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

3
votes
2answers
203 views

Why is this contradiction using axiom of constructibility incorrect?

Today I thought of this paradox and I'm trying to find the wrong assumption that causes it. Does anyone know what is wrong in the following argument: let $$A=\{x\in \mathbb{R}|\exists\phi:\forall ...
0
votes
1answer
61 views

$ V_{ \kappa}$ ( $ \kappa $ inaccessible ) models there is a countable model of ZFC

I think that this statement is very well-known but I am a bit unclear on some of the reasoning. I am aware that $ V_{ \kappa }$ models ZFC when $ \kappa $ is an inaccessible cardinal. Therefore by ...
1
vote
0answers
41 views

Does subsets of a semigroup with strictly smaller cardinality insure infinitely many disjoint translation copies of the set?

Let $S$ be a semigroup with infinite cardinality, $A\subset S$ with $|A|<|S|$. Under what condition we may find a infinite net $\{s_\alpha; \alpha\in \Gamma\}$, such that $s_\alpha A \cap s_\beta A ...
1
vote
1answer
30 views

Is $|2^\omega \cap L| = |(\omega_1)^L|$?

Is $|2^\omega \cap L| = |(\omega_1)^L|$? Here I mean the set of all subsets of $\omega$ by $2^\omega$, but I hope that choosing that particular interpretation does not really matter here, as always. ...
2
votes
1answer
111 views

Is there is limit to what $2^{\aleph_0}$ can be in a ctm?

Is there is limit to what $2^{\aleph_0}$ can be in a countable transitive model? How large can be the value of the continuum in a countable transitive (standard) model of ZFC? For instance, if we ...
4
votes
1answer
125 views

About ordinals and cardinals in structural set theory

Among the most important concepts of set theory for mathematical real life applications are ordinal numbers and cardinal numbers. In material set theory, ordinal numbers are defined as transitive ...
2
votes
1answer
80 views

If $\kappa$ is regular, is $\prod^{\text{fin}}_{\alpha<\kappa}\operatorname{Fn}(\omega,\alpha)$ $\kappa$-cc?

Given an ordinal $\alpha$ let $\operatorname{Fn}(\omega, \alpha)$ be the set finite partial functions from $\omega$ to $\alpha$. Given a cardinal $\kappa$ let ...
9
votes
1answer
230 views

Product of metric outer measures

The problem below has been asked recently already but, as a naive user, I got burned (well singed perhaps) because I asked the question in the wrong place. So if this looks like a redundant question ...
0
votes
2answers
61 views

on the continuum hypothesis: only finitely many cardinals between $\aleph_0$ and $2^{\aleph_0}$

Is it at least known that there are only finitely many cardinals between $\aleph_0$ and $2^{\aleph_0}$?
0
votes
1answer
42 views

Showing extensionality for Mostowski collapse

I have been trying to show that if $\gamma$ is such that there is a real (an element of $2^\omega$) in $L_{\gamma+1} \setminus L_\gamma$, then there are countable $M$ and a surjection $f : \omega ...
4
votes
1answer
185 views

On the Axiom of Choice for Conglomerates and Skeletons

Say that $\mathcal{X}$ is a conglomerate if $\mathcal{X} = \{X_i: i \in I\}$, where each $X_i$ and $I$ are classes. The Axiom of Choice for Conglomerates is the statement: Whenever $\mathcal{X}$ and ...
1
vote
2answers
119 views

Real numbers for beginners

I am thinking about the Wikipedia (I understand disputed) article about “definable real numbers”. It begins to say that, A real number $a$ is first-order definable in the language of set theory, ...
2
votes
1answer
36 views

Measure theory Vitali nonmeasurable set.

On $[0,1]$ I've got the relation $\sim$ defined as: $x \sim y \iff x-y \in \mathbb Q$ this is a relation of equivalence and so: we can make a factor class ...
1
vote
2answers
61 views

König's theorem (set theory) implication

How does König's theorem imply $\quad\aleph_{\omega} \neq \beth_1$?
3
votes
2answers
61 views

Countable Elementary Submodels $M \preccurlyeq H ( \theta) $.

I am reading through some of Kunen's material on Elementary Submodels and am a little unclear on one proof. Here is a part of the claim: Let $ \theta$ be an uncountable cardinal, and let $ M$ be ...
4
votes
0answers
68 views

Why is the intersection of countably many homogeneously Suslin subsets of $\,^{\omega} \omega$ homogeneously Suslin?

I found this assertion in these notes: The derived model theorem (Steel) right in the beginning on page 3, together with the remark that this is 'not too hard to show'. Unfortunately, I'm ...
2
votes
1answer
70 views

Is there a sequence of universes in TG?

Assume we are working in ZFC + Tarski's axiom (Every set is an element of some universe). I wonder, if there is a universe $U$ with a sequence $(U_n)_{n\in \mathbb{N}}$ in $U$, s.t. $U_1,U_2,\dots$ ...
1
vote
1answer
59 views

Show that every proper filter on a set X can be extended to a proper prime filter?

Are the following enough to complete the proof? The union of a chain of filters is a filter. A maximal filter is an ultra-filter. How I can use Zorn's lemma to find the maximal filter?
1
vote
2answers
91 views

Proving infinity vs Axiom of infinity

I am not much of a set theorist, I deal primarely with algebra in my interest and what I study so this is toward set theorists. I am curious as to why cannot infinity be properly proven to exist? I ...
9
votes
4answers
146 views

Does it make sense to define $ \aleph_{\infty}=\lim\limits_{n\to\infty}\aleph_n $? Is its cardinality “infinitely infinite”?

I recently read a book about infinity, which introduced the basic notions of different kinds of infinity. I'm a total layman concerning this topic, and one question fascinated me: Can we, in some ...
2
votes
1answer
71 views

Tarski-Grothendieck set theory [duplicate]

Does Tarski-Grothendieck set theory can prove the consistency of ZFC?
0
votes
2answers
64 views

Equivalence Modulo an Ultrafilter in creating Hyperreals

If we let F be a non-principal ultrafilter on the Natural numbers, and define $a,b \in \mathbb R^{\mathbb N}$ to be real-valued sequences. Then the equivalence relation ~ can be defined as a~b if ...
5
votes
1answer
196 views

More than the real numbers:hyperreals, superreals, surreals …?

I've read something about extensions of the real numbers, as hyperreals, superreals, surreals and, as I can understand, all these extensions contain some new kinds of infinitesimal and infinite ...
1
vote
1answer
75 views

Countable Ordinals

In an intro topology class we briefly brought up ordinal numbers during a conversation of transfinite induction. I believe I understand how the ordinal numbers work, at least up to $\omega^\omega$. ...
0
votes
1answer
59 views

Some questions regarding a theorem of Paul J. Cohen

In his paper "Automorphisms of Set Theory", Paul Cohen proved the following theorem: "There exist models of $ZF$ admitting automorphisms of order two. More exactly, If $M$ is any countable model ...
1
vote
1answer
31 views

Two ways to describe image of a filter under a function

Let $f$ be a function from a set $A$ to a set $B$. Let $\mathcal{A}$ be a filter on $A$. (Note that I do not require that all filters are proper.) It is easy to verify that $\{Y\in\mathscr{P}B \mid ...
0
votes
0answers
50 views

(Set theory, Ordinal number) How to construct explicit isomorpism between epsilon(ordinal number) and subset of Q?

Now I know that there 'is' isomorpism between them, but the problem asks explicit isomopism. How can I do this? (This is problem from Introduction to Set Theory(Hrbacek,Jech) chapter6.)
3
votes
2answers
76 views

Equivalence of categories and axiom of choice

I don't understand much about set theory but nevertheless try to learn category theory. In books like Gelfand Manin, there is the well known theorem that a functor defines an equivalence of categories ...
1
vote
1answer
54 views

Cohen forcing question

Suppose $M$ is a countable transitive model of ZFC and $(x, y, z)$ is Cohen generic point in $\mathbb{R}^3$ over $M$: This means that for every open dense set $U \subseteq \mathbb{R}^3$ in $M$, $(x, ...
1
vote
0answers
26 views

(Ordinal number)How to prove that epsilon-naught is countable without using Axiom of Choice? [duplicate]

How to prove that epsilon-naught is countable without using Axiom of Choice? or, Can we explicitly show that there is isomorpism between epsilon naught and subset of rational number?
5
votes
3answers
118 views

How can we define the set $\{\{\}, \{\{\}\}, \{\{\{\}\}\}, …\}$?

How do you define the set of elements which can be made by repeatedly wrapping the null set in another set: $\{\{\}, \{\{\}\}, \{\{\{\}\}\}, ...\}$? I've tried $A = \{x\ |\ x=\emptyset \lor \exists ...
-2
votes
1answer
59 views

This cardinal exist in ZFC

Does $2^{{\aleph_0}^{\aleph_0}}$ exist in ZFC without additional large cardinals axioms?
0
votes
0answers
46 views

Formulas of LST

I have been slowly teaching myself the fundamentals of Set Theory, but as I get more in-depth, I feel I have a misunderstanding about what formulas of LST are... In the book 'Constructibility' by ...
0
votes
0answers
20 views

Question of Hechler forcing and $\mathbb{LOC}$ notion Localization forcing

Let $\mathbb{D}$ notion Hechler forcing and $\mu$ uncountable regular cardinal. All subalgebra of $\mathbb{D}$ of size $<$ $\mu$ is $\mu$-centered. Let $\mathbb{LOC}$ notion Localitation forcing. ...
5
votes
1answer
85 views

Members of $\Sigma^1_4$ sets

I'm pretty sure this is an easy descriptive set theory question that I'm just blanking on. Is it consistent with large cardinals - say, with a measurable - that every (nonempty) $\Sigma^1_4$ class ...
2
votes
3answers
192 views

Recommendations for Intermediate Level Logics/Set Theory Books

I currently don't know what books I should read after having studied elementary logics/set theory, so I'd be glad if I can get some recommendations. My classes used the following two books for ...
-1
votes
1answer
80 views

Defining the critical point of an elementary embedding when there exist incomparable cardinals.

Assume one has. for $V$ and for some transitive class $M$, an elementary embedding $j$: $V$$\rightarrow$$M$ and that $j$$\neq$$id$, where $id$ is the identity. If $V$ and $M$ satisfy $ZFC$ then the ...
1
vote
1answer
57 views

Induction principle in the proof of the recursion principle

Dudley, in " Real Analysis and Probability", defines and proves the recursion principle as it follows: The passage tagged in yellow is an application of the induction principle, which has been ...
0
votes
2answers
44 views

Using the axiom of choice to construct uncountable non-cocountable subsets of a set

This question is a natural follow up to Constructing a subset of an uncountable set which is neither countable nor co-countable. Let $\Omega$ be an uncountable set. Using the axiom of choice, how can ...
2
votes
1answer
148 views

Constructing a subset of an uncountable set which is neither countable nor co-countable

Let $\Omega$ be an uncountable set. Without assuming the axiom of choice, does there exist a subset $S\subset \Omega$ such that neither $S$ nor $\Omega\setminus S$ is countable?
2
votes
2answers
84 views

Elementary embeddings,V,set theory,L,cardinals

I would like to make clear some properties of $j,M,L$ and $V$ in ZFC. let $j:V \to M$ denote a (nonidentity) elementary emedding and $M$ a transitive $\in$-model of ZFC.Is there an example of ...
4
votes
2answers
113 views

$L$ models set theory and so does $V_\kappa$ for $\kappa$ inaccessible

Background: I have been reading the 1980 edition of Kunen. Theorem VI.2.1 states it is provable from ZF that $\mathbf L$ (Kunen writes classes in bold) is a model of ZF. Also, it is a well-known ...
4
votes
2answers
82 views

Is Set auto-similar?

Let me first define the following notion of auto-similarity (I'm not using the formalism of signature because I'm not really accustomed with it, so let's stay intuitive): we say that a structure $A$ ...
2
votes
1answer
44 views

Axiom of Replacement for Cartesian Products

I am reading through Kenneth Kunen's Foundations of Mathematics in which he invokes the Axiom of Replacement to justify the existence of Cartesian products for two sets. I have two problems, and I ...
3
votes
2answers
96 views

Is $V=L$ a single first-order sentence?

Is $V=L$, the axiom of constructibility, a single first-order sentence? Since $V=L$ really stands for $(\forall x)(\exists \alpha)[\alpha \in \mathrm{On} \wedge x \in L_\alpha]$, so my question might ...
2
votes
1answer
80 views

Follow-up question on Monotonic “Subfunction”

Let $f$: $\mathbb{R}\longrightarrow\mathbb{R}$ be an arbitrary function. Must there exist $E\subseteq\mathbb{R}$ of cardinality $\aleph_1$, such that $f$ restricted to $E$ is monotonic? Assuming CH, ...
1
vote
1answer
35 views

ZF without regularity allows P(A) member of A

Last week, I was doing a bit of reading around on some axiom of regularity(/foundation)-related questions, and found an answer in one of them (which I cannot seem to locate, for the life of me, right ...
46
votes
13answers
4k views

What is the definition of a set?

From what I have been told, everything in mathematics has a definition and everything is based on the rules of logic. For example, whether or not $0^0$ is $1$ is a simple matter of definition. My ...
4
votes
2answers
66 views

Conditions that topologies must have if (only if) the condition “$G_\delta$ iff (open or closed)” holds?

Consider the class of topological spaces $\langle X,\mathcal T\rangle$ such that the following are equivalent for $A\subseteq X$: $A$ is a $G_\delta$ set with respect to $\mathcal T$ $A\in\mathcal ...
5
votes
0answers
84 views

Why does Namba forcing preserve $\omega_1$?

I have seen proofs that under CH, Namba forcing does not add reals, and thus preserves $\omega_1$. How do you prove in ZFC alone that it preserves $\omega_1$? I have also seen the stronger claim ...