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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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Book Recommendation: Topology & Urelements [closed]

I am looking for a book about Topology with urelements. I am wondering whether the matter has ever been researched in the first place, especially since I could not find any trace on the internet: not ...
Fadi Hasan's user avatar
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1 answer
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Kunen lemma 7.9, chapter IV.

In Kunens "Set Theory An Introduction to Independence Proofs", lemma 7.9, chapter IV, we are given the following: Let $G$ be an isomorphism from $A$ to $M$ which respects the $\in$-relation....
Ben123's user avatar
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Constructive mathematics with different computational models

I am interested in understanding how the capabilities of constructive mathematics evolve when different computational models are considered. Specifically, if constructive mathematics traditionally ...
Pan Mrož's user avatar
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Countable Choice from Finite Sets

Consider the following 4 statements: Axiom of countable choice Axiom of countable choice from finite sets Axiom of countable choice from Dedekind finite sets Existence of a choice function for any ...
svq0231's user avatar
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A union of unions needn't be a union? (Sans AC) [duplicate]

For any collection $\mathscr C$ of sets, write $\Upsilon(\mathscr C)$ for the collection of arbitrary unions in $\mathscr C$. Now, I ask the innocent question of idempotency of $\Upsilon$: Is $\...
Atom's user avatar
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Irrational numbers Cardinality.

The real numbers, $\mathbb{R}$, are uncountable and the rational numbers, $\mathbb{Q}$, are countable. We can write $\mathbb{R} = \mathbb{Q} \cup (\mathbb{R} \setminus \mathbb{Q})$. Since $\mathbb{Q}$ ...
Mathstudent123's user avatar
2 votes
1 answer
77 views

Does $A$-fold choice imply $|A| + |A| = |A|$ and $|A|\cdot |A| = |A|$?

Let $A$ be an infinite set. Then Zorn's lemma can be used to conclude that $A\times\{0, 1\}$, $A\times A$ and $A$ are all equinumerous (a proof is presented here). However, I am aware that for $A$ ...
Atom's user avatar
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For $k$-algebras $B_1, \dots, B_n$, $\# \operatorname{Hom}_k( \prod_{i=1}^nB_i, \Omega) = \Sigma_{i=1}^n \# \operatorname{Hom}_k(B_i, \Omega)$?

Let $k$ be a field with $ k \subseteq \Omega$ a algebraically closed field. Let $B_1 , \dots, B_n$ be ( possibly finite local ) $k$-algebras. Then next equality of cardinals holds $$ \# \operatorname{...
Plantation's user avatar
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3 votes
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Does Löwenheim-Skolem require Foundation in any way?

As title states, I'm curious whether Löwenheim-Skolem (in either of its upward or downward versions) necessitates some implicit use of Foundation. The usual presentation makes quite clear the reliance ...
Sho's user avatar
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A question related to the axiom of choice [closed]

I have a reasonable background in Math, but I've been unable to get my mind around the axiom of choice. Here is a statement of the axiom of choice taken from the book Real Analysis and Probability by ...
Mike's user avatar
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1 answer
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Proving the Equality of Infinite Cardinal Products and Powers

Theorem: Let $\Xi$ be an infinite set, $\{\kappa_i\}_{i \in \Xi}$ be a family of cardinal numbers, and $\lambda$ be a cardinal number. Then: $\prod_{i \in \Xi} \kappa_i^{\lambda} = \left(\prod_{i \in \...
Chau Long's user avatar
1 vote
0 answers
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Possible typo in "Introduction to set theory" third edition by T. Jech

In the chapter 8, section 1, page 143 of the book "Introduction to set theory" third edition by T. Jech appears . The issue I have is that I don't understand why it says $\xi \in A$ in the ...
Noobunaga's user avatar
1 vote
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Godel's incompleteness theorem: Question about effective axiomatization

I am studying Godel incompleteness theorems and I am struggling with the definition of effective axiomatization. From Wikipedia: A formal system is said to be effectively axiomatized (also called ...
Tereza Tizkova's user avatar
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1 answer
64 views

Small doubt regarding proof of Replacement under Von Neumann Universe

As title states, I have a small doubt concerning how exactly the Replacament axiom schema is proven to hold for the Von Neumann universe $V = \bigcup_{\alpha \in \mathrm{Ord}}V_\alpha$. As I ...
Sho's user avatar
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Two example of non-$\omega$ models in Kunen.

In Kunens "Set Theory An Introduction to Independence Proofs" page 146, chapter IV, we are given the following: $\ldots$ For example, let $S$ be $\sf{ZF}+ \neg \sf{CON}(\ulcorner \sf{ZF} \...
Ben123's user avatar
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Does every set decorate a graph?

Aczel's Antifoundation Axiom $\mathsf{AFA}$ can be formulated as "every graph is decorated by a (unique) set" (well, more specifically, every directed pointed graph or accessible pointed ...
Sho's user avatar
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2 votes
1 answer
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von Neumann-Numbers in Second Order ZFC with Full Semantics - Eliminating Non-Standard-Numbers

The background: In $ZFC$ the summary of the von Neumann numbers is not in every model a set, because $ZFC$ allows models, in which there are non-standard numbers that cannot be separated with FOL (...
RalfK's user avatar
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Set representation of pi [duplicate]

The Zermelo representation of natural number is following \begin{gather*} 0 = \{\} \\ 1 = \{0\} = \{\{\}\} \\ 2 = \{1\} = \{\{\{\}\}\} \\ \vdots \\ n = \{n-1\}. \end{gather*} But is there a ...
Kamil Kiełczewski's user avatar
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Axiom of choice for function domain reconstruction

I know the following definition of the axiom of choice: For each family $S$ of nonempty disjoint sets, there is a set $V$ (the so-called selector) that contains exactly one element from each of the ...
Kamil Kiełczewski's user avatar
3 votes
1 answer
64 views

Is there a function $f :A^{<\mathbb{N}}\to E$ such that $f (s{}^\frown a)=h(f(s),a,|s|)$ for any $s\in A^{<\mathbb{N}}$ and $a\in A$?

Given a set $A$, define $A^{<\mathbb{N}}:=\cup _{n\in\mathbb{N}}A^n$ with $A^0:=\{\emptyset\} $ and $A^n$ being the $n$-fold cartesian product of $A$. For any $s:=(s_0,\cdots,s_{n-1})\in A^n\...
rfloc's user avatar
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∅ or Φ as the empty set? [closed]

I was just getting into set theory along with my friends and we noticed this confusion with notation of the empty set. Seeing that it is pronounced as "phi" we assumed to use the greek ...
Pranjal Kumar's user avatar
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1 answer
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Platonistic interpretation of Gödel (theorem 14.2, I in Kunen).

On page $41$ in Kunens "Set Theory An Introduction to Independence Proofs", after proving If $\phi(x)$ is any formula in one free variable, $x$, then there is a sentence $\psi$ such that $$...
Ben123's user avatar
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Is every strict ordering by inclusion a well-ordering?

Given a set $s$ which is transitive and completely ordered by inclusion, that is, such that $z \in s \rightarrow z \subset s$ and $\left( x \in s \wedge y \in s \wedge x \neq y \right) \rightarrow \...
Mark Fischler's user avatar
5 votes
3 answers
487 views

Motivation of inventing concept of well-ordered set?

I've started studying set theory for a while. I understand what is an ordered sets but i still fail to see what motivated mathematicians to invent these concept. Could you please enlightment me ? ...
InTheSearchForKnowledge's user avatar
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Is there a model of ZF not C where not every set of reals is Lebesgue measurable? [duplicate]

I know that there is a model of ZF set theory plus the negation of the axiom of choice where every set of reals is Lebesgue measurable. But is there also a model where not every set of reals is ...
user107952's user avatar
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Constructing a choice set for the rational equivalence on the irrational numbers

In the 5th edition of Real Analysis written by Roden and Fitzpatrick, Problem 28 of Chapter 2 requires us to explicitly find a choice set for the rational equivalence relation on $\mathbb{R} \setminus ...
efsdfmo12's user avatar
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0 answers
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What is the difference between the notations $\beth_{\beth_{\beth_\ddots}}$ and $\beth_{\omega_{\omega_\ddots}}$ for the first beth fixed point?

Reading some papers on set theory I have found these two different notations used for the first beth fixed point. $\beth_{\beth_{\beth_\ddots}}$ $\beth_{\omega_{\omega_\ddots}}$ I don't think the ...
Hegel Gehel's user avatar
2 votes
0 answers
36 views

The law of composition with infinitely many elements of algebraic structure

We can define the law of composition with finite elements by induction. But there are some operations such as sum or union, which can be compute with infinitely many, or even uncountably many elements....
Strassss's user avatar
0 votes
1 answer
51 views

Is $\operatorname{rank}(A)\subseteq\operatorname{TC}(A)$?

This question came to me when I was thinking about rank and transitive closures. Let $A$ be a set of rank $\alpha$ and let $\operatorname{TC}(A)$ denote the transitive closure of $A$. Is it true then ...
Anon's user avatar
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Two families of isomorphic structures have isomorphic ultraproduct.

I am trying to prove the following result: Let $(\underline{M}_i)_{i\in I}$, $(\underline{N}_i)_{i\in I}$ be two families of structures such that, for all $i\in I$, $\underline{M}_i \cong \underline{...
WiggedFern936's user avatar
2 votes
1 answer
101 views

$\mathrm{Con}(\sf{ZFC}) \rightarrow \mathrm{Con}(\sf{ZFC}+\neg \exists \alpha(\text{$\alpha$ is strongly inaccessible}))$

On page 133 in Kunens "Set Theory An Introduction to Independence Proofs", we are given the following (paraphrasing), If $\kappa$ is the first strong inaccessible, then $H(\kappa)$ is a ...
Ben123's user avatar
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2 votes
0 answers
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Principle in between BPI and AC

I have searched extensively in the literature, but all references I have consulted always place BPI (the Boolean Prime Ideal Theorem) as a sort of "cover" of the Axiom of Choice as far as ...
Rodrigo Nicolau Almeida's user avatar
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1 answer
101 views

Greek letter "Kappa" use in Set Theory

I have been given a definition that the Greek letter "K" is used when a cardinal and ordinal number of a set are the same. As I have NO knowledge of Set Theory, can someone explain this? I ...
John Bond's user avatar
7 votes
0 answers
72 views

Inequality of cardinal sums [duplicate]

Define $|A|<|B|$ iff there is an injection from $A$ to $B$ but there is no injection from $B$ to $A$. Is it provable in ZF that $|A|<|C|$ and $|B|<|D|$ and $A\cap B=\emptyset=C\cap D$ implies ...
Lucenaposition's user avatar
3 votes
0 answers
54 views

Epimorphism is equivalent to surjective using right-inverses.

I want to show that in the category of sets a function is an epimorphism iff it is surjective. The problem I'm having is with showing that epimorphism $\Longrightarrow$ surjective. I've seen the ...
Dinis P.'s user avatar
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0 votes
2 answers
109 views

Proof of Gödels first incompleteness theorem (as in Kunen)

On page 40 in Kunens "Set Theory An Introduction to Independence Proofs", we are given the following theorem: Gödel. If $\phi(x)$ is any formula with one free variable, $x$, then there is a ...
Ben123's user avatar
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1 vote
1 answer
33 views

Absoluteness of ordinal exponentiation and rank on transitive models of $\sf{ZF}-\sf{P}$.

In Kunens "Set Theory An Introduction to Independence Proofs", page 129-130, we are given a proof that for any transitive model $\mathbf{M}$ of $\sf{ZF}-\sf{P}$, ordinal exponentiation and ...
Ben123's user avatar
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1 vote
1 answer
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Is there something missing in Jech's proof of Zermelo's Well-Ordering Theorem?

Here is the proof from p. 48 of the Millennium Edition, corrected 4th printing 2006: My question: how do we know that there is any ordinal $\theta$ such that $A=\{a_\xi\,\colon\xi<\alpha\}$? ...
Nat Kuhn's user avatar
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5 votes
1 answer
113 views

Absoluteness of inaccessible cardinals

I'm studying large cardinals and I'm hoping to fully understand the proof that says ZFC is not able to prove the existence of inaccessibles (given ZFC is consistent, of course). I've already fully ...
Darsen's user avatar
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4 votes
1 answer
87 views

Why are extensions of countable models of ZFC better behaved than extensions of arbitrary models of ZFC?

This answer hints that certain kinds of extensions are only guaranteed to exist for countable models of ZFC. Why? One intuitive reason i can think of is that the metatheory might not have enough new ...
Carla_'s user avatar
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0 answers
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What exactly are characteristic functions of sets when constructing Boolean models?

Reading J. L. Bell's Set Theory --- Boolean-Valued Models and Independence Proofs. This is how the introduction of Boolean models is motivated: Suppose that for each set $x ∈ V$ we are given a ...
zaq's user avatar
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1 vote
1 answer
62 views

Need to check this proof that the class of models of ZC that fail replacement is not axiomatisable.

Here ZC is ZFC minus Axiom of Replacement. My proof is as follows: Suppose $M$ was axiomatized by a theory $H$. For non-zero limit ordinal $\alpha$, let $T_\alpha$ be the set of the replacement axioms ...
Adam's user avatar
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5 votes
1 answer
99 views

What does consistency of $\frak c=\aleph_\alpha$ for any ordinal $\alpha$ without cofinality $\omega$ mean (ZFC)?

In this answer, Asaf Karagila says that it is consistent with ZFC that $\frak c=\aleph_\alpha$ for any ordinal $\alpha$ without cofinality $\omega$. It is not clear to me what exactly this means. If $\...
Carla_'s user avatar
  • 457
13 votes
5 answers
3k views

How do we define addition?

I've been trying to learn naive set theory through a YouTube series by 'The Bright Side of Mathematics'. So far, I've been able to understand successor maps and the definition of $\mathbb{N}_{0}$. I ...
Spyridon Manolidis's user avatar
2 votes
1 answer
14 views

Formalizing the construction of a Cantor scheme

I'm asking this question here since a similar question asked on math.stackexchange didn't receive an answer. I'm trying to understand the proof of the following theorem: Theorem (Brouwer): The Cantor ...
rfloc's user avatar
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1 vote
0 answers
84 views

Asking the experts in mathematical logic and set theory for survey articles to supplement the standard textbooks.....

I'm asking this question to the experts in mathematical logic and set theory.I'm a former graduate student in mathematics with an additional background in philosophy engaged in serious self study. I'...
Mathemagician1234's user avatar
1 vote
1 answer
86 views

Theorem 5.6, chapter IV in Kunens Set Theory

Boldface-letters (i.e. $\mathbf{A}$,$\mathbf{V}$,etc.) indicate a class. The following is an excerpt from Kunens "Set Theory An Introduction to Independence Proofs" (theorem 5.6, $\S5$ of ...
Ben123's user avatar
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2 votes
0 answers
133 views

Why is the Axiom of Choice Necessary in ZFC

Within the framework of Zermelo-Fraenkel set theory with the Axiom of Choice $(ZFC)$, when we considered the method of constructing the set of natural numbers, we regarded it as the smallest inductive ...
Bezina Taki's user avatar
0 votes
1 answer
30 views

Infinite Ramsey Theorem analogue for infinite complete Bipartite Graphs

The version of the Infinite Ramsey Theorem I am interested in states that for any countably infinite complete graph $K_{\aleph_0}$ edge-coloured by $f: E(K_{\aleph_0}) \to [1, c]$ (with $c \in \mathbb{...
DaumaK's user avatar
  • 3
0 votes
1 answer
43 views

Absoluteness of ordinal multiplication (as defined in Kunen).

In Kunens "Set Theory An Introduction to Independence Proofs", p. 128, theorem 5.5 in chapter IV, he wants to prove the following: For ordinals $\alpha$ and $\beta$, the following statement ...
Ben123's user avatar
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