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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Tychonoff spaces with small weight

Let $\kappa$ be an infinite cardinal. Is there a Tychonoff space $(X,\tau)$ such that $|X| = 2^\kappa$ and $(X,\tau)$ has weight $\kappa$ (= a basis consisting of $\kappa$ elements)?
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36 views

Why is addition of powers of $\omega$ absorptive?

Why is it the case that if $\alpha<\beta$, $\omega^\alpha+\omega^\beta=\omega^\beta$? Is it because ...
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1answer
87 views

Are there axioms for $\mathrm{ZFC}$ that imply that $\aleph_1$ is very large?

A bit of philosophy: under the usual definition of the aleph numbers, ZFC proves the sentence "$\aleph_1$ is an ordinal." However, in some sense $\aleph_1$ isn't really an ordinal (in my opinion), ...
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0answers
34 views

Do large cardinal properties tend to be semiabsolute?

I don't know much about large cardinals; so, I want to get a feeling of the landscape. Hence this question. Definition. Whenever $C$ is a unary predicate in the language of ZFC, let us call $C$ ...
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0answers
56 views

A problem with an assumption in a previous lemma for the proof of Silver's Theorem on SCH in Jech's “Set Theory”

In the Jech's textbook proof of Silvers Theorem, specifically in the Lemma 8.15, there are an assumption in the beginning of the proof. First, the Lemma 8.15 says that, under the assumption that ...
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1answer
69 views

A question regarding non-(Lebesgue)-measurable sets in models of ZFC+$2^{\aleph_0}$=$\aleph_2$

Let $\mathscr V$ represent a set of Vitali's type. It is known that $|\mathscr V|=2^{\aleph_0}$. Does $\mathscr V$ have any measure-theoretic properties in models of, say, ...
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1answer
39 views

Well-foundedness is not a first order property.

In the book 'Logic, Induction and Sets' by Thomas Foster I read the following in page 100 (Section 'The language of predicate logic'): "We can show that well-foundedness is not a first-order property ...
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1answer
36 views

Hereditary cardinality and products of sets

To start, suppose that $\lambda$ is an infinite cardinal and suppose that $\alpha, \beta \in \mathbf{H}_\lambda$, where $\mathbf{H}_\lambda = \{x : \left|\operatorname{trcl}(x)\right| < \lambda\}$. ...
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1answer
125 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 ...
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1answer
148 views

ZF and The Cardinality of The Set of Finite Subsets

In a comment on one of my answers, I claimed that the abelian group generated by a set of $S$ generators, each of order two, could take on any infinite cardinality; this is equivalent to saying that, ...
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4answers
626 views

Was there anybody before Cantor who conjectured existence of infinities of different sizes?

Georg Cantor is formally known as the first one who discovered existence of infinities of different sizes. But the history of thinking about the concept of "infinity" in maths and philosophy goes back ...
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33 views

is there a known set in ZF, such that we can't find a well order on? [duplicate]

is there a known set in ZF, such that we cant find a well order on? since the axiom of choice $(AC)$ and it's negation is consistent with ZF, i wonder if we have a concrete example of a set $A$ that ...
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0answers
22 views

About a function ranging filters

Let $U$ be an (infinite) set and $N$ be an (infinite) index set. I denote $\mathfrak{A}$ the set of filters on $U$ (including the improper filter). Let $f$ be an $N$-ary relation that is a set of ...
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0answers
44 views

Are there any articles (or otherwise) that discuss the idea that the set-theoretic universe should be as “free” as possible?

Question. Are there any articles (or otherwise) that discuss (and perhaps attempt to formalize) the idea that the set-theoretic universe should be as "free" as possible? Discussion. ...
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0answers
72 views

About models of ZFC and models in general.

So I've been attending lectures in Set Theory lately and been struggling with the following. When studying the universe of sets V our approach is: let ZFC be consistent, then a model V of the theory ...
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2answers
139 views

A question regarding the Power Set Axiom in ZFC

It is known that the Axioms of ZFC are not necessarily independent of each other. For example, it can be shown that one can derive Separation from Replacement even though both are listed as axioms of ...
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2answers
177 views

Are non-standard models always not well-founded?

Are non-standard models of ZF set theory by definition always not well-founded? And it seems it is, because it must be. But then, Wikipedia says that when there is a set that is a standard model of ...
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1answer
22 views

How to prove that a union of cardinals is a cardinal [duplicate]

I have this question: Let $\omega_1$ the least uncountable cardinal, and for all $n \in \omega$, $n \geq 1$. Let $\omega_{n+1}$ the least cardianal greater than $\omega_n$. Show that $$\bigcup_{n \in ...
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1answer
43 views

Notation for the class of all cardinals

I have seen the notation for the class of all ordinals to be $\rm Ord$ or $\rm On$, is there an analogous notation for the class of all cardinals?
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1answer
37 views

If $2^{\text{cf } \kappa} < \kappa$, then $\kappa^+$ is the least possible value of $\kappa^{\text{cf } \kappa}$

I would like to show that if $2^{\text{cf } \kappa} < \kappa$, then $\kappa^+$ is the least possible value of $\kappa^{\text{cf } \kappa}$ where $\kappa$ is an infinite cardinal. I'm certain it ...
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1answer
45 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 ...
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0answers
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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 ...
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0answers
61 views

“constructible but not definable”

T. Jech in his monograph Ch. 18, p. 301 writes "... the set T is constructible, but not definable in L." My guess is that this means that T (a subset of, say, some ordinal) is definable in some ...
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1answer
121 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 $f: 2^{\omega} \rightarrow 2^{\omega}$ ...
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1answer
73 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
37 views

Prove that a preorder is not anti symmetric

Let $\prec$ be a relation on the set $ A = Z \times (N \setminus \{0\}) $ in this way: A. $<a,b> \prec <c,d> $ if $ ad \le bc$ Prove that $\prec$ is a Preorder and show it's not ...
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1answer
77 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 ...
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0answers
36 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 ...
2
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1answer
49 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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2answers
52 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 ...
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0answers
47 views

Show that $\omega_1 \leq \mathfrak{a}\leq 2^\omega$ [closed]

Let $\mathfrak{a}$ the least infinite cardinal $\kappa$ such that exists a family of sets of natural numbers $\mathscr{F}$ with cardinality $\kappa$ such that $\mathscr{F}$ is maximal almost ...
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1answer
41 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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1answer
554 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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1answer
54 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 ...
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0answers
25 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 ...
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0answers
43 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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2answers
44 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, ...
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1answer
33 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
81 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 ...
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1answer
44 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?
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2answers
38 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 ...
2
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1answer
47 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
51 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 $2^{\aleph _0}$ ...
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1answer
138 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 ...
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2answers
370 views

How to define countability of $\omega^{\omega}$ and $\omega_1$? in set theory?

How is the ordinal $\omega_1$ defined? I know that it is a supremum of all smaller ordinals, but then $\omega^\omega$ is also a supremum of all smaller ordinals. How can we distinguish these two ...
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3answers
376 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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0answers
46 views

A countably infinite set of prisoners on death row who believe in the Axiom of Choice are locked in a room. [duplicate]

A countably infinite set of prisoners on death row who believe in the Axiom of Choice and who can process an infinite amount of information in a finite time are locked in a room. The ...
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2answers
44 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 ...
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4answers
831 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 ...
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1answer
62 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 ...