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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1answer
51 views

Unordered pair of proper classes

The usual Kuratowski ordered pair function does not work on proper classes, because if $A,B$ are proper classes and $\langle A,B\rangle=\{\{A\},\{A,B\}\}$, then since $A\notin\{A\}$ and so on you get ...
2
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0answers
47 views

Weight of topology related to sizes of open sets?

I'm wondering if there is a notion relating the weight of a topological space (=minimal base cardinality) to the sizes of its open sets. In particular I'm looking for properties of spaces, whose ...
6
votes
1answer
149 views

Turing invariance on large sets

Definition: A function $f: 2^{\omega} \rightarrow 2^{\omega}$ is Turing invariant if $x \equiv_T y \rightarrow f(x)\equiv_T f(y)$. Question I (under $ZFC$): Let $f: 2^{\omega} \rightarrow 2^{\omega}$ ...
5
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1answer
91 views

Separating disjoint sets of size $\aleph_1$ with Borel sets

Question: Is it consistent with ZFC that every pair of disjoint sets $A,B\subseteq\mathbb{R}$, both of size $\aleph_1$, can be separated by a Borel set? This statement is clearly false under CH; take ...
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0answers
38 views

Cardinal exponentiation ambiguity

This is a following passage from Jech on cardinal exponentiation. Am I correct in that $\kappa^\lambda$ means two different things here? Namely, that the first $\kappa^\lambda$ is the set of all ...
4
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1answer
85 views

$H(\kappa)$ a model of all the axioms of ZFC for $\kappa$ not inaccessible

Is this possible? For each cardinal $\kappa$, we can define $H(\kappa) = \{ x \mid |trclx| < \kappa\}$, where trcl is the transitive closure, and $|A|$ is the cardinality of $A$. For every regular ...
2
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1answer
53 views

Can somebody explain (and ideally reference) this strange use/version of the Pressing Down Lemma?

In Stevo Todorcevic's "A dichotomy for P-ideals of countable sets" (link, page 261 at the bottom [page 11 in the pdf]), the following confusing situation comes up: (Context: $\mathcal I$ is a P-Ideal ...
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1answer
46 views

Pressing-Down-Lemma for Jech's notion of stationary sets

So, apparently there is a variant of the Pressing-Down-Lemma (or Fodor's Lemma) for Jech's notion of stationarity, i.e. for sets in $[X]^\lambda$. Does anybody know a citable source for this?
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2answers
56 views

Strong version of Baire Category Theorem

We know that in a complete metric space or compact Hausdorff space the intersection of $\omega$-many open dense sets is dense. In such spaces is the intersection of fewer than $2^\omega$-many open ...
5
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1answer
564 views

Can someone point out the flaw in my proof of AC?

I have a fake proof of the axiom of countable choice. Obviously it is not correct, but I cannot see my flaw. Forgive me, I am only learning set theory. Let $\{A_n : n \in \mathbb{N}\}$ be a countable ...
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0answers
30 views

Complete lattice without greatest element

Is there any term for "complete lattice without greatest element" (because the lattice is too big to have the greatest element). A typical example would be the lattice of all small (in Grotendieck's ...
1
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1answer
56 views

Show that any $n$-linked family is subset of one $n$-linked maximal family.

A $\mathscr{F}$ family of finite subsets in $\mathbb{N}$ is called $n$-linked if for each $X_1,...,X_n\in \mathscr{F}$, $X_1\cap...\cap X_n$ is infinity. Show that any $n$-linked family is subset of ...
2
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2answers
56 views

Showing that a Transitive Set of Transitive Sets is an Ordinal

My definition of an ordinal is a transitive set that's well ordered by $\in$. Let $\alpha$ be a transitive set all of whose elements are transitive sets. Since every element of $\alpha$ is transitive, ...
3
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1answer
34 views

Question about a line in the proof of Zorn's lemma in Jech.

The proof of Zorn's lemma in Jech is the following Proof. We construct (using a choice function for nonempty sets of P), a chain in P that leads to a maximal element of P. We let, by induction, ...
3
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1answer
85 views

Without AC, it is consistent that there is a function with domain $\mathbb{R}$ whose range has cardinality strictly larger than that of $\mathbb{R}$?

I stumbled across this question earlier, and the top comment on the bottom answer asserts two claims: Without the Axiom of Choice, It is consistent that there exists a function with domain ...
0
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1answer
46 views

No injection $Ord \to A$

Let $A$ be a set. Why is it true that there is no injection $Ord \to A$ without the axiom of choice, where $Ord$ is the class of all ordinals?
2
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1answer
54 views

Cardinality of a set of positive Lebesgue measure

I have pretty no knowledge in set theory, so likely the question has a trivial answer. All countable subsets of $[0,1]$ have Lebesgue measure of zero, thus all sets of positive Lebesgue measure are ...
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2answers
42 views

Applying the Axiom Schema of Separation for the property $x = \{x\}$

On a past exam paper in a set theory module I am taking I am asked the question: Express as a first-order sentence in the language of set theory, the instance of the Axiom Schema of Separation for ...
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2answers
58 views

Partition of set of size more than $2^{\aleph _0}$ [closed]

Can every set of size more than $2^{\aleph _0}$ be partitioned into subsets, such that each is non-singleton and each has size at-most $2^{\aleph_0}$? Can every set of size more than $2^{\aleph _0}$ ...
5
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1answer
141 views

Prenex form of the power set axiom

I began teaching myself Zermelo-Fraenkel Set Theory today, and decided to test myself by writing down all the axioms I have read about without looking at the notes. On the axiom of power set, I wrote ...
6
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3answers
391 views

Mean value theorem and the axiom of choice

There's this theorem in Spivak's book of Calculus: Theorem 7 Suppose that $f$ is continuous at $a$, and that $f'(x)$ exists for all $x$ in some interval containing $a$, except perhaps for ...
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2answers
49 views

unbounded class of ordinals not a set

A class $C$ of ordinals is unbounded just in case ∀α∈ORD (the class of all ordinals), there exists a β ∈ $C$ with α ∈ β. How would I show that no unbounded class of ordinals is a set? Do I need to ...
2
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1answer
68 views

Hereditary Cardinality and Rank

For a set $x$, we have its hereditary cardinality defined as $$\textrm{hcard }x=|\textrm{trcl }x|\ ,$$ where trcl $x$ denotes the transitive closure of $x$. Meanwhile, the rank of a set is defined ...
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1answer
275 views

An extension of the axiom of choice needed?

Given a set $\mathscr F$ of nonempty sets. Is it possible to define the set of all choice functions on $\mathscr F$?
2
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1answer
37 views

Countable transitive model of ZFC and $\mathcal{P}(\omega)$

Let $\mathbb{M}$ be a countable transitive model of ZFC. I understand that $\omega^\mathbb{M} = \omega$ but $\mathcal{P}(\omega)^\mathbb{M} \neq \mathcal{P}(\omega)$, for ...
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4answers
861 views

How can a set contain itself?

In Russell's famous paradox ("Does the set of all sets which do not contain themselves contain itself?") he obviously makes the assumption that a set can contain itself. I do not understand how this ...
2
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0answers
66 views

Inaccessible Cardinals and Grothendieck Universes

I'm trying to prove that the following statements are equivalent: 1.$\forall\alpha\in\mathbb{ON}\ \exists\ \kappa>\alpha$ is an inaccessible cardinal. 2.$\forall x\ \exists\ U\ x\in U$ and $U$ is ...
5
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1answer
135 views

Problem in Kunen - suitable representation of ZF proves the consistency of ZF?

I tried to prove the exercise problem in Kunen (Chapter IV, problem 36.) Problem. Show that there is a formula $\chi(x)$, such that $\chi$ represents ZF; i.e.,$$\phi\in \mathsf{ZF}\to ...
4
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1answer
63 views

$\omega_2$ is a not countable union of countable sets without AC

This is an exercise from Jech 3.13 I would like to show that $\omega_2$ is not a countable union of countable sets without AC. I'm given the following hint: I'm not sure how to define the mapping. ...
2
votes
1answer
57 views

For every infinite class C of sets in V the universe is there an infinite set $x$ such that $x\subset C$?

For every infinite class C of sets in V the universe is there an infinite set $x$ such that $x\subset C$? I wasn't sure about how to phrase the question, I could have also asked, is V closed ...
4
votes
1answer
75 views

Is the existence of such a transitive model $M$ of ZFC consistent?

Questions. Q0. Does anyone know of a refutation, in ZFC, of the following statement? Q1. If not, does ZFC plus large cardinals prove its consistency? Statement. There exists a transitive model $M$ ...
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3answers
141 views

What is the difference between $\omega$ and $\aleph_0$?

The book I'm using says that the cardinality of a set $X$ is the least ordinal $\alpha$ such that $|X| = |\alpha|$. So then $\omega = \aleph_0$, but $\omega + \omega \ne \omega$, while $\aleph_0 + ...
2
votes
1answer
46 views

About Mostowski's Collapse, how can you explain the starting set not being transitive?

The Theorem as it has been presented to me states as follows: Let $r$ be a relation on a set $M$ that is well founded and extensional (satisfies then axioms of foundation and extension), then there ...
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0answers
41 views

Existence(?) of a set whose cardinality cannot be determined in ZFC

(First, I apologize if I display any fundamental misunderstanding of how set theory works.) I had a question regarding the limitations of ZFC (assuming its consistency, of course.) Is there any ...
2
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1answer
144 views

Does the ordered set from Zorn's lemma have a lowest element?

The ordered set in the Zorn's lemma needs to have supremums for all chains, that is, including the empty chain. The supremum of an empty chain is its lowest upper bound. Since any element is an upper ...
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0answers
56 views

Is there an axiom for ZFC that totally orders Cichon's diagram without collapsing it?

Is there a known axiom for ZFC that: Fixes a total ordering on the cardinal numbers appearing in Cichon's diagram. Is "natural looking" - in particular, its not allowed to be a conjunction of ...
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0answers
37 views

Applications of Infinitary Matrices in Set Theory

Matrices have a natural generalization to infinitary context. There are few known applications of such matrices in set theory. For example one may use Ulam matrices to show that real-valued measurable ...
3
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1answer
48 views

Continuum Hypothesis for closed sets

In A Beginner's Guide to Modern Set Theory [page 48], the author says: [Cantor] did prove that every closed uncountable subset of $\mathbb R$ has cardinality $2^{\aleph_0}$... ... but I cannot ...
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3answers
665 views

Do we really need Choice to pick socks?

It is said that you need the Axiom of Choice to pick one sock from each of infinitely many pairs, but that you don't need it for shoes, since you can just pick all the left shoes. But Choice is ...
2
votes
1answer
43 views

for $x\in V_{\omega}$ is then $\operatorname{rank}(\operatorname{tc}(x))=\operatorname{rank}(x)$?

Let $x\in V_{\omega}$, is $\operatorname{rank}(\operatorname{tc}(x))=\operatorname{rank}(x)$? while $\operatorname{tc}(x)$ is the minimal transitive set ${a}$ such that $x\subseteq a$ i what to show ...
8
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2answers
220 views

Are there non-equivalent cardinal arithmetics?

‎Generalizing a concept in mathematics is always a problematic situation. In most cases there are several ways to generalize a notion and it is not easy to decide if a particular generalization is ...
0
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1answer
49 views

Topological , Homeomorphic version of $|S \times S|=|S| $

Give example of a subset $A$ of $\mathbb R$ such that with respect to some topology , $ A$ is homeomorphic to $A\times A$ . In set theory ZF it is known to be equivalent to A.C. that for any ...
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1answer
66 views

understanding cardinal numbers arithmetic

I have a question about notation in a book I'm reading on set theory and beside of my question I will be glad for a recommendation for a good book that explains well cardinal numbers arithmetic. If ...
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2answers
58 views

Countable collection of countable sets and Axiom of choice

Do we need Axiom of choice(or weaker version axiom of countable choice) to say countable Cartesian product of countable sets is nonempty? I think yes. I read somewhere answer no giving argument: each ...
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0answers
53 views

Carlson's translatability - are theses characterisations equivalent?

Given a translation-invariant ideal $\mathcal{I}$ on a commutative group $G$ and it's dual filter $\mathcal{I}^*$, I am trying to show that $$ (\forall I \in \mathcal{I})(\exists I' \in ...
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1answer
39 views

$\langle \mathfrak{c},\mathfrak{c}\rangle$-Independent Matrix

Given cardinals $\lambda,\kappa$, an $\langle \lambda,\kappa\rangle$-independent matrix on $X$ is a colection $\mathcal{A} = \{A_{\alpha}^{\beta}:\alpha<\lambda\wedge \beta<\kappa\}$ sattisfying ...
2
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1answer
47 views

Constructing almost disjoint families

Let $\mathcal A$ be an almost disjoint family of subsets of $\omega$ and let $\Psi (\mathcal A)$ be the Mrówka space (definition here). Let $$\mathcal I (\mathcal A)=\{X\subseteq \omega : X\subseteq ...
4
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2answers
109 views

Ordinal exponentiation identity with natural sum of exponents

This is related to a previous question on How to think about ordinal exponentiation? One possible definition for the natural product $\alpha\otimes\beta$ of ordinals is based on Cantor Normal Forms ...
6
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2answers
86 views

Embedding $\omega_1$ in the hyperreals

I've been trying to show that there exists a subset of the hyperreals which has order type $\omega_1$, that is, that there exists a subset $A \subseteq {}^*\mathbb{R}$ for which there is an order ...
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0answers
40 views

Least ordinal $\alpha$ that $\omega^\alpha > \beta$ is a successor ordinal

Let $\beta$ be an ordinal. The least ordinal $\alpha$ that $\omega^\alpha > \beta$ is a successor ordinal. This is true, but I'm not sure why. Can someone give me a hint?