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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Any known nontrivial undecidable/independent problems

Given ZF + standard model ℕ, are there any nontrivial, non-self-referential statements that are known to be independent of ZF + ℕ other than the Axiom of Choice? The halting problem isn't one. While ...
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1answer
36 views

Set partitioning in ZFC

Does ZFC allow the partitioning of a set by claiming that a and b are in the same subset if f(a,b)? Cause I've once seen this tehnique being used in a proof but I can't see how it is consistent with ...
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1answer
38 views

Indiscernibility of indiscernibles in second order logic

It is not clear to me if the statements in $0^\#$ remain indiscernible when we move to second order logic. Or are there second logic formulas that can discriminate between first order indiscernibles?
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1answer
94 views

Statements comparable with Axiom of Choice in ZF

Let $AC$ denote any fixed statement of the Axiom of Choice in $ZF$. Consider the set of statements $\phi$ in the language of $ZF$ such that either $ZF+\phi$ proves $AC$ or $ZF+AC$ proves $\phi$. The ...
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0answers
41 views

What is an undefinable ordinal? How large are the least undefinable ordinal and supremum of all undefinable ordinals?

What does it mean to say that a particular ordinal/cardinal number is "definable"? Where and how do we define this "definability"? If there are undefinable ordinals/cardinals, how large are the least ...
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1answer
85 views

Examples of Forcing in Model Theory

My question is exactly my title: What are some examples of (set theoretic) forcing in model theory? I have been studying (combinatorial) set theory and model theory (independently of one another) for ...
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0answers
35 views

If full semantics higher order logic is set theory, which set theory is it?

I've been trying to get a handle on how higher order logic interacts with set theory. It's been stated convincingly that higher order logic with full semantics is set theory in sheep's clothing. For ...
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1answer
67 views

2 Questions regarding Relative Consistency Proofs

First Question: Let IC be the statement "There is an inaccessible cardinal." I have read that one cannot prove (in ZFC) the relative consistency of ZFC + IC w.r.t. ZFC. i.e. $ Con(ZFC) \rightarrow ...
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0answers
26 views

A question about HOD[A] and HOD(A)

HOD[A] (the class of sets hereditarily ordinal definable from $A$ where $A$ is some set) is known to be a transitive model of ZFC. HOD(A) (the class of sets hereditarily ordinal definable over $A$) is ...
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0answers
73 views

On the contradictory nature of large cardinals & choice-like axioms

Compare these two results: Theorem (Scott): $ZFC+V=L\vdash \nexists~\text{Measurable cardinal}$ Theorem (Kunen): $ZFC+AC\vdash \nexists~\text{Reinhardt cardinal}$ Now compare these two ...
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4answers
1k views

Why do we need to learn Set Theory?

I was planning to write some article for the Mathematics magazine of our college and it occurred to me that it will be a good idea to write about the impact and importance of Set Theory. I plan ...
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1answer
118 views

Examples of Extreme Anti-Choice Axioms

Axiom of Choice has many variants like the followings: There is a choice set for every family of non-empty sets. All sets are well-orderable. Of course in many cases one don't need AC to ...
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1answer
31 views

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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0answers
46 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 ...
2
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1answer
109 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
38 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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1answer
43 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 ...
2
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1answer
41 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
103 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, ...
6
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1answer
155 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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0answers
34 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
23 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
46 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
75 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
146 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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1answer
23 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
44 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 ...
2
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0answers
71 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
47 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
45 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
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1answer
129 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
78 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

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 ...
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1answer
82 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
50 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
42 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
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 ...
5
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1answer
556 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
28 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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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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2answers
48 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, ...
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1answer
83 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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1answer
48 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
40 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
52 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
139 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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3answers
377 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 ...