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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2
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2answers
221 views

Does the set of all ordinals strictly dominated by a given set exist in ZF?

How do I prove that $$ \{\alpha\in\mathsf{On}\,|\,\alpha\prec A\}\in V,$$ assuming $A\in V$? I know that if AC is assumed, this set is equal to $\mbox{card}(A):=\mu_\alpha(\alpha\approx A)$, and ...
0
votes
1answer
701 views

Every finite partially ordered set has a maximum length chain.

Every finite partially ordered set, $(A, \leq)$, has a maximum length chain. A chain is a sequence of distinct elements $a_1 \leq a_2 \leq .......\leq a_n$ with relation $"\leq"$ where $a_i \in A$ ...
5
votes
1answer
112 views

$\varepsilon$-numbers

An ordinal $\xi$ is an $\varepsilon$-number (where does this name come from?) if $\omega^\xi = \xi$. Is the set of all countable $\varepsilon$-numbers closed in $\omega_1$? In other words, does an ...
5
votes
2answers
529 views

If every nonempty subset of a set $S$ has a least and greatest element, is $S$ finite?

Some sets are well-ordered; all of their nonempty subsets have least elements. You can also have sets where all of their nonempty subsets have greatest elements. Some sets have both of these ...
15
votes
2answers
419 views

A question about a two player game and axiom of choice

Suppose two players 1 and 2 play the following game: Player 1 starts by playing the set of reals $\mathbb{R}$. Player 2 plays an uncountable subset $Y_1$ of $\mathbb{R}$. Then player 1 plays an ...
1
vote
3answers
140 views

Question regarding well-ordering theorem

The well-ordering theorem states that every set can be well-ordered. That is, there is a total order on the set in which every non-empty subset has a least element. I was wondering whether it's also ...
5
votes
2answers
355 views

Showing there is only one isomorphism between well ordered sets using transfinite induction

I need to show specifically using transfinite induction that given two well-ordered sets $\left(A,<_{1}\right)$ and $\left(B,<_{2}\right)$ there is only one isomorphism between them. To do ...
4
votes
2answers
151 views

Two questions regarding Ordinal Numbers.

I'm trying to prove that there is an uncountable ordinal all which members are countable ordinal. This is fairly easy if I can state that the class of all countable ordinals is a set and then take the ...
3
votes
1answer
453 views

Question about Hausdorff Maximal principle and antichain

I have this prob and still haven't figured about Let (A, $\leq$) be a partially ordered set. A subset B of A is called an antichain if and only if for any two distinct elements x and y in B, ...
11
votes
4answers
636 views

Clarification of a remark of J. Steel on the independence of Goldbach from ZFC

On page 424 of the following paper: S. Feferman, Harvey M. Friedman, P. Maddy and John R. Steel, ``Does Mathematics Need New Axioms?'' The Bulletin of Symbolic Logic, Vol. 6, No. 4 (Dec., 2000), pp. ...
5
votes
2answers
177 views

How does Fraenkel's urelement proof show choice is independent of ZF?

I understand the actual proof Fraenkel gives but I can't see how it proves choice independent of the full ZF because he works in a very restricted universe. Can anyone show how to connect one to the ...
4
votes
2answers
333 views

Axiom of choice for infinite strictly descending chains of subsets.

Let $X$ be an infinite set. Then there are infinite stictly decreasing chains of subsets of X. Does this fact require axiom of choice? If yes, what is the choice of function?
5
votes
0answers
157 views

Interpretation of impure set theory within pure set theory?

I recently came across this paper, where the author (in section 3.1) gives an interpretation of ZFU within ZF. The author of the paper goes on to show what he calls the "synonymy" (i.e., that ZF and ...
1
vote
2answers
228 views

infinite sequence of sets $\{X_i\}$ that for each $i$, $X_i\in X_{i-1}$

I need to show that the following infinite sequence $\{X_i\}$ doesn't exist: for all $i$, $X_i \in X_{i-1}$ I really don't know where to start. The only thing in my mind is the axiom : for every non ...
6
votes
4answers
395 views

Axiom of choice in set theory

Just as the title stated, what is the main point of axiom of choice? I do not quite understand what is written in the axiom. The axiom that I know is: Given any collection of non-empty sets, there ...
7
votes
1answer
235 views

pronunciation of “Kunen” [closed]

How should the last name of set theorist Ken Kunen be pronounced? I have heard both koo-nen and kyoo-nen from other people. Probably the best answer would be to say how he himself pronounces it, but ...
1
vote
1answer
48 views

Problem understanding definition of ordering on stationary sets

In the book "Set Theory" of Thomas Jech, it is defined for any two stationary subsets $S,T$ of a regular uncountable cardinal $\kappa$, $S<T$ if and only if $S\cap\alpha$ is stationary in $\alpha$ ...
2
votes
1answer
161 views

Infitive distributive law in boolean valued models

I'm posting the problem 2.14 and 2.15 of the book "Set theory" of J.L. Bell. These problem are proposed after the forcing relation chapter and I'm new in this kind of stuff, so I have some little ...
3
votes
2answers
134 views

Computing $\kappa^{<\lambda}$, for cardinals $\kappa$ and $\lambda$

I'm trying to show that, for $\lambda$ an infinite cardinal and $\kappa$ any cardinal, that $$\kappa^{<\lambda} = \sup\{\kappa^\theta:\theta<\lambda\land\theta\text{ a cardinal}\},$$ where $\...
4
votes
3answers
4k views

Does the set of all sets that contain themselves contain itself?

We always hear about the paradox of the set of all sets that don't contain themselves and whether it contains itself or not. What about the set of all sets that do contain themselves? Is that an ...
3
votes
4answers
237 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
841 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
274 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
415 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
322 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
587 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
804 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
190 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
361 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
140 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
226 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
225 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
119 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
341 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
298 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
254 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
143 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
218 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
149 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 ...