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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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Lemma for proving Zermelo's theorem

I'm trying to understand the following lemma in Bourbaki's set theory (chapter III, §2,no. 3,Lemma 3): Lemma 3: Let $E$ be a set, let $S$ be a subset of $P(E)$, and let $p$ be a mapping of $S$ into $...
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30 views

Can we have a Choice-AntiChoice chain?

Can we have a $\kappa$ sized sequence $\mathcal S$ of transitive domains $\mathcal M_i$ of models of $``\text{ZF + negation of choice}"$, where $\kappa$ is inaccessible, such that for all $i < \...
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2answers
30 views

Why do we need axiom of choice to prove that there does not a exist definition of $P(A)$, defined for all subsets $A \subset [0, 1]$

I'm reading "A First Look At Rigorous Probability" by Jeffrey S. Rosenthal. On chapter one there is a proof which I can't fully understand. Suppose, to the contrary, that $P(A)$ could be so defined ...
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1answer
41 views

Borel Hierarchy

i'm in trouble with an exercise on Kechris, Classical Descriptive Set Theory. The Theorem 22.4 shows $\Sigma_\xi^0(X)\neq\Pi_\xi^0(X)$ for each ordinal $\xi\lneq\omega_1$ and uncountable polish space $...
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0answers
69 views

How is it possible to have a model of a set theory? [duplicate]

I am trying to understand the basics of model theory. Before getting too deeply into it, I would like to know how it is even possible to construct a model, i.e. a structure that satisfies axioms of a ...
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1answer
48 views

Why is there a unique ordinal $\alpha$ for every infinite cardinal $\kappa$ such that $\kappa = \aleph_\alpha$?

For finding the $\alpha$ I literally can't get any further than writing out the definitions. For the second part: Suppose $\aleph_\alpha = \aleph_\beta$ and $\beta \neq \alpha$. Then either $\alpha \...
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1answer
58 views

Is there an embedding theorem for non-second-countable manifolds?

It is well known although I don't know how to prove that any second-countable topological manifold of dimension $n$ can be embedded into $\mathbb{R}^{2n}$(we consider Hausdorff manifolds only). I ...
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1answer
65 views

Why is the “axiom of extension” an axiom? [duplicate]

I think this is a definition of = because it explains the meaning of the new symbol =. However, I wonder why the set theory thinks this as an axiom rather than a definition.
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1answer
61 views

Linearly ordering the power set of a well ordered set with ZF (without AC)

As the title says, my question is, how one can use only ZF-theory to prove that the power set of A, whereby (A, <) is a well-ordering, can be linearly ordered?
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1answer
53 views

finding bijection such that $|\{ x\in A : x \neq f(x)\}| =\mathfrak{c} $

Let $|A| = 2^{\mathfrak{c}}$. I am finding function $f$ is bijection from $A$ to $A$ such that $|\{ x\in A : x \neq f(x)\}| =\mathfrak{c} $. Any ideas? I will try to prove it later.
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Which classes of subsets are absolute under forcing?

Let $X$ be a ‘definable’ Polish space (in the day-to-day, not necessarily the set-theoretic sense, though possible the latter generalises this). Consider a complexity class $\Gamma(X)$ of subsets of $...
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1answer
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Countable set of sequences - is there a sequence where every element is greater or equal?

I'm looking at this question: $\mathrm { N } ^ { \mathrm { N } } : = \{ f : f : \mathrm { N } \rightarrow \mathrm { N } \}$ is the set of all sequences of natural numbers. Let $A= \left\{ f _ { n }...
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Cardinality of k-Partitions [duplicate]

Let $k,\nu$ be two infinite cardinals, $\nu \le k$. Evaluate $$|\{\mathcal{P}:\text{$\mathcal{P}$ is a partition of $k$ in $\nu$ pieces }\}|$$
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1answer
23 views

Cardinality of k-bijections

Suppose that $k$ is an infinite cardinal, how can I prove that $|\{f:k\rightarrow k :\text{$f$ is a bijection}\} | = 2^{k}$ ?
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2answers
28 views

What does: “for all free variables shown” mean in Set Theory.

I am reading the definition of what it means for a class $A$ to model a formula of the Language of Set Theory. It begin, Let $A$ be a class and $\phi(x_1,\ldots, x_n)$ be a formula of the Language ...
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0answers
49 views

What goes wrong shooting a club on $\kappa >\aleph_1$?

I'm reading through Jech's book, and in the section Stationary Sets in Generic Extensions (pages 444 and 445), he remarks that the poset used in shooting a club through a stationary $S\subseteq\...
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1answer
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Avoiding choice in proving “Sequential compactness implies Lebesgue Number Lemma”

The standard proof can be found in ProofWiki. From what it is shown on that, it uses the Axiom of Countable Choice when choosing the subsequence $\{x_n\}$ to produce a contradiction. And normally, as ...
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2answers
307 views

Why can't we just add “nothing else is a set” as an axiom?

The axioms of ZF define what a set is by: $\omega$ is a set If $x$ and $y$ are sets, then $\{x, y\}$ is a set If $x$ is a set, then $\bigcup x$ is a set If $x$ is a set, then $\mathcal{P}(x)$ is a ...
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1answer
43 views

Possible use of choice in proving “Compactness implies limit point compactness”

A standard proof can be found here. Basically, the idea is to prove the contrapositive: Let $A\subseteq X$. If $X$ is compact and $A$ doesn't have any limit point, then A is finite. Since A has ...
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0answers
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weakly normal filters

Kanamori (Ultrafilters over Uncountable Cardinals) in his Phd Thesis defines a filter $\mathcal F$ as weakly normal whenever every function $f$ such that $\{\xi<\kappa\mid f(\xi)<\xi\}\in\...
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What does it mean to draw a point from a distribution with non-singleton atoms?

This is a philosophical/pedagogical question. Ordinarily, if $(\Omega,\mathcal{A},\mu)$ is a probability space, then "drawing random point $X\in\Omega$" means that for each $A\in\mathcal{A}$, the ...
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2answers
130 views

Well-definedness of uncomputable functions.

I have been reading about Rayo's function and uncomputable functions in general, and have gotten very confused. There is apparently concern over the well-definedness of Rayo's function, but I never ...
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1answer
90 views

Serre says open covers do not form a set, why? Directed sets and limits.

The following selection is from Serre's FAC (Chapter 1, §3, n°22, page 26). The relation `$\mathfrak{U}$ is finer than $\mathfrak{V}$' (which we denote hencforth by $\mathfrak{U} \prec \mathfrak{...
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1answer
38 views

Can elementary embeddings make ordinals jump over strongly inaccessible cardinals?

Let $j:V\to M$ be an elementary embedding. Assume $\xi\geq crit(j)$ is strongly inaccessible, and $\alpha<\xi$. Is it possible that $j(\alpha)\geq\xi$ ? If yes - does the additional assumption $...
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1answer
79 views

What is the status of the Axiom of limitation of size? (adrift for almost a century now)

On reviewing the wiki article Axiom of limitation of size I come away with the impression that there are issues surrounding this 'maybe too powerful' principle/heuristic/doctrine/axiom that haven't ...
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0answers
33 views

The use of the Axiom of Choice in this proof of the sequence lemma [duplicate]

Here is a proof of the sequence lemma I lifted from this post. It seems to me naively that the axiom of choice is being used to selected the $x_n$'s. Is the axiom of choice being invoked in this ...
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1answer
76 views

What is the $\text{♢-Axiom}$? (now known to be the 'Diamond Principle / Axiom')

While skimming over some research papers I found this abstract Mathematics > Operator Algebras (link here) Large irredundant sets in operator algebras Clayton Suguio Hida, Piotr Koszmider (...
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0answers
21 views

Cardinality of initial segments [duplicate]

I've seen in several places that that given a well-ordered set, $X$, any initial segment $S = \{x | x < a\}$ may not be order isomorphic to $X$. Is it still possible that there is a non-order-...
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1answer
53 views

Models of $\mathrm{V\neq HOD}$

This is basically a proof verification question. I want to find a model $\mathrm{ZFC+V\neq HOD}$ where $\mathrm{HOD}$ is the class of heriditarilly ordinal definable sets. The idea is to add one Cohen ...
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1answer
67 views

Ultraproduct of simple structures

Definition. 1) A complete first-order $\mathcal{L}$-theory $T$ is said to be simple if each type does not fork over some subset $A$ of its domain where $|A|\leq |T|$. An $\mathcal{L}$-structure $\...
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1answer
45 views

Let $X$ and $Y$ be no empty sets, then $|X|\le |Y|$ if and only if there exists $f\colon Y\to X$ onto

Theorem. Let $X$ and $Y$ be no empty sets. Then $$\big(|X|\le |Y|\big)\quad\text{if and only if}\quad\big(\exists\;f\colon Y\to X\;\text{onto}\big).$$ Proof.$(\Rightarrow)$ Let $|X|\le|Y|$, then ...
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2answers
92 views

Let $X$ and $Y$ be set, then $|X|\le |Y|$ or $|Y|\le|X|.$

Theorem. Let $X$ and $Y$ be set, then $|X|\le |Y|$ or $|Y|\le|X|.$ Proof. We consider the family $$\mathcal{F}=\left\{(A,f)\;\middle |\; A\subset X,\;f\colon A\to Y\;\text{injective}\right\}.$$ We ...
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1answer
117 views

Meaning of “absolute zero” in set theory

With great pleasure I am reading "Believing the axioms" by Penelope Maddy after someone linked to it here on MSE. (https://www.jstor.org/stable/2274520). However on page 495 there is a sentence I don'...
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1answer
75 views

Measurable cardinals: non-trivial two-valued measures

while doing some exercises about measurable cardinals, I got stuck on this one: If $κ$ is the minimal cardinal that carries a non-trivial two-valued measure, then how can one prove that $κ$ is ...
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1answer
101 views

Reining in the Axiom of Power Set in ZF

Given the powerset operator $\mathit P$, we have the following mapping $\tag 1 \mathcal \Phi: \mathbb N \to \Phi(\mathbb N) $ $\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \...
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1answer
44 views

Equivalent condition for a cardinal number to be of cofinality $\aleph_0$

Let cf$(\alpha)$ denotes the co-final of the transfinite cardinal number $\alpha.$ For every successor cardinal $\alpha$ we have cf$(\alpha)=\alpha\neq \aleph_0.$ Thus cf$(\alpha) = \aleph_0,$ implies ...
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27 views

restricted 3- space can be a field?

if the topological boundary set is non-empty; we can put a field structure. for example 3-space is not a field but R^3-{Ox,Oy,Oz} is a field. i want to know can this be a theorem?
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Prove that (if $R$ is set-like, then its transitive closure is set-like) implies the axiom of replacement

This is exercise I.9.6 from Kunen's Set Theory (2011). $R$ is set-like on the class $A$ iff $y \in A$ implies $\{ x \in A: xRy \}$ is a set. I am not so sure about this, but it is implied that this ...
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1answer
91 views

Constant functions in set-theory

I need some help with an exercise in set theory, which is about certain constant functions. Let $S$ be a stationary subset of a regular uncountable cardinal $\lambda$. Given an ordinal $\alpha$, let $...
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1answer
65 views

Question about the proof of the Recursion Theorem

I have been studying the book, Introduction to Set Theory written by Hrbacek and Jech. In this book the Recursion Theorem is stated as follows: For any set $A$, any $a\in A$, and any function $g: ...
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What features of a logic make possible the proof of Downward Lowenheim-Skolem?

The Downward Lowenheim-Skolem Theorem asserts that if a countable first-order theory has an infinite model, then it has a countable model. Although associated with first-order logic, the result also ...
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Is there any “reasonable” definition of cardinality which agrees with the standard definition on finite sets but disagrees on infinite sets?

Sometimes I talk about math to my friends, and some (with an engineering background) aren't used to the idea that in math, we have to define stuff. For example, they may not be used to the idea that $\...
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1answer
56 views

How is this definition of closedness compatible with the order topology?

Let $\kappa$ be a limit ordinal. Taken from the definition of a closed unbounded set, we say a subset $C\subseteq\kappa$ is closed in $\kappa$ if and only if $\sup(C\cap\alpha)=\alpha<\kappa\...
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2answers
78 views

Example of a set of real numbers that is Dedekind-finite but not finite

Without assuming $AC$, can we find an explicit example of a subset of $\mathbb{R}$ such that it is not finite but it is Dedekind-finite?
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68 views

Proof of the Solovay Theorem in Jech

The Solovay Theorem says: Let $\kappa$ be a regular uncountable cardinal. Then every stationary subset is the disjoint union of $\kappa$-many stationary subsets. [Jech, Theorem 8.8, p. 95] One ...
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2answers
32 views

Let $α$ and $β$ be ordinals such that $α>1$ and $β\neq0$. Let $γ$, $ζ$, and $η$ be ordinals st $γ<β$, $ζ<α$, and $η<α^{γ}$. Prove $α^γ ζ+η<α^{β}$.

Let $\alpha$ and $\beta$ be ordinals such that $\alpha>1$ and $\beta\neq0$. Let $\gamma$, $\zeta$, and $\eta$ be ordinals such that $\gamma<\beta$, $\zeta<\alpha$, and $\eta<\alpha^{\gamma}...
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1answer
51 views

Finite chain condition - Variation of Martin's Axiom statement

In the following $k$ and $w$ will be cardinal numbers. Consider the classical statement $MA(k)$: For any partial order $P$ satisfying the countable chain condition (hereafter $ccc$) and any family ...
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1answer
32 views

The relation between the limit cardinal $\alpha$ and a sequence of cardinal numbers strictly less than $\alpha$

For a limit cardinal $\alpha$ can we find a sequence of sets $(X_n)$ with $card X_1< card X_2<...< card X$and $card X= card X_1+ card X_2+...?$
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2answers
47 views

Is every limit cardinal $\alpha$ of the form $\alpha=card(X)+card(P(X))+cardP(P(X))+…$, for some set $X$?

A limit cardinal is a cardinal number $\alpha$ such that if $\beta<\alpha,$ then there is a cardinal number $\gamma$ with $\beta<\gamma<\alpha.$ Now if $\alpha$ is a limit cardinal then can ...
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1answer
56 views

$\alpha + \beta = \sup_{v<\gamma} (\alpha + \beta_{v})$? [closed]

On page 124 of Introduction to Set Theory, Jech claims that ordinal functions below are continuous in the second variable: If $\gamma$ is a limit ordinal and $\beta= \sup_{v<\gamma} \beta_{v}$ then ...