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 ...

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1answer
62 views

Hereditarily finite/countable/small sets in ZF

I want to determine which ZF axioms are satisfied by each of these. HF It seems obvious to me that extension, empty-set, pair-set, union, and power-set hold for these. I'm not sure about replacement ...
0
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1answer
55 views

inner model notion

I am confused by some notation, and perhaps theory, associated with inner models. I have seen an inner model, $M$, of $N$ defined by the formula, $F$, as given by the pair: $M = \langle n, e^{m ...
3
votes
1answer
64 views

Transitive models of ZF without Power Set

I've read at different places that there are transitive classes that don't satisfy the power set axiom but I don't understand these classes look like. Is there a simple example? It's easy to break ...
0
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1answer
27 views

Large ordered family of set inclusions complete?

I'm not an expert on set theory, so this might be trivial: Let $\dots \subseteq X_{\alpha} \subseteq X_{\alpha+1} \subseteq \dots$ be a chain of set inclusions, indexed by ordinals(!), s.t. ...
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1answer
30 views

Set Theory: Graphs and $k$-Colorings

Let $G = (V, E)$ be a graph with $V = \omega$. Show that if for all $n < \omega$, the graph $G_{n} = (n, E \cap [n]^{2})$ is $k$-colorable, then $G$ is $k$-colorable. I know how to prove this ...
3
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2answers
73 views

$\omega \ \oplus 2 \neq 2\ \oplus \omega $ in NBG set theory

I'm struggling to rigorously prove $\omega \ \oplus 2 \neq 2\ \oplus \omega$ in context of NBG set theory. I haven't really seen a full proof anywhere besides the basic structure. Please direct me ...
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0answers
66 views

Minimal model of ZF with $0\sharp$

We know that the constructible universe $L$ is an absolute and minimal model of ZF (every standard model of ZF contains "an" $L$, and it is actually the same $L$ for all of them). It is also my ...
2
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1answer
55 views

Properties of the constructible closure of a set

Given any set $A$ define $L_0(A)=\{A\}\cup \operatorname{tr cl}(A)$ $L_{\alpha+1}(A)=\mathcal{D}^+(L_\alpha(A))$ $L_\gamma (A)=\bigcup_{\alpha<\gamma}L_\alpha(A)$ if $\gamma$ is a limit ordinal. ...
2
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1answer
62 views

Exercise on relative constructibility

Given a set $A$ in Kunen's set theory book (page 143) we can find the following definition of the relative constructible univers $L[A]$. $L[A](0)=\{A\}\cup \operatorname{tr cl}(A)$ ...
5
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2answers
116 views

Proving the Powerset Axiom for hereditarily finite sets

Consider $\mathsf{ZF}$, and relace the Axiom of Infinity with its negation. This gives us the theory of hereditarily finite sets. Its universe is $V_\omega$. Intuitively, I feel that I can construct ...
1
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1answer
50 views

Problem on infinite cardinal number

If $e$ is an infinite cardinal number and $d$ is a cardinal number satisfing $2 ≤ d ≤ 2^e$. I need to prove the following $$d^e= 2^e$$ Any help will be appreciated. Thank you in advance. .
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1answer
100 views

Existence of infinite sets in ZF

If I need the existence of an infinite set for a proof, for instance the smallest set $x$ such that $(\emptyset\in x)\land(\forall y,z\in x)(y\cup\{z\}\in x)$, how do I know such set exists in a model ...
7
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1answer
83 views

Fixed points in the enumeration of inaccessible cardinals

Let inaccessible cardinal mean uncountable regular strong limit cardinal. Consider $\mathsf{ZFC}$ with an additional axiom: For every set $x$ there is an inaccessible cardinal $\kappa$ such that ...
4
votes
1answer
105 views

Permutation of ZF model

I want to make sure I understand how these work. Can someone please check my answers to the following exercise // answer my questions. :) Let $(V, \in)$ be a model of ZF, and let $\sigma$ be a ...
0
votes
1answer
44 views

Are there (non obvious) obstructions for divisibility of sets? [closed]

Assume $|A|=\omega $. Assume $\omega_1 \times \omega_2\leq \omega $. Are there $\omega_1$ disjoint subsets of $ A $ each of size $\omega_2$?
4
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0answers
91 views

The Diamond Principle plus implies a Kurepa family.

I've been looking at Kunen's (2011, p. 227) proof that the Diamond Principle plus: $(\Diamond^+)$ There is a sequence $\mathcal A = \langle A_\alpha:\alpha<\omega_1\rangle$ such that $A_\alpha$ is ...
4
votes
2answers
313 views

Regular sets in set theory

I have come across the following definition in different books on set theory: A set $x$ is regular iff $(\forall y)(x\in y\implies(\exists w\in y)(w\cap y=\emptyset))$. I don't grasp this ...
9
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1answer
84 views

Relative consistency of slightly modified GCH

Is it consistent with ZFC that $2^\kappa=\kappa^{++}$ for all regular $\kappa$ and $2^\kappa=\kappa^{+}$ for all singular $\kappa$?
4
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1answer
68 views

Combinatorially equivalent polyhedron with vertices from a given dense set

In this question we are only interested in convex polyhedra in the Euclidean space $\mathbb R^3$. Polyhedra $P$ and $P'$ are said to be combinatorially equivalent iff there is a bijection between ...
5
votes
1answer
77 views

ordinal isomorphism theorem

The only proof of the theorem stating that every well-order is order-isomorphic to a unique ordinal of which I am aware relies on the Axiom of Replacement. Is this necessary?
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0answers
75 views

Incompleteness theorems in encoding schemes other than Gödel numbering

Gödel's proof of his incompleteness theorems makes use of Gödel numbering, which is a device that allows a theory of arithmetic $S$ (e.g. PRA) to express and reason about metamathematical statements ...
2
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1answer
89 views

Exercise with transitive models of ZFC

I was trying to solve the following exercise from Kunen's new book but I'm not sure if my answer its right or less formal than would it be. The exercise says that assuming the consistency of ...
11
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1answer
107 views

How does equality $\mathfrak{m}^\mathfrak{m} = 2^\mathfrak{m}$ in $\sf{ZF}$ relate to the axiom of choice?

Usually, the equality $\mathfrak{m}^\mathfrak{m} = 2^\mathfrak{m}$ for infinite cardinal $\mathfrak{m}$ is proved like that: $$ 2^\mathfrak{m} \leqslant \mathfrak{m}^\mathfrak{m} \leqslant ...
2
votes
1answer
62 views

Substituting $\Sigma_1^{(n)}$ functions into $\Sigma_1^{(n)}$ formulae

Although my question is very short, I feel the obligation to introduce the relevant notation first: Let $M = (J_\alpha^A,B)$ be an acceptable $J$-structure with $n$-th projectum $\rho^n$ (where ...
2
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0answers
64 views

the set of all cardinal numbers

I want to prove the following theorem. Theorem. There is no set $\Omega$ such that for every set $A$, there exists a $M\in\Omega$ with the same cardinality as $A$, that is: $|A|=|M|$. In order to ...
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4answers
205 views

Does one need to learn set theory before learning category theory?

I am having a course in Algebraic Topology and learning some basic category theory. But I only have a very limited understanding of basic set theory. I have no idea what is ZFC, and stuff like that. ...
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vote
3answers
72 views

Axiom of choice - Equivalence relation - Representatives

Let $ X $ be a set and $ \sim $ an equivalence relation on $ X $. In many proofs, a set of representatives of equivalence classes of $ X $ is used (e.g. coset or orbit representatives in groups, ...
0
votes
1answer
121 views

Ordered set of cardinality continuum that has more than continuum many initial segments

The book I am reading has the following has the following exericse: "Show that there is an ordered set of cardinality continuum that has more than continuum many initial segments," but I don't see how ...
1
vote
1answer
92 views

Why do we call “comprehension” and “regularity” to the axiom schemas in Set Theory?

I have several Set Theory books in my "shelve" but I have found the justification for the names in none. Usually all of them just state something like: "The axiom schema of ...
0
votes
2answers
43 views

Introducing a term in a first-order theory

Consider, in the first-order NGB theory of sets, the following axioms: $$\vdash\exists x\forall y(y\notin x)$$ and $$\vdash\forall y(y\notin\varnothing)$$ The second one defines the constant ...
4
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0answers
34 views

Can we associate values to elements in a poset? [duplicate]

My question comes from personal investigation. Suppose you have a poset $(X, \le_X)$. I would like to associate to all elements $x \in X$ a value $v(x) \in V$, where $(V, \le_V)$ is a totally ordered ...
3
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2answers
96 views

Countable ordinal

I haven't done ordinal algebra for a long time and I can't remember how to prove that $\omega + \omega$ is a countable ordinal. Precisely what is the bijection between $\omega$ and $\omega \cdot 2$ ? ...
2
votes
2answers
99 views

Isomorphisms of well ordered sets

Let $A$, $B$ be well ordered sets. If there exist order-preserving functions $f : A \to B$ and $g : B \to A$ (do not need to preserve initial segments), are $A$ and $B$ isomorphic? I know this is not ...
2
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1answer
43 views

Ordinal enumeration in ordered Mostowski model - does it not need the global choice?

In Jech's axiom of choice he proves following lemma (lemma 4.5(b) in his book): There is a injective mapping from $M$ to $\mathrm{Ord}\times \operatorname{fin}(A)$, where $A$ is a set of atoms and ...
2
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2answers
122 views

Different axiomatizations of set equality

I've seen two definitions (or axioms?) of set equality: $a=b \Leftrightarrow (\forall x : x \in a \Leftrightarrow x \in b)$ $a=b \Leftrightarrow (\forall x : a \in x \Leftrightarrow b \in x)$ That ...
3
votes
1answer
64 views

In what sense does forcing increase the width of a set-theoretic hierarchy?

I've often read that, whereas top extensions increase the height of the cumulative hierarchy, forcing extensions increase its width (see, e.g., the answer to this question where's it's said that ...
0
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2answers
110 views

uncountable well-ordered chain in $(\mathcal{P}(\mathbb{N}),\subseteq)$ without $AC$

If we assume $AC$ we can construct an uncountable well-ordered chain of subsets of $\mathbb{N}$ by well-ordering the reals and then using the same Dedekind's cut construction as in this question. ...
2
votes
2answers
69 views

Does the following condition implies full outer measure?

Let $X \subseteq 2^{\omega}$ be a set of positive Lebesgue measure. Suppose that for every $\eta, \nu \in 2^{<\omega}$ of the same length, the measure of $X$ above $\eta$ is the same as the measure ...
18
votes
5answers
816 views

Is the set of all topological spaces bigger than the set of all metric space?

I was wondering right that since the notion of a topology is much more general than that of a metric, and that "neighborhodness", if you will, and the concept of continuity, is generalized by the ...
5
votes
1answer
1k views

Which is the most powerful language, set theory or category theory? [closed]

As far as I know, mathematics is written based on a language which can be for example set theory or category theory. My concern is about the power of these languages. How can we realize which language ...
0
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1answer
30 views

If $R$ be a Union of zero measure sets , what is the cardinal of index set? [duplicate]

If $R$ be a Union of zero measure (lebesgue) sets , what can we say about the cardinal of index set? Does this question related to continuum hypothesis? Thanks.
1
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1answer
70 views

Does every diagonal intersection contain $0$?

This might even be a notational nuisance, but here it goes. Let $\kappa$ be a cardinal, $X_\alpha\subseteq\kappa$ for all $\alpha<\kappa$. As you know, the diagonal intersection of ...
1
vote
1answer
125 views

Subsets of the reals when the Continuum Hypothesis is assumed false

If one assumes that the continuum hypothesis is false then there are subsets of the reals of intermediate cardinality, uncountable but smaller than the continuum. What can be said about the necessary ...
0
votes
1answer
64 views

Ordinary Infinity and Dedekind Infinity

Does anyone know a good proof for the claim that one can add a finite number of elements to a set A, which is an ordinary infinity (take their union), and A will still be equipollent to this new set. ...
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0answers
32 views

Complete atomic boolean algebras as coalgebras of some endofunctor on Set

I was hoping to use the fact that CABAs are powersets with extra structure on the morphisms to find an endofunctor $F:\text{Set}\to\text{Set}$ with $\text{Set}^{op}\simeq\text{Coalg}F$. I started by ...
9
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0answers
189 views

An exercise in Fine Structure of constructible universe concerning projectum patterns

This question assumes some familiarity with Jensen's fine structure analysis of the constructible universe L (https://en.wikipedia.org/wiki/Jensen_hierarchy, ...
0
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1answer
205 views

How to simulate power sets in structural set theory (ETCS)?

How to simulate power sets in structural set theory (ETCS)? (nlab) It turns out that one of the primary attributes of a structural set theory is that the elements of a set have no “internal” ...
6
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1answer
242 views

Well-foundedness of cardinals and the axiom of choice

Without axiom of choice it is not generally true that the class of all cardinal (in this question we consider Scott cardinal rather than cardinals as ordinals) is not well-founded under the ordinary ...
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vote
3answers
584 views

Why is the Axiom of Infinity necessary?

I am having trouble seeing why the Axiom of Infinity is necessary to construct an infinite set. According to a professor of who's mine teaching a class on "infinity," the Peano axioms are only ...
3
votes
1answer
242 views

Theorems not Formulable in Set Theory

Several sites I have been reading say that set theory is a good foundation for mathematics because virtually every theorem can be cast into a theorem in set theory. What is an example of a theorem ...