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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3
votes
4answers
239 views

Is it possible to make the set of all sets of cardinality $\aleph_0$?

I know that in ZFC that some collections of objects cannot be gathered together into a set (for example, the "set of all sets") does not exist, nor is "the set that just contains itself." Is it ...
3
votes
2answers
852 views

Cantor-Bendixson theorem proof

I am looking for a proof of Cantor-Bendixson theorem involving transfinite numbers (I am interested only in the case of real line). I fact, I have already seen one but I have a trouble in ...
4
votes
1answer
278 views

Problem understanding proof of Solovay's Theorem on stationary sets

Solovay's Theorem on stationary sets states that any stationary subset of a regular uncountable cardinal $\kappa$ is the disjoint union of $\kappa$ stationary subsets. In Jech's "Set Theory", it is ...
5
votes
2answers
121 views

Weak cardinal powers and singular cardinals

Suppose $\kappa > \operatorname{cf}(\kappa)$. Show that: i) if $\kappa$ strong limit then $\kappa^{<\kappa} = \kappa^{\operatorname{cf}(\kappa)}$ ii) if $\kappa$ not strong limit then $2^{<\...
2
votes
1answer
417 views

Paradox: Any set theory without universe set is not a model of itself

Because a model of a first order theory is not allowed to use a proper class as its domain, we can't use the universe of the set theory from the "meta-level" directly as a model for a first order ...
2
votes
1answer
326 views

$\omega$ is a set

I'm trying to show from the $ZFC$ axioms that $\omega$ is a set and I have something that I think could be a proof but I'm not sure about one of its parts. In particular, I want to use the axiom that ...
6
votes
2answers
596 views

Differences between pure and impure set theory?

What are some differences between pure and impure set theory? For example, this paper references the result that ZFC with urelements is categorical if you assume that the urelements form a set. ZFC, ...
18
votes
3answers
814 views

For any two sets $A,B$ , $|A|\leq|B|$ or $|B|\leq|A|$

Let $A,B$ be any two sets. I really think that the statement $|A|\leq|B|$ or $|B|\leq|A|$ is true. Formally: $$\forall A\forall B[\,|A|\leq|B| \lor\ |B|\leq|A|\,]$$ If this statement is true, ...
8
votes
2answers
191 views

Non-measurable subset of $\omega_1$

Consider $\omega_1$ equipped with the order topology. Then Borel subsets of $\omega_1$ are precisely those which contain a closed and unbounded set or the complement contains such a set. There must be ...
5
votes
4answers
363 views

Is the powerset of every Dedekind-finite set Dedekind-finite?

Is the powerset of every Dedekind-finite set Dedekind-finite? I think this statement can be written in $\textbf{Set}$: If every mono (=injection) $f: A \to A$ is iso (=bijection), then every mono $g:...
1
vote
1answer
141 views

Question on a mapping between a Boolean algebra and an algebra of sets

On page 81, Set Theory, Jech(2006), to prove the Stone's Representation Theorem, a mapping $\pi$ is defined as Let $B$ be a Boolean algebra. We let $$S=\{p:p \text{ is an ultrafilter on }B\}.$$ ...
9
votes
2answers
228 views

Axiom of Determinacy

It is quite easy to see that $ZF + AD$ (the Axiom of Determinacy) implies the countable axiom of choice ($AC_\omega$), yet $AC$ is inconsistent with $AD$. The dependent choice principle $DC$ is ...
4
votes
3answers
1k views

What properties are allowed in comprehension axiom of ZFC?

I am trying to understand the axioms of ZFC. The axiom schema of specification or the comprehension axioms says: If A is a set and $\phi(x)$ formalizes a property of sets then there exists a set C ...
6
votes
2answers
226 views

Ordinals definable over $L_\kappa$

Suppose $\kappa$ is an uncountable cardinal, with $L_\kappa$ an admissible set (i.e. a model of Kripke–Platek set theory). Let $<_\gamma \subseteq \kappa \times \kappa$ denote a wellordering of $\...
3
votes
1answer
121 views

Is Choice an assumption or determined by category?

Is the axiom of choice an assumption, that one may "freely" choose (eg, ZFC) or discard (eg, ZF, ZF+AD), or is it determined by the nature of the categories being considered? The latter view is ...
1
vote
2answers
79 views

Limiting of Power Sets

This is, I believe, a relatively simple set-theoretical question. I am ,however, not sure of the answer. If we take a set, say $A$, and if we call the power set of $A$, $P_{1}(A)$, and we define $$...
1
vote
2answers
349 views

What is the cardinality of the countable ordinals?

Every countable ordinal $\alpha$ can be written uniquely in Cantor canonical form as a finite arithmetical expression, say $C(\alpha)$. We thus have the 1-1 correspondence between the countable ...
2
votes
2answers
96 views

What does it mean that $f$ is unbounded modulo $D$?

Given an ultrafilter on $\omega$ and a function $f:\omega\longrightarrow\omega$, what does it mean that $f$ is unbounded modulo $D$? Thanks
4
votes
1answer
81 views

Problem with definition of Rudin-Keisler equivalence

I'm trying to do exercise 7.11 of Jech's "Set Theory": If $D$ and $E$ are ultrafilters on $\omega$, then $D\leq E$ and $E\leq D$ implies that $D\equiv E$, where $\leq$ is the Rudin-Keisler ordering, ...
5
votes
1answer
300 views

Absoluteness of $ \text{Con}(\mathsf{ZFC}) $ for Transitive Models of $ \mathsf{ZFC} $.

Is $ \text{Con}(\mathsf{ZFC}) $ absolute for transitive models of $ \mathsf{ZFC} $? It appears that $ \text{Con}(\mathsf{ZFC}) $ is a statement only about logical syntax. Taking any $ \in $-sentence $ ...
19
votes
6answers
3k views

Book on the Rigorous Foundations of Mathematics- Logic and Set Theory

I am asking for a book that develops the foundations of mathematics, up to the basic analysis (functions, real numbers etc.) in a very rigorous way, similar to Hilbert's program. Having read this ...
3
votes
2answers
122 views

Question about passage in Halbeisen's book

I am looking at the following passage in Halbeisen's book "Combinatorial Set Theory" (p 260 at the bottom): What is the role of $\Phi$? It seems to me that a finite fragment is the same as a ...
2
votes
2answers
256 views

Dedekind finite union of Dedekind finite sets is Dedekind finite

I could use some help proving the following: Let $A$ be a Dedekind Finite set of pairwise disjoint Dedekind finite sets $\left(\mbox{i.e each}\, a\in A\,\mbox{is a Dedekind Finite set}\right)$ ...
2
votes
1answer
144 views

Question about the proof of $GCH$ holds in $\mathbf L$

I have a question about the proof of the following: (Lemma 37) Assume $\mathbf V = \mathbf L$, and let $\kappa$ be a cardinal. Then $\mathcal P (\kappa ) \subseteq L_{\kappa^+}$. Assume we ...
7
votes
2answers
219 views

Example of a set that is in $\mathbf V$ but not in $\mathbf L$

Let $\mathbf V$ denote the cumulative hierarchy and let $\mathbf L$ denote Gödel's constructible universe. We then have $\mathbf L \subseteq \mathbf V$. Would someone give me an example of a set that ...
5
votes
1answer
150 views

What is a “set-like class”?

Just / Weese contains the following theorem (p 126): Theorem 5: Let $\mathbf Z \neq \varnothing$ and let $\mathbf W \subseteq \mathbf Z \times \mathbf Z$ be a wellfounded set-like class. Let $\mathbf ...
2
votes
1answer
147 views

Assuming $(GCH)$, strongly inaccessible and weakly inaccessible coincide

My book says "... If $GCH$ holds, then the notions of strongly inaccessible and weakly inaccessible cardinals coincide, ..." In $ZFC$ I can prove this. But the paragraph from which I have excerpted ...
1
vote
1answer
113 views

Explicit choice functions for finite sets in topological spaces

When dealing with finite nonempty sets of real or natural numbers it is always possible to define a explicit choice function, that choose one (arbitrary, but well defined) element out of that set: ...
1
vote
2answers
184 views

Defining strong limit cardinals in $ZF$

I do not understand the following passage/footnote in the book I am currently reading: An initial ordinal $\lambda$ is called a strong limit cardinal if $2^\kappa < \lambda$ for every $\kappa ...
1
vote
1answer
152 views

Proving Axiom Schema of Replacement holds in $H_\lambda$

I think I proved the following, can you tell me if my proof is correct? Exercise 23: Show that if $\lambda$ is a regular infinite cardinal then $\langle H_\lambda , \overline{\in} \rangle$ satisfies ...
1
vote
0answers
68 views

A sheaf of cumulative hierarchies

I've recently read a paper about sheaf forcing in which a sheaf of cumulative hierarchies was defined (defintion 5.3 on page 30). The same object is described in this English paper (defintion 3.1 on ...
3
votes
3answers
279 views

How to break power set for non-transitive models?

Apparently, $\varphi (x,y) : y = P(x)$ where $P$ denotes the power set is not absolute for transitive models. We call a formula $\varphi(v_1, \dots v_n)$ in $L_S$ absolute for a class $\mathbf X$ ...
0
votes
3answers
171 views

Extensionality of a relation and Axiom of Extensionality

What's the difference between the Axiom of Extensionality $(A1)$ and an extensional relation? The definitions are $(A1) \forall x,y ( x = y \leftrightarrow \forall z ( z \in x \leftrightarrow z \in ...
2
votes
3answers
232 views

Is there a bounded-quantifier formula in ZF without Foundation that defines the naturals?

Does there exist a first-order set-theoretic formula $\phi(x)$ using only bounded quantifiers such that, in ZF without assuming the Axiom of Foundation/Regularity, $\phi(x)$ is true if & only if $...
1
vote
4answers
174 views

A question about standard models

As I understand it a standard model is a model in which the relation is the $\in$ on the actual set of sets constituting the model. (i) Hence theories that aren't in the language of set $L_S$ ...
4
votes
2answers
335 views

A non-well-ordered set where principle of transfinite induction holds?

A theorem in my textbook says: Let $(A, < )$ be a totally ordered set. Set A has a least element and principle of transfinite induction holds in A if and only if A is well ordered. I understand ...
10
votes
4answers
586 views

How to introduce advanced set-theoretical objects to philosophy students?

First, I apologize if MSE is a bad fit for this question. I'm going to give a course as the last course of "elementary set theory" (the previous courses were not given by me). I planed to introduce ...
3
votes
0answers
186 views

A question about a passage in Just/Weese's Basic Set Theory

I have a question regarding the following passage in Just/Weese (p 192), $\mathbf{V}$ denoting the cumulative hierarchy: "But consider the following situation: Where in "($\beta$)" does the $\...
5
votes
1answer
214 views

Books on Axiom of Dependent Choices?

Are there any books about the axiom of dependent choice? There are books about the axiom of choice, e.g. Herrlich or Jech. But I can't seem to find any on the axiom of dependent choices. As far as I ...
2
votes
1answer
105 views

SDR for an infinite set of sets

Let $F$ be a set of nonempty sets. A set $R\subseteq\cup F$ is said to be a system of distinct representatives (SDR) of for every $B\in F$ there exists unique $x\in R$ such that $x\in B$ and $\forall ...
7
votes
2answers
377 views

Relationships between AC, Ultrafilter Lemma/BPIT, Non-measurable sets

How is it possible to reconcile the following... In 1970, Solovay constructed Solovay's model, which shows that it is consistent with standard set theory, excluding uncountable choice, that all ...
6
votes
1answer
447 views

Bourbaki Proof of Zorn's Lemma in Lang's Algebra

Serge Lang in his book Algebra has a nice appendix on set theory at the end of the book. In particular, in paragraph 2, pp. 881-884 he provides a proof of Zorn's Lemma from "other properties of sets ...
6
votes
2answers
218 views

Prove that the statement implies the Axiom of Choice

Prove that the following statement implies the Axiom of Choice: Let $ C $ is a set (of sets) and $ B $ is a set such that for all $ c \in C $, there exists a $ b \in B $ such that $ b \not\in c $. ...
4
votes
3answers
382 views

Defining cardinals without choice

According to Wikipedia if we assume AC we define a cardinals number to be an ordinal that is not in bijection with any smaller ordinal. Without AC, one takes the cardinality of a set $X$ to be the ...
0
votes
1answer
88 views

What is a conservative/intersective function?

I can't find any information on what a conservative or intersective function is. ...
12
votes
1answer
659 views

Strength of the statement “$\mathbb R$ has a Hamel basis over $\mathbb Q$”

I would like to know if there are "interesting" equivalences to the statement "$\mathbb R$ has a Hamel basis over $\mathbb Q$". I am not interested in more general statements, like "every vector space ...
2
votes
1answer
229 views

Bijection Definition Clarification

I was reading A Walk Through Combinatorics: An Introduction to Enumeration And Graph Theory (Miklós Bóna) and came across a definition for bijection: Let $X$ and $Y$ be two finite sets, and let $f ...
3
votes
2answers
180 views

Axiom of Choice and Ascending Chain Condition

Matsumura, in his "Commutative Ring Theory" p. 14 proves that "A partially ordered set $\Gamma$ satisfies the ascending chain condition $\Leftrightarrow$ every nonempty subset of $\Gamma$ has a ...
1
vote
2answers
255 views

Axiom of Choice - Naive Counterexample [duplicate]

Possible Duplicate: Axiom of choice question I know there is a lot of discussion on the axiom of choice and, in fact, I attended once a lecture on it, but I still cannot understand the ...
2
votes
1answer
151 views

Proving properties of enumeration of infinite cardinals

I am doing the following exercise from Just/Weese: where $F$ is defined as follows: (a) I'm not sure whether the following passes as "convince myself": Apparently $F$ is defined for all ...