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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Can transitive reflection be proved in absence of regularity or powersets?

Is transitive reflection a theorem schema of $\sf ZF-Reg.$? Is transitive reflection a theorem schema of $\sf ZF-Powersets$? Transitive reflection is the schema that for any formula $\varphi$ in $\...
Zuhair's user avatar
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What exactly makes the ordinals an indefinitely extensible concept?

I understand the principles of generation that cantor used to create the ordinals but I cannot see what exactly is the property that makes the ordinals an indefinitely open plurality and not the ...
Arianit Gashi's user avatar
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1 answer
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Asymmetric relation is antisymmetric and irreflexive as well

A relation is asymmetric if and only if it is both antisymmetric and irreflexive. I read this in Wikipedia's article about binary relation. So my problem with the statement is that though I can ...
Mystic mystic's user avatar
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2 answers
45 views

Exploring the Equivalence between Infinite Sets and the Existence of Subsets under Certain Mappings

I recently found this equivalence between the two propositions in a book $( E )$ is an infinite set $\forall f \in E^E. \exists A \in \mathcal{P}(E) \setminus \{\emptyset, E\} \, :f(A) \subset A$ ...
Bezina Taki's user avatar
5 votes
1 answer
103 views

Confused by this proof in Jech's set theory

In Jech's Set Theory, Chapter 11, the universal set $U$ is defined as: For each $\alpha \geq 1$, there exists a set $U \subset \mathcal{N}^2$ such that $U$ is $\Sigma_{\alpha}^0$ (in $\mathcal{N}^2$) ...
Link L's user avatar
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Construction of an m-step function used in the Recursion Theorem

In Introduction to Set Theory by Hrbacek and Jech, the Recursion Theorem is stated as: For any set $A$, any $a\in A$, and any function $g: A \times \mathbb{N} \rightarrow A$, there exists a unique ...
coffeez's user avatar
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6 votes
1 answer
70 views

$L[a] \cap 2^{\omega}$ is $\Sigma_2^1$

I have the following question: Let $a\in \mathbf{R}$ sucht that $X = L[a] \cap 2^{\omega}$ is uncountable. Why is $X$ is a $\Sigma_2^1$ set? $L[a]$ is the inner model that can be built by ...
Caro Meier's user avatar
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71 views

Are there infinite sets whose aleph and beth numbers are both unknown?

Is there a set $S$ that is definable in ZFC and known to be infinite, but for which we know neither the aleph nor beth number? For example, we do not know the aleph number of $|\mathbb R|$, but we ...
WillG's user avatar
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114 views

Forcing as a Quotient

I'm reading Jech and following his Boolean algebra models approach to it. I'm wondering if I've got the right idea here. Let $M \models \mathrm{ZFC}$ and $B \in \mathbf{CompBoolAlg}$. We construct $M^...
ASheard's user avatar
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An ordinal $\nu$ is a natural iff there is no injection $f$ of $\nu$ into $X$ in $\mathscr P(\nu)\setminus\{\nu\}$.

Let's we prove the following theorem. Theorem An ordinal $\nu$ is a natural if and only if there is no injection of $\nu$ into $X$ in $\mathscr P(\nu)\setminus\{\nu\}$. Proof. Let's we assume there ...
Antonio Maria Di Mauro's user avatar
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1 answer
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What are generic ideals and dense sets, intuitively?

Someone commented under one of my previous posts that, intuitively, a generic set isn't supposed to have any "conspicuous properties". I wonder what the precise meaning of that comment was ...
zaq's user avatar
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3 votes
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ZFC + Grothendieck universes vs Mac Lane's One universe

First note that I'm not an expert in set theory, but do have an interest in the topic in that I want to be able, at least in principle, to reduce my mathematics to the underlying axioms of my chosen ...
kaba's user avatar
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1 vote
1 answer
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Can this presentation of reflection be considered foundational?

Working in the first order language of set theory. By $R$-bounded quantifiers its meant those of the form $\forall x \ R \ a \, ( \cdots) $ , or $ \exists x \ R \ a \, ( \cdots) $, and these are ...
Zuhair's user avatar
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2 votes
2 answers
130 views

Why subset W of cartesian product $X\times Y$ cannot be X

Let $W\subseteq X\times Y$ where $\forall x\in X , \exists ! y\in Y , (x,y) \in W $. How to prove that if $W\neq \emptyset$ then always $W \neq X$ ? I try to prove that $x = (x,y)$ cannot exist - ...
Kamil Kiełczewski's user avatar
1 vote
1 answer
60 views

Theorem of Shelah about the existence of an inaccessible cardinal

There is a theorem of Shelah, stated in the following way: If all $\Sigma_3^1$ sets of reals are measurable, then $\aleph_1$ is an inaccessible cardinal in $L$. In some textbooks (for example ...
C_M's user avatar
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2 answers
74 views

Axiom of Regularity and nested set

I have some trouble witch the axiom of regularity. I wolud like to show that $$x = (x,y)$$ for any y, not exists. As example pair definition based on set I use Kuratowski definition - so: $$x=\{\{x\},\...
Kamil Kiełczewski's user avatar
1 vote
1 answer
104 views

$\omega$-th or $(\omega + 1)$-th when putting odd numbers after even numbers?

The following clip is taken from Chapter 4 - Cantor: Detour through Infinity (Davis, 2018, p. 56)[2]. When putting odd numbers after even numbers, what should the index for the first odd number ($1$ ...
Yif's user avatar
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0 answers
48 views

Example of a family of Boolean subalgebras such that their union is not a Boolean algebra

I read that given a family of Boolean subalgebras, their union is not in general a Boolean algebra except if the family is directed. What is an example of a family of Boolean subalgebras whose union ...
Link L's user avatar
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4 votes
0 answers
68 views

Forcing with the generic multiverse

Fix a countable transitive model $M$ and consider the collection $\mathbb{P}$ of all forcing extensions of $M$ (i.e., the generic multiverse of $M$), ordered by reverse containment. What happens if we ...
Lxm's user avatar
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1 answer
89 views

How to prove that CC($\mathbb R$) is true in permutation models

I've familiarised myself with models of set theory and am beginning to understand the basics, but am still very far away from being a proper model theorist. I currently live under the impression that ...
Cloudscape's user avatar
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0 answers
18 views

Extending a consistent set of sentences to a complete, witnessing, consistent set of sentences

I'm attempting to unpack the following paragraph from a proof of the Model Existence Theorem (p.44 on this set of notes): Suppose we have a consistent $S$ in the language $L = L(Ω, Π)$. Extend $S$ to ...
Sam's user avatar
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70 views

Countable union of a countable many collection is the power set of that collection

i'm trying to prove that cannot exists a countable sigma algebra (not trivial). So, the statement is the following: Let $X$ a set and let $\mathcal{M}$ a sigma algebra on X. Show that $\mathcal{M}$ is ...
Sigma Algebra's user avatar
1 vote
0 answers
149 views

Prove that any cardinal has a successor without using ordinals

Can one show that any cardinal has a successor without using ordinals? I mean: cardinals are not defined as special ordinals, but have a naive definition, as equivalence classes of equipotent sets, ...
abellaic's user avatar
0 votes
1 answer
80 views

Different ways to describe Hilbert's Hotel

In math, Hilbert's Hotel (e.g. https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel) is a famous paradox that describes a situation where a hotel with infinitely many rooms and full ...
firstpassage's user avatar
1 vote
1 answer
64 views

Subset of integers called by a somewhat ill-defined property

I know there were some issues with set theory that involved self-reference and famous example being that of Russell's example. Here, I have a set that I use a somewhat vague term "use" to ...
Mahammad Yusifov's user avatar
1 vote
1 answer
79 views

Finite Content Not Continuous on the Empty Set

I have come across a lemma which states: Let $\mu$ be a finite content on an algebra $\mathcal{A}$. $\mu$ is a pre-measure if and only if $\mu$ is continuous on the empty set, i.e., for every sequence ...
SineOfTheTimes's user avatar
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0 answers
40 views

Is every class on a set also a set?

Using the axioms of $ZF$, you ensure that from a set or multiple sets, you can also create a set. In my opinion, which I do not claim to be accurate, in a number of finite uses of these axioms, it can ...
Bezina Taki's user avatar
3 votes
1 answer
54 views

A countable ordinal which is $\Sigma_n$-definable in first-order ZFC, but not $\Sigma_m^1$-definable in full second-order arithmetic

Let us say that a $\Sigma_m^1$-formula $\phi$ defines a countable ordinal $\alpha$ if it defines a type-$1$ object (i.e. a real) $x$ that encodes a well-ordering of $\mathbb{N}$ of order type $\alpha$ ...
lyrically wicked's user avatar
4 votes
4 answers
203 views

Is every subset of a set also a set?

Using the axioms of $ZF$, you ensure that from a set or multiple sets, you can also create a set. However, the question that arose in my mind is whether all subsets of this set that was created are ...
Bezina Taki's user avatar
-1 votes
0 answers
85 views

What's the largest (known) ordinal number that every smaller ordinal is definable?

What's the largest ordinal number that every smaller ordinal could be accurately described, say in ZFC, in a finite length formula? Obviously, it's greater than $\omega_1^{CK}$, but smaller than $\...
user23013's user avatar
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1 vote
2 answers
186 views

Example of an uncountable subset of $\mathbb R$ which cannot be proved to have the same cardinality as $\mathbb R$

I am new to mathematical logic so forgive me if this is a bad question. I understand that the Continuum Hypothesis (CH) is independent of ZFC and therefore there exist models of ZFC in which the CH is ...
Oliver's user avatar
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1 vote
0 answers
63 views

Defining ordered tuples of arbitrary length in NBG

How can we define in NBG or in any other axiomatization of set theory that uses classes, the concept of ordered tuples of proper classes. For example the classical Kuratowski definition does not work ...
Shthephathord23's user avatar
0 votes
0 answers
68 views

Is $F(x) \in z$ absolute

For an absolute Function F (meaning that the formula y=F(x) is absolute), is there a proof that $F(x) \in z$ is an absolute Formula? This Lemma from Kunen applied for $\Phi(w,z)$ being $w \in z$ and $...
Rubids's user avatar
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6 votes
0 answers
84 views

When is the Cayley embedding for infinite groups optimal?

By Cayley's theorem, any group $G$ naturally injects into the symmetric group $\mathrm{Sym}(G)$ of its underlying set via the Cayley embedding. Let's say this embedding is optimal for $G$, if there is ...
Tim Seifert's user avatar
3 votes
0 answers
78 views

What is $\epsilon_0 \cdot \omega$?

I'm a bit stuck on telling what the ordinal $\epsilon_0 \cdot \omega$ is. $$\epsilon_0 = \sup \{1, \omega, \omega^{\omega}, \omega^{\omega^{\omega}}, \dots \}$$ so $$\epsilon_0 \omega = \sup \{\omega, ...
Robin's user avatar
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2 votes
2 answers
141 views

Do we really need the axiom of regularity? [duplicate]

I don't think I could ever say I understand ZFC if I don't get to the bottom of this question. It all started when we had extensionality and comprehension. Then Russell finds a paradox and here comes ...
The curious amateur's user avatar
0 votes
0 answers
53 views

NBG the intersection of a set and a class

In this post I will assume the NBG set theory For all classes $X$ let $M(X)$ mean $(\exists T)(X \in T)$ i.e. $X$ is a set and not a proper class. Prove that $\left\{X \subseteq Y, M(Y) \right\} \...
Shthephathord23's user avatar
0 votes
3 answers
135 views

Does $(\bigcap \{z:z\notin x\} = \emptyset)$ hold for every set $x$ in ZF$-$AF?

I asked about this and the answer is true if we assume AF. As a more interesting question, does it still hold without AF? According to the suggestion in the comment, I should post it as another ...
Y.X.'s user avatar
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-2 votes
2 answers
137 views

Prove that under ZF, $\forall x (\bigcap \{z:z\notin x\} = \emptyset)$

I am taking a view on the chapter on set theory in Hinman and discovered this statement as Exercise 6.1.27. May I please ask for a proof of it? The claim is a bit interesting, it says ``for every set $...
Y.X.'s user avatar
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-1 votes
1 answer
57 views

Increasing limit in sequence of set [closed]

I am reading Foundations of Modern Probability by Kallenberg and stuck on the meaning of increasing limit $A_n\uparrow A$ for sequence of set (pageno 1), I read this question, but I have to confirm ...
zia badar's user avatar
2 votes
2 answers
149 views

Can we include a metric space into its completion?

Let $(X,d)$ be a metric space and let $(X^*,d^*)$ denote its completion (via equivalence classes of Cauchy sequences in $X$). Let $f:X \to X^*$ be an isometry such that $f(X)$ is dense in $X^*$. Now ...
Alphie's user avatar
  • 4,720
2 votes
1 answer
51 views

Equivalent condition for measurable filters.

It's a known fact (see for example Bartoszynski and Judah: Set Theory, On the Structure of the real Line) that for a filter $F$ on $\omega$ the following conditions are equivalent: $F$ is Lebesgue ...
Caro Meier's user avatar
0 votes
0 answers
49 views

Cofinality of $\beth_\lambda$ and increasing of beth sequence

Premise. I was given the following definition of beth-numbers: $$ \beth_\alpha := \begin{cases} \beth_0 = \aleph_0 \\ \beth_{\alpha + 1} = 2^{\beth_\alpha} \\ \beth_{\lambda} = \bigcup_{\alpha < \...
leluch_l8r4's user avatar
3 votes
1 answer
115 views

Symmetric submodels of $M[G]$ are exactly the classes $(HOD(M[x]))^{M[G]}$

I'm trying to understand Grigorieff's proof (Theorem 3, page 478, Intermediate Submodels and Generic Extensions in Set Theory) that symmetric submodels of $M[G]$ are exactly $(HOD(M[x]))^{M[G]}$ for $...
JLB's user avatar
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-4 votes
0 answers
90 views

Bijection between ZxZ and ZxZxZ [closed]

Here : https://math.stackexchange.com/questions/7643/produce-an-explicit-bijection-between-rationals-and-naturals#= I have found a bijection between $Z$ and $Q$. I'm thinking about a bijection between ...
Aurelian Florea's user avatar
2 votes
1 answer
88 views

Questions about Namba forcing

I am looking at Jech's proof of properties of Namba forcing. The conditions consist of perfect subtrees of $\omega_2^{<\omega}$, where perfect means every node has $\omega_2$ many (not necessarily ...
new account's user avatar
2 votes
1 answer
82 views

Where does the term 'dense' used in forcing/Martin's axiom come from?

There are some common meanings to 'dense' in Mathematics. In Topology, a subset $S\subseteq X$ of a topological space $(X, \tau)$ is dense if the intersection of every non-empty open set with $S$ is ...
Chad K's user avatar
  • 4,296
1 vote
2 answers
82 views

Model Theory in the Language of Peano Arithmetic

Most introductory textbooks on model theory establish the theory based on the ZF set theory (e.g. [1]). In particular, a structure is defined to be a 4-tuple of sets, and so on. In [2], I came to ...
Student's user avatar
  • 1,812
1 vote
1 answer
72 views

How can different representations of the same integer be equivalent?

I recently read about a way to define the set of integers as the set of all equivalence classes for some equivalence relation $\simeq$ satisfying $(a,b)\simeq(c,d)$ for $(a, b),\;(c,d)\in\mathbb{N}\...
naytte2's user avatar
  • 321
4 votes
3 answers
176 views

Understanding Metatheory and the Broader Picture of Foundational Set Theory

So I'm trying to put together a clearer picture of what is going on when we study set theory. I'll describe my current picture which I'd appreciate some feedback on, and I'll ask some specific ...
space_kale's user avatar

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