F}_{2}=\{0, 1\}$ and $\mathrm{A}\subset \mathbb{F}^{n}_{2}$ be such that $\textbf{b}+\mathrm{A}...
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Equivalents of "union" and "intersection" for setoids?

Context: I'm trying to figure out how much theory one can carry over from sets to setoids. A setoid $A$ here consisting of a "carrier" set $S_A$ and an equivalence relation $\sim_A$ on $S_A$....
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2 answers
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Can we have this sequence of transitive models of ZFC?

This comes in follow up of this posting. Can we have a transitive model $M_\omega$ of $\sf ZFC$ in which there exists a sequence $(M_n)_{n \in \mathbb N}$ of transitive models of $\sf ZFC$ such that $...
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Proving that the is no set of all sets with $n \in \mathbb{N}$ elements in ZFC

Let $n\in \mathbb{N}$ be a cardinal distinct from zero. How to prove in ZFC that there is not set containing all sets of $n$ elements? $\textbf{Edit 1.}$ If I suppose that the is such a set that ...
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Division by countable infinities of comparable magnitudes

Suppose we have a countably infinite set, like $\mathbb{N}$ where $|\mathbb{N}| = \infty$. Does it make sense to say that $|\mathbb{N}|/|\mathbb{N}| = 1$? How about something like this? Construct a ...
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ZFC: Sets are classes - notational question

I am following some lecture notes I found online to study axiomatic set theory, more precisely ZFC. The lectures have also been recorded and can be found here. The first lecture is mostly an ...
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1 vote
1 answer
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Is this infinite sequence of transitive models of ZFC equiconsistent with Con(ZFC)?

Is it provable in $\sf ZFC$ that the existence of a transitive model of $\sf ZFC$ implies the existence of a sequence $(M_n)_{n \in \mathbb N}$ of transitive models of $\sf ZFC$ such that $M_m \in M_n$...
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Find result of indexed set from 0 to n

For each $n \in \mathbb{N} $, let $A_n = \{0, 1, 2, 3,..., n\}$. I don't know why $\bigcup_{i\in\mathbb{N}}A_i=\{0\}\ \cup\ \mathbb{N}$ while $\mathbb{N}$ has been from zero already.
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Order-preserving injection between intervals of ordinals

Let $[\alpha, \alpha')$ and $[\beta, \beta')$ be two intervals of ordinals such that $\alpha' - \alpha \leq \beta' - \beta$. Is there an order-preserving injection $f : [\alpha, \alpha') \rightarrow [\...
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2 answers
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Resolving Russell's Paradox in NBG

Looking back at the form of the paradox we see that we now have a way out. In order to derive $ R \in R$ we would need the extra assumption that $R$ is a set... contin Topoi, a categorical analysis of ...
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2 votes
1 answer
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Closure of extender ultrapowers

Some elementary embeddings $j : V \to M$ can be defined as ultrapower embeddings by extenders. Extenders are defined using finite indices, and as I've noted in a previous question, that makes it not ...
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Are there only these two multiplicative functions with this property?

Let $f$ be an arithmetic function that is simultaneously completely multiplicative and strongly multiplicative. I want to show that the set $f(N)=\{ f(n): n \in \mathbb{N} \}$ hat at most 2 elements. ...
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Trouble with chessboard set definition

The following equation is supposed to define a set of points that produces a square grid (like a chessboard). s is the distance between each square in the grid w and h are the dimensions of the board ...
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