# Tagged Questions

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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $$S=\{p:p \text{ is an ultrafilter on }B\}.$$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $\left(\mbox{i.e each}\, a\in A\,\mbox{is a Dedekind Finite set}\right)$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $\mathbf X$ ...
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### 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 $L_S$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $b \not\in c$. ...
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### 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 ...
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### What is a conservative/intersective function?

I can't find any information on what a conservative or intersective function is. ...
### 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 ...