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 ...

learn more… | top users | synonyms

10
votes
5answers
528 views

Natural isomorphisms and the axiom of choice

The definitions of "natural transformation", "natural isomorphism between functors", and "natural isomorphism between objects" captures - among other things - the intuitive notion of "an isomorphism ...
4
votes
1answer
171 views

Cohen forcing and bounds on GCH violation

I recall (though it might be my faulty memory) an exercise in some book (perhaps Jech?) that was along the lines "Suppose $\mathrm{GCH}$ holds in $V$ and $P$ is a Cohen forcing that adds $\kappa$ ...
6
votes
2answers
735 views

Cardinality of sets of subsets of $\mathbb{N}$

If we dont assume CH, is there a procedure to construct or define a set of subsets of $\mathbb{N}$ such that we cannot prove it to be of cardinality $\aleph_0$ or $\aleph_1$? Or if we assume not CH, ...
5
votes
1answer
118 views

The cardinality of $L_\alpha$ in $L$

It's true (in $V$) that for any infinite ordinal, $|L_\alpha|=|\alpha|$. My question: Is it also true in $L$? i.e., does $L$ itself satisfies $|L_\alpha|=|\alpha|$ for any infinite ordinal $\alpha$?
5
votes
2answers
270 views

Number of Ramsey ultrafilters

Hereafter, "ultrafilter" is intended to mean "nonprinicpal ultrafilter on the set, $\omega$, of natural numbers." An ultrafilter $U$ is Ramsey if for every partition $P$ of $\omega$ into sets not in ...
6
votes
2answers
792 views

Diamond Principle and its variants - two questions

The standard formulation of the Diamond Principle $\Diamond$ is as follows: There exists a sequence $\langle f_\alpha:\ \alpha<\omega_1\rangle$ of functions $f_\alpha:\alpha\to2$ such that for ...
5
votes
1answer
112 views

Exercise about Galvin-Hajnal rank

I'm stuck at exercise 9a of chapter 2.2 of Introduction to cardinal arithmetic by Holz, Steffens and Weitz. It is as follows: Assume that $\kappa > \omega$ is a regular cardinal, $I$ is the ...
64
votes
1answer
4k views

How do we know an $ \aleph_1 $ exists at all?

I have two questions, actually. The first is as the title says: how do we know there exists an infinite cardinal such that there exists no other cardinals between it and $ \aleph_0 $? (We would have ...
4
votes
1answer
218 views

Are these sets stationary?

Assume $\kappa > \omega$ is a cardinal of uncountable cofinality and $S$ is a stationary set in $\kappa$. If $\alpha < \kappa$ is a successor ordinal then the set $S+\alpha=\{\sigma+\alpha : ...
2
votes
2answers
175 views

Accessible formal specification and explanation of First Order Logic?

I am trying to get good at proofs by working through How To Prove It. Unfortunately I am very bothered by the fact that I do not understand all the formalities in First Order Logic + set theory ...
5
votes
2answers
2k views

Cantor-Bernstein-like theorem: If $f\colon A\to B$ is injection and $g\colon A\to B$ is surjective, can we prove there is a bijection as well?

I've been trying to find this proof: If there exists $f \colon A\to B$ injective and $g \colon A \to B$ surjective, prove there exists $h \colon A \to B$ bijective. I thought of using ...
4
votes
2answers
164 views

Independence of the comprehension axiom

What's wrong with the following line of reasoning, if so? The comprehension axiom of Zermelo's set theory would be provable by the other axioms, if the following was provable: ($*$) All ...
11
votes
1answer
642 views

The categories Set and Ens

I've recently started reading Categories for the Working Mathematician and I'm a little puzzled about the distinction between $\textbf{Set}$ and $\textbf{Ens}$. On the one hand, it seems like ...
4
votes
0answers
178 views

Property of Galvin-Hajnal rank

In what follows, $\|\Phi\|_I$ means the Galvin-Hajnal rank of a function $\Phi:\kappa \to Ord$ with respect to a $\omega_1$-complete ideal $I$ of $\kappa$. Lemma 2.2.5 of Introduction to Cardinal ...
4
votes
2answers
513 views

Model existence theorem in set theory

From the FOM newsgroup I learned: It's a theorem of (first-order) set theory that every consistent first-order theory has a model. What's the exact formulation of this theorem in purely ...
5
votes
3answers
465 views

Is the Notion of the Empty Set Relative or Absolute?

Suppose we specify subsets of a reference set by pairs, where the first co-ordinate specifies a member of the universe of discourse, and the second co-ordinate specifies the value that the ...
1
vote
1answer
171 views

Galvin-Hajnal ranks of certain functions

Galvin-Hajnal rank of an ordinal function $f: A \to Ord$ with respect to $I$ is defined as $|f|_I=\sup_{g <_I f} (|g|_I+1)$ where $I$ is an ideal of $A$ and $g <_I f$ iff $\{a \in A: f(a) \leq ...
3
votes
2answers
124 views

Help me construct a transfinite bijective enumeration, with restrictions

Let $E$ be an infinite set of cardinality $\aleph_\alpha$, and let $\mathfrak{F}\subset\mathfrak{P}(E)$ such that $\bigcup\mathfrak{F}=E$, $|\mathfrak{F}|=\aleph_\alpha$, and $|A|=\aleph_\alpha$ for ...
6
votes
3answers
499 views

Does every locally compact second countable space have a non-trivial automorphism?

Does every locally compact second countable space have a non-trivial automorphism? The motivation for this question comes from something I'm thinking about in logic.
1
vote
1answer
262 views

kunen exercise about ccc which is not separable

my question is about: show that if $\kappa > 2^\omega$, then the space $2^\kappa$ is not seperable. (kunen, page 86, exercise 4) – ıf there exist a countable dense set $D$ where is the ...
4
votes
1answer
326 views

Regressive function on an ordinal

I'm trying to prove the following statement. Let $0<\overline{\alpha}\leq\alpha$ be two ordinals such that $\omega_{\overline{\alpha}}$ is the cofinality of $\omega_\alpha$. Let $f$ be a mapping ...
0
votes
1answer
190 views

Sets invariant under sections

Let $X$ be a compact in the Polish space (metric, complete, separable) and $G\subseteq X\times X$ is open. For $x\in X$ we define the section of $G$: $$ s(x) = \{y\in X|\langle x,y \rangle \in ...
14
votes
2answers
702 views

Relationship between Continuum Hypothesis and Special Aleph Hypothesis under ZF

Special Aleph Hypothesis AH(0) is the claim $2^{\aleph_0}=\aleph_1$, i.e. there is a bijection from $2^{\aleph_0}$ to $\aleph_1$. Continuum Hypothesis CH is the claim $\aleph_0 \leq \mathfrak{a}< ...
4
votes
3answers
430 views

Set Theory Notation

I am reading Enderton's Set Theorey, in which he showed proof for a theorem: There is no set to which every set belongs. In the proof, he wrote: Let A be a set; we will construct a set not belonging ...
5
votes
2answers
358 views

Where is the Axiom of choice used?

In Reid's commutative algebra, there is a proof of equivalent conditions of Noetherian rings, especially (1) The set of ideals of $A$ has the a.c.c. $\Rightarrow$ (2) Every ideal in $A$ is finitely ...
7
votes
4answers
698 views

Simple (even toy) examples for uses of Ordinals?

I want to describe Ordinals using as much low-level mathematics as possible, but I need examples in order to explain the general idea. I want to show how certain mathematical objects are constructed ...
7
votes
1answer
171 views

Sequence of surjections imply choice

I am reading a paper where a side remark said that if a sequence $\langle g_\beta\colon\omega\to\beta\mid\beta<\omega_1\rangle$ is a sequence of surjections then $\omega_1$ is regular. I have ...
3
votes
2answers
260 views

How to prove that if $\kappa>\omega$ then $|H(\kappa)|>2^{<\kappa}$

I could not see $|H(\kappa)| > 2^{<\kappa}$. It is a question in Kunen book. The other part is answered, this part may be clear but I could not see. Also, $2^{<\kappa}$ is not clear for ...
0
votes
2answers
410 views

How can I determine the cardinality of a set of polymorphic functions?

It seems obvious to me that the set of functions with the signature $\forall A. A \rightarrow A$ is "once-inhabited", i.e. there is only one such polymorphic function which "works" for any set $A$, ...
11
votes
2answers
505 views

When does $V=L$ becomes inconsistent?

In a wonderful course I'm taking with Magidor we are finishing the proof of the Covering Theorem for $L$. The theorem, in a nutshell, says that $V$ is very close to being $L$ if and only if $0^\#$ ...
3
votes
0answers
183 views

Can Egoroff's Theorem be Strengthened for A Sequence of Smooth Functions?

I have posted this question on MO, and I'll raise the question here in a more concrete way. To simplify the situaton, we assume the measure space we are dealing with right now is a closed interval $I$ ...
0
votes
3answers
461 views

Infinite paths that connect two vertices?

This is a follow-up to another question concerning infinite paths which was admittedly ill-posed. I hope this question is posed better. The graph $N$ with vertex set $V(N) = \mathbb{N}$ and $(x,y) ...
6
votes
2answers
221 views

Boolean algebras without atoms

Why is the theory of Boolean algebras without atoms $\omega$-categoric?
5
votes
2answers
285 views

$\omega$-saturation of $(\mathbb{R},<)$

Could anyone of you explain me why $(\mathbb{R},<)$ is $\omega$-saturated? EDIT: do you know also why the theory of Boole algebras without atoms is $\omega$-categoric? Added: The added question ...
-2
votes
3answers
434 views

Infinite shortest paths in graphs

From Wikipedia: "If there is no path connecting two vertices, i.e., if they belong to different connected components, then conventionally the distance is defined as infinite." This seems ...
9
votes
2answers
204 views

Easy Question about P-names

Notation(from Kunnen): $M$ is countable transitive model of $ZFC$, $P$ is a partial order that belongs to $M$, $G$ is a $P$-generic filter over $M$, $\hat{b}$ will be a $P$-name for $b\in M$, and ...
2
votes
1answer
200 views

Show that for any A, there exists { dom R | R in A }

I want to show that (a) $ \forall \alpha \; \exists \beta$ such that $\beta = \{ dom R \; | \; R \in \alpha \}$ I'm having difficulty proving this axiomatically. I'm not even certain that it can be ...
7
votes
1answer
322 views

powers of singular cardinals

I am trying to solve the following two problems: 1) if $\beta <\omega_1$, $2^{\aleph_1}<\aleph_{\omega_1}$, and $\aleph_\alpha^{\aleph_0} \leq \aleph_{\alpha +\beta}$ for a stationary set of ...
9
votes
2answers
274 views

How to show the existence of an infinite set of independent undecidable sentences?

How to show the existence of an infinite set of independent undecidable sentences? By "independent" I mean that no implication between any two elements is provable. A finite set that satisfies the ...
8
votes
1answer
234 views

Understanding an Easy Relative Consistency Proof

When proving that $(ZF-Reg)\vdash CON(ZF - Reg)\rightarrow CON(ZF)$ we start by defining the class $WF=\cup_{\alpha \in ORD} V_{\alpha}$. Then we prove with $ZF-Reg$ that actually $WF$ is a model for ...
5
votes
2answers
440 views

Why is cofinality important? And some help in proving absoluteness for $R(\kappa)$

I am starting to read the Kunen's book set theory. I could not understand the why we need cofinality. Why is it important? Also, I'm trying to solve some exercises (on page 146) about absoluteness ...
6
votes
1answer
104 views

Basic constructibility question

I'm currently reading J. D. Hamkins' paper "Unfoldable cardinals and the GCH," and I've run across a comment that I think I ought to find trivial, but I don't. On page 1187, he says that ...
12
votes
2answers
464 views

Any good decomposition theorems for total orders?

I'm learning about set theory and partial orders, well orders, total orders, etc. I'm curious to know of any good decomposition theorems that apply to all (or very general types of) total orders. ...
7
votes
1answer
556 views

No uncountable ordinals without the axiom of choice?

In Uncountable ordinals without power set axiom Francois Dorais explains that without the Power-set Axiom we cannot prove the existence of uncountable ordinals. I am guess that the power set of an ...
3
votes
1answer
577 views

Non-standard models of arithmetic for Dummies (2)

I've learned that there are (at least) three types of countable non-standard models of arithmetic, depending on the primitive operations: successor only → $\mathbb{N}$ followed by a copy of ...
5
votes
1answer
287 views

If you randomly choose a subset of the real line, what is the probability that it will be measurable?

Suppose that you are working with the axiom of choice. If you randomly choose a subset of the real line, what is the probability that it will be measurable?
4
votes
3answers
226 views

Hausdorff ultrafilters

I know that Ramsey ultrafilters are Hausdorff ($\mathcal{U}$ is Hausdorff iff for every $f,g:\mathbb{N}\rightarrow\mathbb{N}$ $f(\mathcal{U})=g(\mathcal{U})$ then $f\cong_\mathcal{U} g$ $\;$). So if ...
3
votes
1answer
111 views

Small question on proof that any ordinal is the index of some initial ordinal

I'm reading a proof of a theorem, but an apparently trivial case has tripped me up. The theorem goes as For any ordinal $\alpha$, there is a initial ordinal $\phi$ such that $i(\phi)=\alpha$. Here ...
8
votes
1answer
289 views

Universal sets in metric spaces

Today I saw in the class the theorem: Suppose $X$ is a separable metric space, and $Y$ is a polish space (metric, separable and complete) then there exists a $G\subseteq X\times Y$ which is open and ...
18
votes
4answers
863 views

Mathematical statement with simple independence proof from $\mathsf{ZF}$

Is it possible for someone with little set-theoretic knowledge (e.g., me) to understand the proofs that either $\mathsf{CH}$ or $\mathsf{AC}$ is independent of $\mathsf{ZF}$? I am looking for any ...