# 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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### If $A\subseteq B$, $B-A$ is infinite, A is countable, then $B-A\sim B$

Question is as stated in the title. Here $\sim$ means "as numerous as", which is a stronger condition compare to the just being "coutable/uncountable", i.e. different uncountable ...
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1 vote
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### non-wellordered linear order that doesn't contain a copy of $\omega^*$ in ZF?

Obviously a wellorder cannot contain a copy of $\omega^*$ (the dual order of $\omega$), and this can be proven in ZF. In ZFC, it is easy to prove any linear order which is not a wellorder does contain ...
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### What is cardinality of $\mathbb{R}^{\mathbb{R}}$? [duplicate]

What is cardinality of $\mathbb{R}^{\mathbb{R}}$ ? I think it is $\aleph_1$ or is it just continuum? And what is the argumentation for it?
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### Find all positive numbers $m$ and $n$ to satisfy properties

Find all positive integers $m$ and $n$ such that there exists finite sets of integers $X$ and $Y$ where, for every $x\in X$, $x+m \in Y$, and for every $x\in Y$, $x^2 - n \in X$. Currently, I have ...
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### Cantor's Diagonalization Process(CDP) and North-east to South-west Diagonalization Process(NSDP)

While looking at the function $f:\mathbb{N}\rightarrow \mathbb{N} \times \mathbb{N}$, I accidentally made an inversion in my labelling for $f(x)$ What I mean is, we go with a zig-zag path in CDP(...
48 views

### Show $f^n=\mathrm{id}_A$ for some $n$ where $A$ is a finite set and $f : A \to A$ is a bijection

Let $A$ be a finite set and let $f: A \to A$ be a bijection. Show that $f^n=\mathrm{id}_A$ for some $n > 0$. I don't really understand how to prove it. And what does the notation $\mathrm{id}_A$ ...
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### Prove that A\(A\B) is a subset of B for all subsets A and B of x

Let X be a set. For subsets C and D of X, we define C\D = { $x\in X$ : $x\in C$ but $x \notin D$ } Prove that A(A\B) is a subset of B for all subsets A and B of X. My try: x∈A(A\B)is equivalent to 𝑥 ∈...
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First of all, I would like to say that I've only recently started researching infinitary logics. I have read some segments of Barwise - related papers on the topic, and I could not find anything ...
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### cardinals such that $\alpha < \alpha ^ \beta$

Let $F$ be a (class) function that assigns at each cardinal $\alpha$ the minimum cardinal $\beta$ such that $\alpha < \alpha ^ \beta$. I have to prove that $\beta \in \text{Im}(F)$ iff $\beta$ is ...
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### Is it possible to define powersets without first defining the subset relation?

This ties back to a previous question I asked, which was about the notion of "to define $X$, you must first define $Y$". The current question is about a specific application of that notion. ...
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### Does the Pigeonhole Principle apply for sets of Infinity?

I once heard as a kid that any random assortment of letters is a name for some number. The thinking there was because there are infinite numbers, any random unnamed number would be labeled by an ...
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### Proving that product of two-point sets is compact

I am reading Jech's Axiom of Choice, and I want to prove: For a non-empty set $I$, if $\{0,1\}^I$, the generalized Cantor space, is non-empty compact, then $\prod_{i\in I}A_i$ where $|A_i|=2$ for all ...
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### Problems with the exact formal definition of the Brauer group of a field [duplicate]

My question is not about the algebra of this thing, but related to formal set theory. I think that usually one will read that the Brauer group of the field K is made of equivalence classes of algebras ...
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### Square principle and continuum hypothesis

I know that the Jensen $\diamondsuit$-principle implies tha continuum hypothesis, but the $\square_{\omega_1}$ implies the continuum hypothesis? Searching about these principles I found the $\clubsuit$...
134 views

### "Infinitely deep" sets

In the Von Neumann Universe, a generation such as $V_{\omega+\omega}$ seems to occur "after" applying the power set operation infinitely many times, so intuitively it feels like there must ...
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### How large is the supremum of all countable ordinals that have absolute definitions?

This answer contains the following statement: it's easy to check that $\omega^2$, $\omega^\omega$, $\epsilon_0$, and even "big" countable ordinals like $\omega_1^{CK}$ (= the first "...
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### When does $|X^n|\leq |k^X|$ without the axiom of Choice?

We're working in $\sf{ZF}$ set theory, we assume $n$ is a finite cardinal, and the set $X$ is not finite. My question is: what's the smallest cardinal $k$ for which $\sf{ZF}$ proves $|X^n| \leq |k^X|$?...
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### $(\kappa \times 2,<_{lex}) \cong (\kappa,\in )$ for cardinal numbes $\kappa$.

A cardinal number $\alpha$ is a ordinal number such that $\forall \beta < \alpha : \beta \not\approx \alpha$. Now I want to show that for a cardinal number $\kappa$ the set $\kappa \times 2$ with ...
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### Modifying the axiom of reducibility in Friedman's K(W)

Let RED2 be the axiom schema $$\forall \bar{x} {\in} U (\phi \to \exists u {\in} U \phi[U := u])$$ where $\phi$ is a formula with free variables among $\bar{x}$. In Friedman's K(W), how does replacing ...
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### In the ZFC Set Theory. In the Axiom of Infinity, Why the emptyset must be an element of the set? What's the point of that? [closed]

$$\exists S(\emptyset\in S\land \forall y\in S((y\cup\{y\})\in S))$$ $$\exists S(\forall y\in S((y\cup\{y\})\in S))$$ Why must need the condition $\emptyset\in S$?
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### Is this the Axiom of Infinity?

While studying elementary Set Theory, I came across the Axiom of Infinity. This comes before the section on ZFC, so I'm not convinced that it is necessarily the same as the typical definition. My ...
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### Show that if $\delta$ is a ordinal number then $\bigcup{\delta} \neq \delta \iff \exists \gamma \in \mbox{Ord} : \gamma + 1 = \delta$.

An ordinal number $A$ is a pair $(A,<)$ with a well-ordering $<$ such that $\forall a \in A$ we have $a = A_a = \lbrace x \in A | x < a \rbrace$. Now I want to show that for an ordinal number ...
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