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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11
votes
1answer
367 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 ...
66
votes
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 ...
1
vote
2answers
82 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, ...
3
votes
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 ...
12
votes
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 ...
7
votes
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.
16
votes
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 ...
2
votes
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 $V$ ...
5
votes
0answers
81 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
votes
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
votes
0answers
64 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
votes
2answers
140 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 ...
1
vote
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 ...
1
vote
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 ...
0
votes
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 ...
0
votes
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
votes
1answer
86 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 ...
0
votes
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
vote
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
votes
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
votes
1answer
61 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
votes
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 ...
4
votes
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
vote
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
votes
1answer
33 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 $G$ ...
4
votes
2answers
94 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 ...
0
votes
2answers
82 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
votes
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 ...
66
votes
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
votes
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 $\cong$ ...
2
votes
0answers
102 views

Are there any good documentary films about the continuum hypothesis?

Are there any good documentary films about the continuum hypothesis? I'm looking for something slightly more serious than the usual "Cantor showed that infinity plus one equals infinity and then went ...
1
vote
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 ...
1
vote
1answer
77 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 ...
5
votes
2answers
167 views

About proving that the Continuum Hypothesis is independent of ZFC

In Mathematical Logic, we were introduced to the concept of forcing using countable transitive models - ctm - of $\mathsf{ZFC}$. Using two different notions of forcing we were able to build (from the ...
6
votes
1answer
194 views

New Axioms of Infinity

Axiom of Infinity says there is an inductive set (i.e. a set which includes $\emptyset$ and is closed under successor operator). Formally: $Inf:\exists x~(\emptyset\in x~\wedge~\forall y\in ...
0
votes
1answer
49 views

proving set intersection is possible with infinite collections

I'm currently going through Enderton's Elements of Set Theory. I have two questions, but they need some context. Here are the subset axioms introduced in the book: For each formula _ not ...
1
vote
0answers
129 views

“Dual cardinality” in the graphs $(V_\alpha,\in_\alpha)$

04-25-2014 Enriched with more details Let define the graphs $(V_\alpha,\in_\alpha)$ in $ZFC$ $V_\alpha$ can be finite too $\in_\alpha \subseteq V_\alpha \times V_\alpha$ and ...
9
votes
0answers
109 views

Sets of Cardinals without choice

I have a theory that the axiom of choice is equivalent to the statement that sets of distinct cardinals are well ordered by cardinality. I can prove that the axiom of choice implies this. However I am ...
5
votes
1answer
193 views

Asymmetric roles that are symmetric in every instance

This is similar to something else I posted, but this time we'll pretend we've never heard of infinite sets or infinite series. \begin{align} & \sin(\alpha+\beta+\gamma+\delta+\varepsilon+\zeta) ...
5
votes
1answer
116 views

Why do $2^{\lt\kappa}=\kappa$ and $\kappa^{\lt\kappa}=\kappa$ when the Generalized Continuum Hypothesis holds?

I'm going to assume GCH here. If that holds, then why do we have the equalities $2^{\lt\kappa}=\kappa$ for every $\kappa$, and $\kappa^{\lt\kappa}=\kappa$ for all regular $\kappa$?
4
votes
3answers
106 views

Can the truth value of an independent property be changed at will by enlarging the model?

Let $\phi$ be a property that's independent of $ZFC$, so that there are strcutures ${\mathfrak A}=(A,{\in}_A)$ (where $A$ is a set or class and ${\in}_A$ is a binary relation on $A$) that are models ...
2
votes
1answer
59 views

Ramsey Theory: Showing the existence of a special set of natural numbers.

Show that there is an infinite set of natural numbers such that the sum of any two elements has an even number of prime factors. My attempt: Define a coloring on the doubletons of $\mathbb{N}$, that ...
1
vote
2answers
65 views

Axiom of Choice (Naive Set Theory, Halmos)

I'm currently reading Naive Set Theory by Paul Halmos and I'm not quite understanding what he means in sec. 15, The Axiom of Choice. Suppose that $\mathscr{C}$ is a non-empty collection of ...
3
votes
3answers
105 views

Confused about Axiom of Choice

(1) I understand that if I have a non-empty set $A$, choosing an element $\alpha$ from $A$ does not require the Axiom of Choice. (2) I also understand that if I have a finite collection of ...
0
votes
0answers
25 views

A Sequnce from an infinte set [duplicate]

We know if $A$ is infinite set then we can choose a sequence from $A$. But I don't know how this requires AC.(or countable AC?) Thanks in advance for any suggestion. Maybe the approach depend on ...
5
votes
1answer
90 views

Landing between $\beth_\lambda$ and $\beth_{\lambda+1}.$

Main Question. Is it consistent with ZFC that there exists a limit ordinal $\lambda$ and a cardinal number $\nu$ satisfying $$\beth_\lambda < \beth_\lambda^\nu < \beth_{\lambda+1}?$$ I am also ...
9
votes
0answers
74 views

Choice in a Particular HOD Type Model

Let $V \models ZFC$. Let $P$ be a forcing and $G \subseteq P$ be generic over $V$. Let $x \in V[G]$. Let $M$ be the class of set that are hereditarily definable (in $V[G]$) using as parameters $x$ ...
4
votes
1answer
64 views

Regarding Thomas Jech's demonstration of Zorn's lemma via induction

Thomas Jech, such as many other mathematicians, demonstrates $AC \rightarrow ZL$ via transfinite induction. He says: Proof. We construct (using a choice function for nonempty sets of P), a chain ...
19
votes
1answer
267 views

What are some good open problems about countable ordinals?

After reading some books about ordinals I had an impressions that area below $\omega_1$ is thoroughly studied and there is not much new research can be done in it. I hope my impression was wrong. ...