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
37 views

Model for replacement

If $R_{k} $ is a model for replacement, is necesarly k a strong limit cardinal? And if k is a regular cardinal, are then equivalent the sentences: k is a strongly inaccesible cardinal if and only if ...
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1answer
46 views

About the definible sets $L_\alpha$

Let $\alpha$ be an ordinal number. Is that true that $\alpha$ = $\beth_\alpha $ is equivalent to the statement $|L_\alpha|=|R_\alpha|$, where $L_\alpha $ is the $\alpha$-th stage of the constructible ...
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1answer
44 views

Ordinal arithmetic question

The question is: find $\delta \le \omega_1$ and $\rho \lt \omega$ such as $\omega_1 = \omega \cdot \delta + \rho$. My guess is that $\delta = \omega_1$ and $\rho = 0$, abut I don't know how to prove ...
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2answers
149 views

Is $\mathrm{ZFC}^E$ outright inconsistent?

From $\mathrm{ZFC},$ define a new theory $\mathrm{ZFC}^E$ by adjoining a constant symbol $E$ together with axioms to the effect that: $E$ is countable and transitive $(E,\in)$ is an elementarily ...
7
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1answer
57 views

Every elementary submodel of $H(\aleph_1)$ is transitive

I am trying to solve the first part of Exercise II.17.30 in Kunen's Foundations of Mathematics which asks: Prove that every $A \preccurlyeq H(\aleph_1)$ is transitive. Here we are working over ...
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1answer
69 views

Existence of countable transitive models.

I have read that $\mathrm{ZF}$ has a transitive model iff it has a countable transitive model. I am interested in generalizations of this result. In particular: Question. Let $\varphi$ and $\Psi$ ...
103
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1answer
3k views

Does the open mapping theorem imply the Baire category theorem?

A nice observation by C.E. Blair1, 2, 3 shows that the Baire category theorem for complete metric spaces is equivalent to the axiom of (countable) dependent choice. On the other hand, the three ...
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0answers
21 views

Choosing from two index families of sets

Let $a$ be a (nontrivial) ultrafilter and $n$ be an infinite set. Let also $U$ be an infinite set. Define $n$-ary relation $\phi$ on $\mathscr{P}U$ by the formula $L\in\phi \Leftrightarrow \forall ...
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2answers
168 views

Random reals and Martin-Löf randomness

My questions are about the relationship between the following notions of randomness: A real $r$ is random over the model $M$ if $r\notin B$ for every null Borel set $B$ coded in $M$. A real $r$ is ...
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1answer
76 views

Ultraproducts of models of ZFC

Let $D$ be a non-principal ultrafilter over $\mathbb{N}$. Let $A_i$ for $i\in\mathbb{N}$ be (countable) models of ZFC such that $A_i\models \mathfrak{c}=\aleph_i$. Then, what is the size of the ...
7
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2answers
142 views

Is it possible that $2^n=3^n$ for some Dedekind-finite cardinal $n\gt0$?

Is it possible that $2^n=3^n$ for some Dedekind-finite cardinal $n\gt0$? I think the question speaks for itself, but let me try and satisfy the "quality standards" algorithm by padding it. Yes, I ...
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2answers
1k views

For every infinite $S$, $|S|=|S\times S|$ implies the Axiom of choice

How to prove the following conclusion: [For any infinite set $S$,there exists a bijection $f:S\to S \times S$] implies the Axiom of choice. Can you give a proof without the theory of ordinal ...
4
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0answers
50 views

Can existence of aleph one be proved without the power set axiom? [duplicate]

Cantor's construction of $\aleph_1$ is to notice that all ordinals constructed after $\omega$ are countable, take the union of all countable ordinals, and then show that this union can not be ...
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1answer
69 views

Is there such a thing as “completion with respect to the axiom of choice?”

Completions abound in mathematics. For example: the completion a metric space with respect to Cauchy sequences the algebraic closure of a field the Stone-Čech compactification of a topological ...
5
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1answer
140 views

Example of an interesting theorem that fails in intuitionistic set theory but is classically valid?

I'm interested in intuitionistic set theories at the moment. I know that lots of principles imply LEM and so fail intuitionistically, and also a few basic principles - linear ordering of ordinals, for ...
4
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2answers
128 views

CH imples the existence of a function

I was studying an article and the author stated that CH implies that there exists a function from $\omega_1 \setminus \omega$ onto the set of all countable subsets of $\omega_1$ such that for each ...
4
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3answers
143 views

Are there other approaches for the foundations of mathematics, other than logic and set theory?

Are there other approaches for the foundations of mathematics, other than logic and set theory? And why does set theory begin talking about objects and groups of objects. Is it proven somewhere that ...
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2answers
41 views

Cardinal of $V_{\omega+\alpha}$

I do not understand why $card(V_{\omega+\alpha})=\beth_\alpha$. The steps in the recursion for $0$ and succesor ordinals are quite easy, but I do not manage to prove it for limit ordinals. I see that ...
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1answer
368 views

Set of All Groups

In my undergraduate Group Theory class, while discussing the impossibility of equivalence relations on groups, my professor said that the set of all groups does not exist. Are there any group ...
67
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10answers
3k views

Different kinds of infinities?

Can someone explain to me how there can be different kinds of infinities? I was reading "the man who loved only numbers" and came across the concept of countable and uncountable infinities, but ...
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2answers
83 views

What do you need to know about ordinals to understand Zorn's Lemma's proof?

I'm trying to understand the proof of Zorn's Lemma but the one which does not use ordinals (Halmos' proof) is extremely long and I really feel I get lost somewhere along the way. On the other hand, ...
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2answers
100 views

Universe cardinals and models for ZFC

I'm reading through Joel David Hamkins' set theory lecture notes. On page 14, on the subject of inaccessible cardinals and submodels of ZFC in $V$, he defines a universe cardinal to be a cardinal ...
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5answers
1k views

What is a good text in intermediate set theory?

I've been working my way through Enderton's Elements of Set Theory for a while, and I feel I have a decent grasp on some of the basics of elementary set theory. My question is, where should I look to ...
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2answers
1k views

Best book on axiomatic set theory.

Which is the best book on axiomatic set theory? I am interested in a book that is suitable for graduate studies and it is very mathematically rigorous.
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6answers
2k views

Textbooks on set theory

I want to do a survey of textbooks in set theory. Amazon returns 3582 books for the keywords "set theory". A small somewhat random selection with number of references in Google scholar is the ...
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2answers
66 views

How do we define recursive class functions like $\alpha \mapsto V_\alpha$ and $\alpha \mapsto L_\alpha$ in the language of set theory?

It is well-known that ZFC proves that everything is an element of $V$. Symbolically, $\forall x(x \in V).$ However, I can't figure out how to translate this into the language of ZFC. We know that $V$ ...
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0answers
82 views

Jech, “Set theory” exercises 12.11 - Is my proof right?

I try to prove the Jech's "Set theory", exercises 12.11: 12.11. If $\kappa$ is an inaccessible cardinal, then $V_\kappa\models \text{there is a countable model of ZFC}$. My attempt. Since ...
2
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1answer
88 views

Sentence $\varphi$ of set theory that is satisfied by all well-founded models of ZFC, but which is not a theorem of ZFC.

I think I read somewhere the following. If a first-order sentence $\varphi$ in the language of set theory holds for every well-founded model of ZFC, then nonetheless: $\varphi$ may fail for a ...
2
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0answers
66 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 ...
3
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2answers
142 views

Axiom of Regularity - Transitive set

I just managed to confuse myself completely while studying for Set Theory. We have the Axiom of regularity: $$\forall S (S\not= \emptyset \rightarrow (\exists x\in S)(S\cap x=\emptyset))$$ Now a set ...
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1answer
43 views

Well-Ordering Theorem without Axiom of Regularity

Personally, I am not fond of the Axiom of Regularity. Some alternative models in set theory use the negation of the Axiom of Regularity as an axiom (non well-founded theories). I am curious if the ...
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1answer
85 views

$V_\omega$, $\mathcal V^{B}_\omega$, $\mathcal V^{*B}_\omega$ and $\mathcal S^{B}_\omega$: alternative superstructures and properties

I was not able to find a beginner introduction to superstructures and the cumulative hierarchy that makes me able to answer to some of my questions about them so I tried to ask here and I apologize ...
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1answer
38 views

Inductive posets, exercise from Notes on set theory.

OK, this book is driving me nuts. This is exercise 16 from chapter 6. For each set $E$, the set $P = E* ∪ (ℕ → E)$ of finite and infinite sequences from E is an inductive poset, under $⊆$. For ...
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2answers
45 views

Finite set with unknown number of elements

Does there exist anything like a finite set whose cardinality cannot be established by means of an algorithm or mathematical proof? I think ZFC should have no problem in accomodating such an object, ...
3
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1answer
87 views

Has anyone considered axioms to the effect that: “The axiom of constructibility fails very very badly?”

If I'm not mistaken, the axiom of constructibility basically says that the universe has no (non-trivial) inner models. Has anyone considered axioms of the opposite flavour, basically asserting that ...
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0answers
27 views

Addition on well ordered sets not-commutative by showing $[0,1) +_o \mathbb{N} =_o \mathbb{N} \neq_o \mathbb{N} +_o [0,1)$

My goal is to show that addition on well ordered sets are non-commutative by showing that, $[0,1) +_o \mathbb{N} =_o \mathbb{N} \neq_o \mathbb{N} +_o [0,1)$ Some definitions (let A and B be ...
1
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1answer
154 views

Lemma required for Cantor-Bernstein-Schroeder Theorem

Let $A$ and $B$ bet sets such that $A \subseteq B$. If there is an injective function $f: B \rightarrow A$, then there is a bijective function $h:B \rightarrow A$. I understand how to prove this ...
4
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2answers
51 views

countably closed forcing cannot add a branch to a $\aleph_2$-tree if $\neg\mathsf{CH}$

I'm reading this survey. In it the author states the following result (fact 5.3) which is attributed to Silver: If $2^{\aleph_0}>\aleph_1$, countably closed forcing cannot add a new branch to ...
3
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1answer
62 views

Truth values in Boolean valued models

In Devlin's "The Joy of Sets" the author introduces the Boolean valued model $V^{{\mathcal B}}$ based on a given Boolean algebra ${\mathcal B}$ and describes how to assign a truth value ...
6
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1answer
67 views

Conservativity of $\mathrm{ZFC}+\varphi$, where $\varphi$ contradicts CH.

It is well-known that ZFC with the continuum hypothesis is a $Π^2_1$-conservative extension of ZFC. General question. What is known about the conservativity of $\mathrm{ZFC}+\varphi$ over ...
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1answer
102 views

A question regarding the status of CH in the Gitik model

Consider models of ZF+"Every uncountable cardinal is singular" (eg. Moti Gitik: "All uncountable cardinals can be singular", Israel journal of Mathematics, 35(1-2): 61-88, 1980). How should CH be ...
1
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1answer
41 views

Object set of a clonal category.

I read the statement that "a clonal category has a small set of objects", which I don't quite agree about. In the definition of clonal category, at least as it is given in that context, it is required ...
2
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1answer
34 views

Do Different Generic Filters Give Different Generic Extensions?

Let $\mathbb{P}$ be a forcing. If $G \subseteq \mathbb{P}$ and $H \subseteq \mathbb{P}$ are two $\mathbb{P}$-generic filters over $V$ and $G \neq H$, does this imply that $M[G] \neq M[H]$. If $G$ ...
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2answers
97 views

Cardinality of an algebra

Suppose that $B$ is the Boolean algebra of all Lebesgue measurable sets in $I=[0,1]$ modulo Null sets. I am asking (1) What will be the cardinality of $B$. Does it have to be $|B|=\mathfrak ...
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2answers
83 views

Consistency of ZFC and the key assumption [closed]

I recently read this answer to a MathOverflow question that got me thinking. Very roughly, here's what the author of that answer says: Gödel's second incompleteness theorem implies that if there ...
5
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2answers
79 views

What is gained by internalizing LST (the language of set theory)?

I'm reading up on Gödels constructible universe L in the book "Constructibility" by Devlin, and by comparing his text with texts like Kunen and Jech, there is one thing in particular that he's doing ...
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15answers
6k views

Why did mathematicians take Russell's paradox seriously?

Though I've understood the logic behind's Russell's paradox for long enough, I have to admit I've never really understood why mathematicians and mathematical historians thought it so important. Most ...
3
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2answers
86 views

Are there situations outside of set theory where it would be useful if $\mathrm{ICF}$ were true?

Write $\mathrm{ICF}$ for the "injective continuum function hypothesis" i.e. the sentence of ZFC expressing that $$2^X \cong 2^Y \rightarrow X \cong Y$$ for all sets $X$ and $Y$, where $\cong$ ...
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0answers
53 views

Is the “set” of all algebraic extensions a set? [duplicate]

Consider a field $K$. Now, consider the class of all algebraic extensions of $K$. Is this a set? Since I think it isn't, how to prove it isn't? If the class were of all extensions of $K$, I think I ...
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1answer
80 views

Hall's marriage Theorem and Tychonoff Theorem

I was reading this paper. In particular the second point. He proves the Hall's marriage Theorem for infinite family using the Tychonoff theorem on topological product of compact $T_2$ spaces and the ...