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
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0answers
15 views

if [a,b] is a subset of(c,d) what relationships exist between a,c and b,d? [on hold]

I want to know what is the relationships I couldn't figure it out???? if $[a,b]$ is a subset of $(c,d)$ what relationships exist between $a$, $c$ and $b$, $d$?
3
votes
3answers
67 views

Is the critical point of an embedding of a model of set theory inaccessible in it?

Can we find an elementary embedding $j:M\to N$ with $M,N$ transitive $ZFC$-models, $\kappa$ being the critical point, so that $\kappa$ is not inaccessible in $M$ ? ($\kappa$ is regular in $M$.) I ...
3
votes
0answers
36 views

Looking for extender axioms

Consider the following extender construction: Given an elementary embedding $j:V\to M$, where $M$ is transitive, with critical point $\kappa$, we can for each $a\in j(V_{\kappa})$ define a ...
2
votes
1answer
30 views

$\omega_1$-closedness and fullness for $\searrow$ $\omega$-sequences

Let $\pi$ is a $\Bbb{P}$-name for a partial order, i.e. there is a name $\pi'$ and $\pi''$ such that $$1\Vdash_\Bbb{P} \pi '' \in \pi\land (\text{$\pi'$ is a partial order of $\pi$ with largest ...
3
votes
2answers
79 views

How can choice fail in ZF?

I don't see how the Axiom of Choice can fail in ZF. By transfinite induction you can demonstrate larger and larger ordinals, using union and pairing to show the successor and limit steps, so that for ...
3
votes
0answers
71 views

Facts on elementary submodels

In the paper of "Aspero, Larson, Moore - Forcing Axioms and the CH" three facts are stated as well-known. As i have not read them before, they are not that obvious to me. Maybe good references to ...
2
votes
1answer
40 views

Looking for a clarification of the Suslin $\mathcal{A}$-Operation with a (finite) example

I have a problem concerning the output of (and the intuition behind) the Suslin $\mathcal{A}$-Operation. More specifically, I really don't see exactly what the output of it really is (even if I can ...
5
votes
3answers
320 views

Definition of infinite tree in set theory

Really basic question concerning trees in set theory. What is the definition of an infinite tree? I ask the following because, rather peculiarly, neither in Kechris classical book on descriptive ...
5
votes
1answer
71 views

Sets Forced to be Equal in All Extensions

My question is: Let $\mathbb{P}$ be a forcing and $\tau \in V^\mathbb{P}$ is a name. Suppose that $$1_{\mathbb{P} \times \mathbb{P}} \Vdash_{\mathbb{P} \times \mathbb{P}} \tau_\text{left} = ...
0
votes
0answers
34 views

Cohen Forcing in Set Theory - Proof that Forcing is Equivalent to intersection of Dense Sets

Cohen's book "Set Theory and the Continuum Hypothesis" on Page 126/127 (see below) shows that the existence of a completed new set a' is equivalent to its intersection with all dense subsets in M. I ...
0
votes
0answers
56 views

A question regarding a theorem of Erdos and Hajnal

Consider the following theorem of Erdos and Hajnal: Definition. For any set $x$, a function $f$ is called ${\omega} $-Jonsson iff $f$: $^{\omega}x$ $\rightarrow$ x and whenever $y$$\subseteq$$x$ and ...
2
votes
0answers
67 views

Closeness of measures on a cardinal

Given an uncountable $\kappa$ and a $\kappa$-complete nontrivial non-normal ultrafilter on $\kappa$, and some $g:\kappa\to\kappa$ with $<_{U}$-rank $\kappa$ (where $f_0<_Uf_1$ iff ...
2
votes
0answers
101 views

Higher infinities without Set Theory

Apart from Cantor's diagonalization argument, there are a number of ways to show that cardinality of R is greater than that of N (eg: Baire Category theorem, path connectedness of R and so on). Are ...
2
votes
0answers
62 views

Measuring the set-theoretical complexity of sets/spaces encountered in general analysis

In analysis, it is common to encounter subsets of $\mathbb R$ (or even $\mathbb R^n$) which appear to be "well-behaved", especially with regard to properties like being measurable, compactness, etc. ...
3
votes
1answer
64 views

Independence of existence of inaccessible cardinals

Let $I$ be the formula which states that there exists strongly inaccessible cardinals. My question is regarding the proof of $ZFC\nvdash I$ appearing in Jech (part of theorem 12.12). He starts by ...
13
votes
3answers
1k views

Motivation for different mathematics foundations

I've been studying an introductory book on set theory that uses the ZFC set of axioms, and it's been really exciting to construct and define numbers and operations in terms of sets. But now I've seen ...
4
votes
1answer
67 views

extending automorphisms in complete boolean algebras

Suppose $A$ is a complete subalgebra of a complete boolean algebra $B$. Suppose $f : A \to A$ is an automorphism. Then $f$ can be extended to an automorphism of $B$. I can see this using the fact ...
2
votes
0answers
65 views

Countably closed Boolean algebra of subsets of the real plane,

The following problem was in The American Mathematical Monthly : A generalized rectangle is $E \times F$ for any subsets $E,F$ of $\Bbb R$ (the reals). If $\mathscr{B}$ is the smallest countably ...
2
votes
0answers
30 views

Bijection of power sets and cardinality [duplicate]

Background We are proving that two free groups on sets $S_1,S_2$ are isomorphic iff the sets have the same cardinality. If they are finite, it's easy to show, by the freeness of the groups, that ...
5
votes
2answers
91 views

Number of equivalence classes of functions of real variable with the a.e relation.

What is the cardinal of the set $\mathcal{F}(\mathbb{R};X)/ \sim$ where $\sim$ is the relation $f\sim g \iff \mu(\{x\in \mathbb{R};f(x)\ne g(x)\})=0$ and $|X|=|\mathbb{R}|$? I guess that is ...
1
vote
1answer
53 views

Countable choice and totally bounded metric spaces

Can we prove that the following statement is equivalent to the axiom of countable choice (CC)? If every sequence in a metric space $X$ has a Cauchy subsequence, then $X$ is totally bounded. ...
5
votes
3answers
88 views

Infinite set always has a countably infinite subset

I'm trying to show that one infinite always has a countably infinite subset, but I'm confused with something on the proof. Let $S$ be one infinite set. In that case, to show it has one countably ...
2
votes
1answer
33 views

ZFC : If $I$ is a set, and $A_i$ is a set for every $i$, is $A = \{A_i\}_{i \in I}$ necessarily a set?

I just started to look into the ZFC axioms, and I'm not sure what is the answer to the following question: If $I$ is a set, and $A_i$ is a set for every $i$, is $A = \{A_i\}_{i \in I}$ necessarily a ...
0
votes
0answers
67 views

Proof for this equivalent statement to $2^{\aleph_0}=2^{\aleph_1}$ ??

This statement is some kind o a weak form of diamond and I am looking for a proof for its equivalence to $2^{\aleph_0}=2^{\aleph_1}$. $2^{\aleph_0}=2^{\aleph_1}$ is equivalent to the following ...
3
votes
1answer
58 views

Von Neumann universe in ZC

Why doesn´t the von Neumann hierarchy to $V_{\omega_1}$ exist in Zermelo set theory if with Scott´s trick you can "count" to $ \omega_1 $
1
vote
1answer
41 views

Binary tree of splitting that separates point over every set?

Is the following true? Let I be any set. For me a binary tree of splitting of I will be the following: start with $I_0=I$, at the step $n+1$ take the set of step $n$ and split each of them in two ...
5
votes
0answers
122 views

Where does the term “mouse” (in set theory) come from?

Does the term mouse stand for "Model Of Universe with Sequences of Extenders" or perhaps mice stands for "Model Including Cardinal Extensions"? A quick review: Gödel's L-universe is a core model ...
3
votes
1answer
83 views

What else does ZFC prove about the “spectrum” of a cardinal number?

An auxiliary definition: Definition 0. Given an infinite set $X$ and a filter $\mathcal{F}$ on $X$, let $\sim_\mathcal{F}$ denote the equivalence relation on $\mathcal{P}(X)$ defined as follows: ...
2
votes
3answers
50 views

Relation between topological denseness and denseness over poset

In the theory of forcing, the notion of dense set is important. Formally, a subset $D$ of a poset $P$ is dense if, for any $p\in P$ we can find some $q\in D$ with $q\le p$. Intuitively, denseness of ...
2
votes
0answers
75 views

On the paper “Forcing and the CH” by Aspero/Larson/Moore

Forcing Axioms and the CH by Aspero/Larson/Moore On page 11 of this paper I struggle with the beginning of the proof of Lemma 3.5. Coding: For $r \in 2^\omega$ and $A \in H(\aleph_1)$ let us say ...
1
vote
2answers
75 views

Does the “special continuum hypothesis” imply the axiom of choice?

In $\mathsf{ZF}$ set theory, does the "special continuum hypothesis" imply the axiom of choice, or is the axiom of choice independent of it? Here, by the "special continuum hypothesis" we mean the ...
1
vote
1answer
32 views

Sets,transversals,PT property,cardinals

A transversal of a family $S$ of sets is an injective choice function. $PT(\lambda,\chi)$ means, if $S$ is a family of $\lambda$ sets,each of cardinality $<\chi$,and every subfamily with ...
3
votes
1answer
62 views

Are the hyper-reals countably transitive?

A hyper-real field is $ R^*=(R^N)_{/U}$ where $U$ is a free ultrafilter on $N$. If A and B are any countable order-isomorphic subsets of $R^*$, is there an order-automorphism of $R^*$ that maps $A$ ...
4
votes
3answers
45 views

some basic cardinal arithmetic on $\text{cf}(\aleph_{\omega_1})$

I'm reading The Joy of Sets by K. Devlin, by self-study. I've just seen a statement $\text{cf}(\aleph_{\omega_1})=\omega_1$ without proof, but I think this is slightly harder to prove than more ...
2
votes
1answer
31 views

Construction of a Ramsey ultrafilter

I am having difficulties with the proof in Jech Set Theory concerning the existence of Ramsey filters in case the continuum hypothesis holds. A similar question about the same proof was asked here ...
4
votes
1answer
128 views

Can the generalized continuum hypothesis be disguised as a principle of logic?

A cool way to formulate the axiom of choice (AC) is: AC. For all sets $X$ and $Y$ and all predicates $P : X \times Y \rightarrow \rm\{True,False\}$, we have: $$(\forall x:X)(\exists y:Y)P(x,y) ...
1
vote
1answer
78 views

Difference between model and interpretation

In the book Mathematical Logic by J. Shoenfield, the author uses the concept of a interpretation of set theory to proof consistency results, while the other texts on set theory (e.g. Kunen and Jech) ...
5
votes
1answer
87 views

Iterated Forcing, to force $2^{\omega}=\kappa$ and $2^{\omega _1}=\lambda$

Hellow i'm stuck on some details in this iterated forcing exercise. Let $M$ be a countable transitive model of $ZFC+GCH$ and assume that $\kappa<\lambda$ are cardinals with $\aleph _0 ...
5
votes
1answer
69 views

Given any set of consistent axioms, is it always possible to find a model for these axioms in ZFC set theory?

If not, are there any conditions under which there must be a model under ZFC theory? Alternatively, is there any set of axioms for which this does hold true? If so, can we drop some of the axioms and ...
5
votes
2answers
96 views

Is there a turing machine for which halting is equivalent to the Axiom of Choice or its negation?

As seen in "A Turing machine for which halting is outside ZFC", Gödel's incompletness theorem can that there a turing machines for which halting can not be decided. My question is, is there a turing ...
0
votes
2answers
50 views

To prove two sets are isomorphic is it sufficient to find 2 injections or 2 surjections?

To prove two sets are isomorphic is it sufficient to find 2 injections $f:A\to B$ and $g:B\to A$ or 2 surjections $h: A\to B$, $j:B\to A$?
8
votes
1answer
133 views

Why Does The Reflection Principle Fail For Infinitely Many Sentences?

I've read the proof that ZF cannot be finitely axiomatized via the reflection principle and the 2nd incompleteness theorem. Since ZF can be countably axiomatized, the finiteness requirement in the ...
6
votes
0answers
157 views

Cutting a Banach-Tarski Cake

I was reading a cake-cutting problem here (not really related, so I won't link to it), and for some reason, this variation occurred to me. I have no idea whether this problem is even well-formed: ...
3
votes
1answer
64 views

Why does a proof of $\exists f: X\to Y$ injection $\iff \exists g: Y \to X$ surjection requires the axiom of choice?

Why does a proof of $\exists f: X\to Y$ injection $\iff \exists g: Y \to X$ surjection requires the axiom of choice? This question is answered here: There exists an injection from $X$ to $Y$ if and ...
4
votes
2answers
62 views

Proving that $(\omega_n)^\omega=\omega_n$ providing CH but not GCH

This is an exercise from a book from Kunen - SET THEORY, An Introduction to Independence Proofs Assume CH but don't assume GCH. Show that $(\omega_n)^\omega=\omega_n$ for $1 \le n < \omega$. I ...
3
votes
1answer
40 views

Number of subsets of $\mathfrak c$ that are different no matter how high you go?

I am interested in collections $\mathcal X \subseteq \mathcal P(\mathfrak c)$ such that for any distinct $X,Y\in\mathcal X$ and $\alpha<\mathfrak c$ we have $X\cap [\alpha,\mathfrak c]\neq Y\cap ...
2
votes
1answer
39 views

Is the following a legitimate proxy for the Axiom of Replacement?

I'm working on an interface between set theory and plural logic. Here's my question: If one were to endow set theory with the expressive resources of plural quantification, could the following count ...
4
votes
0answers
44 views

Over ZF does “every non-seperable Hilbert space has an orthonormal basis” imply “there exists a non-Lebesgue measurable set”?

I know from this question that it's an open problem whether or not the existence of a dense orthonomral basis for every real or complex Hilbert space $(\text{B}_\text{orth})$ implies the axiom of ...
10
votes
1answer
190 views

Is there a non-trivial countably transitive linear order?

A linear order $<$ on a set S is countably transitive iff, whenever A and B are order-isomorphic countable subsets of S, there is an order-automorphism of S which maps A onto B. Does such an order ...
4
votes
2answers
74 views

Intuition behind constrution of the Hyperreals

Just want to attempt to check if my understanding/intuition for the construction of the Hyperreal numbers via an ultraproduct is correct. Appreciate any corrections or help. So Hyperreals are ...