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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Are $\sf ZF+Con(\sf ZF),\sf ZFC+Con(\sf ZF)$ equiconsistent?

It's very well known that (over base theory being $\sf ZF$) theories $\sf ZF$ and $\sf ZFC$ are equiconsistent. Is the same known to be true about $\sf ZF+Con(\sf ZF)$ and $\sf ZFC+Con(\sf ZF)$? How ...
166
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1answer
5k views

Does the open mapping theorem imply the Baire category theorem?

A nice observation by C.E. Blair1, 2, 3 shows that the Baire category theorem for complete metric spaces is equivalent to the axiom of (countable) dependent choice. On the other hand, the three ...
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1answer
37 views

Total ordering on $\mathcal P(\Bbb R)$

Is there a total ordering on $\mathcal P(\Bbb R)$, the set of all subsets of $\Bbb R$, such that the set of countable subsets is dense in it? (Given a total ordering $(X,>)$, a set $A\subseteq X$ ...
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0answers
57 views

Cardinality of ultraproduct [on hold]

Let $\mathcal{U}$ be a non-principal ultrafilter on $\omega$. Let $S:\omega\rightarrow\omega$ be monotone and unbounded. Let $T_{\mathcal{U},S}=\prod\limits_{\mathcal{U}}S(n)$ the ultraproduct as set. ...
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0answers
52 views

Limiting the size of near-coherence classes in $\omega^*$

We say that two ultrafilters in $\omega^*$ (i.e., two non-principal ultrafilters on $\omega$) are nearly coherent if there is a finite-to-one mapping $\varphi:\omega\to\omega$ such that $\beta\varphi$ ...
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3answers
121 views

Help understanding incomparable cardinalities

Given a set $A$ with cardinality $c$, there is a subset of $A$ having any cardinality less than $c$. There is an injection from the subset $B$ to the set $A$, namely, the identity in which each ...
3
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1answer
30 views

Does every set have choice sequences as long as the original set?

Given a set $X$, we say that $X$ has choice sequences of length $|I|$, denoted $CS(|X|,|I|)$, if for any $f:I\to{\cal P}(X)\setminus\{\emptyset\}$ there is a function $g:I\to X$ such that $g(x)\in ...
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1answer
122 views
+50

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 ...
3
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3answers
125 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 ...
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42 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 ...
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120 views

Infinite Chain of Implication Statements - Can it have a Conclusion?

First, does an infinite string of implication statements have a conclusion? If so, is there a such thing as a "closure" of such a beast, giving a conclusion? I am also trying to determine if the ...
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3answers
91 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
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1answer
34 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 ...
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2answers
223 views

Is transfinite induction needed to remove all the elements from an uncountable set?

Given any uncountable set S, would I need to use transfinite induction to prove if I remove single elements recursively, I will be left with the empty set? It seems like this can be thought of as an ...
3
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2answers
88 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 ...
2
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1answer
44 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
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3answers
328 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 ...
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1answer
75 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} = ...
9
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6answers
729 views

Countable axiom of choice: why you can't prove it from just ZF

This is a follow-up question to the discussion about the finite axiom of choice here. Suppose we have a countable collection of non-empty sets $\{A_1, A_2, A_3,\cdots\}$ Reasoning as indicated in ...
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0answers
36 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 ...
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3answers
1k views

Banach Tarski — any demonstration?

Is there any where to watch a video of a ball being decomposed into 5 pieces that are then translated and rotated to create two balls? How is this even possible without stretching? Is it possible ...
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0answers
71 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 ...
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0answers
57 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 ...
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0answers
109 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 ...
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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
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1answer
66 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 ...
3
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1answer
112 views

What is wrong with this proof that $\text{card}(\mathcal{P}(\mathbb{N}))=\aleph_1$?

I'm reading books on set theory and I came up with the following 'proof' that $\text{card}(\mathcal{P}(\mathbb{N}))=\aleph_1$. What is wrong with it? I really cannot tell! By ...
13
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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
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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 ...
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68 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 ...
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1answer
544 views

Do sets, whose power sets have the same cardinality, have the same cardinality?

Is it generally true that if $|P(A)|=|P(B)|$ then $|A|=|B|$? Why? Thanks.
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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
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2answers
94 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 ...
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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. ...
2
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1answer
34 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 ...
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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
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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 $
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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
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0answers
124 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
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1answer
64 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 $A$ ...
3
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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: ...
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3answers
52 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 ...
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0answers
77 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 ...
3
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1answer
117 views

How do I show the existence of a weakly inaccessible cardinal is not provable in ZFC?

So I know that if we assume the existence of inaccessible cardinals than we can show ZFC is consistent. How do I show that the existence of such cardinals is not provable in ZFC?
2
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2answers
98 views

Example of a $\kappa$-long sequence of disjoint club subsets of regular cardinal $\kappa$

I'm self-studying set theory and got stuck on this exercise: Let $\kappa$ be a regular cardinal. Give an example of a sequence $\langle C_\alpha\mid\alpha<\kappa\rangle$ such that $C_\alpha$ is ...
11
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1answer
176 views

Sets of Cardinals without choice

I have a theory that the axiom of choice is equivalent to the statement that sets of distinct cardinals are well ordered by cardinality. I can prove that the axiom of choice implies this. However I am ...
1
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1answer
72 views

Ultrapower construction of the Hyperreal numbers

I've been learning about non-principal ultrafilters with the overall aim of understanding the ultrapower construction of the Hyperreals. Couple of things I'm confused on: Firstly the Ultrafilter ...
13
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2answers
427 views

What's the difference between saying that there is no cardinal between $\aleph_0$ and $\aleph_1$ as opposed to saying that…

... $\aleph_1$ is the immediate successor of $\aleph_0$? I was reading the wiki article on $\aleph_1$ where a distinction is made between the two. If there's isn't a cardinal between $\aleph_1$ and ...
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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 ...
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2answers
76 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 ...