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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2
votes
1answer
51 views

Elementary subsctructure of $L_{\omega_1}$

Let us consider $A$ the family of all elements of $L_{\omega_1}$ definable without parameters. I was trying to prove that assuming $V=L$ then $A$ is an elementary substructure of $L_{\omega_1}$ by ...
7
votes
1answer
132 views

$|A \cup B| = \mathfrak{c}$, then $A$ or $B$ has cardinality $\mathfrak{c}$.

$A$ and $B$ are sets. $|A \cup B| = \mathfrak{c}$, prove that $A$ or $B$ has cardinality $\mathfrak{c}$. This is an exercise problem from my textbook. It's easy if I assume CH to be true. But how can ...
0
votes
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 ...
7
votes
5answers
223 views

Summable enumerations of $\Bbb Q$

We say that a set of natural numbers $A$ is summable if $\sum_{n\in A}\frac1n$ is finite. It is not hard to see that $\{A\subseteq\Bbb N\mid A\text{ is summable}\}$ is an ideal on $\Bbb N$: Subsets ...
0
votes
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
62 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 ...
3
votes
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 ...
1
vote
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 ...
1
vote
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
votes
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. ...
1
vote
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 ...
5
votes
2answers
115 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 ...
2
votes
1answer
61 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 $L[A]$. $L[A](0)=\{A\}\cup \operatorname{tr cl}(A)$ ...
1
vote
1answer
49 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. .
4
votes
1answer
119 views

How to prove that $\omega_1 \leq \mathfrak p$

I am trying to understand the stated above Theorem. $\mathfrak p$ is the smallest cardinality of any family $\mathcal F \subseteq [\omega]^\omega$, which has the strong finite intersection property, ...
64
votes
8answers
18k views

Can you explain the “Axiom of choice” in simple terms?

As I'm sure many of you do, I read the XKCD webcomic regularly. The most recent one involves a joke about the Axiom of Choice, which I didn't get. I went to Wikipedia to see what the Axiom of ...
7
votes
1answer
80 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 ...
11
votes
1answer
201 views

Cardinality of power sets decides all of cardinal arithmetic?

Assuming ZFC, is it possible to have two models which agree on the cardinality of all the power sets, but disagree on the cardinality of some other cardinal exponentiation (meaning that they agree on ...
9
votes
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
votes
0answers
90 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 ...
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 $\omega_2$?
6
votes
1answer
232 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 ...
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 ...
4
votes
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 ...
13
votes
0answers
305 views

Fields whose algebraic closure cannot be constructed without the axiom of choice

One can show that the statement that every field has an algebraic closure requires the axiom of choice. However, for almost all "everyday" fields, it seems that one can actually produce an algebraic ...
2
votes
0answers
121 views

About models of ZFC and models in general.

So I've been attending lectures in Set Theory lately and been struggling with the following. When studying the universe of sets V our approach is: let ZFC be consistent, then a model V of the theory ...
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?
2
votes
1answer
84 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 ...
1
vote
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 ...
7
votes
2answers
377 views

mahlo and hyper-inaccessible cardinals

Wikipedia states that a Mahlo cardinal is hyper-inaccessible, hyper-hyper-inaccessible, etc. Is this a characterisation of Mahlo? If not what about "alpha = hyper^alpha-inaccessible" with the obvious ...
2
votes
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. ...
11
votes
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 ...
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 ...
2
votes
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 ...
1
vote
3answers
71 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, ...
2
votes
2answers
121 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 ...
1
vote
1answer
90 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 ...
7
votes
2answers
445 views

A few questions on nonstandard analysis

I know that nonstandard analysis is analysis plus the existence of infinitesimal numbers. Does it mean that nonstandard analysis is the same theory as $ZF+\exists$infinitesimal numbers? From what I ...
4
votes
3answers
722 views

Axiom of Choice and Cartesian Products

According to Wikipedia one formulation of AC is The Cartesian product of any family of nonempty sets is nonempty. If I consider an cartesian product $\prod_{i} X_i$ of nonempty sets $X_i$, then ...
3
votes
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 $\omega \cdot 2$ ? ...
4
votes
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 ...
2
votes
2answers
98 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
votes
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 ...
9
votes
0answers
178 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, ...
1
vote
3answers
572 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
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
votes
2answers
108 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. ...
18
votes
5answers
814 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 ...