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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Why does $x \in A \cdots x \in B$ imply $A \subset B$ and not $B \subset A$?

1 Why does $x \in A \cdots x \in B$ imply $A \subset B$ and not $B \subset A$? When we show that x is in A, then we show it is in B. SO why does that imply $A\subset B$? 2 And how do we prove ...
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56 views

What is an Empty set?

We define the term "Set" as, A set is a collection of objects. And an "Empty set" as, An empty set is a set which contains nothing. First problem I encountered: How the definition of ...
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21 views

First order logic and first order set theory [duplicate]

(1) I understood that in the language of first order logic quantification cannot be over sets (there are no predicate variables that can be bound by quantifiers), whereas the language of second-order ...
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Exercise 23.10 of Jech's book

I would need a hint for Exercise 23.10 of Jech's Set Theory (third edition), which states: If $\kappa$ is a regular cardinal, then there exists a strongly almost disjoint family ...
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3answers
52 views

How to show there exists $E$ such that $E \cap K_n$ is dense for every $n$?

Let $\Omega$ be a region (nonempty connected open subset of the complex plane). Let $K_n$ be a sequence of compact sets whose union is $\Omega$, such that $K_n \subset \mathring{K_{n+1}}$ (the ...
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Definability in $L(\omega_1)$

I'm trying to solve problem II.6.35 in Kunen's book, which asks to prove that if $V=L$, then the set $B$ of $\beta<\omega_1$ such that $L(\beta)\models ZF-P$ and every element of $L(\beta)$ is ...
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3answers
245 views

Does every cover have an irredundant subcover?

While composing an answer for this question, I got troubled by a technical point. I wanted to assert the existence of an irredundant subcover of a given open cover, but realized I'm not sure how to ...
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2answers
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show that there is a $\mathbb P$-name $\sigma$ such that $M[G]\vDash \exists x\phi (x) \iff M[G]\vDash \phi (\sigma [G])$

I want to show that there is a $\mathbb P$-name $\sigma$ such that for every $G$ a generic filter we will have $$M[G]\vDash \exists x\phi (x) \iff M[G]\vDash \phi (\sigma [G])$$ while $\phi (x)$ is a ...
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Isn't the axiom of determinacy inconsistent with ZF? What am I overlooking?

I'm sure there's something I'm missing here; probably a naive confusion of mathematics with metamathematics. Regardless, I've come up with what looks to me like a proof that (first-order) ZF+AD is ...
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How can nontrivial elementary embeddings of the universe to some inner model be surjective?

Consider $\kappa$ to the least measurable cardinal, or equivalently $\kappa$ is the critical point for an elementary embedding $j:V \rightarrow M$ from the universe $V$ to an inner model $M$ (critical ...
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Adding bounded quantifiers increases complexity in $L$

I'm reading Devlin's Constructability to learn about $L$. Following the proof that $L_\alpha$ has a $\Sigma_1$ skolem function for limit $\alpha>\omega$ (II.6.5), the author notes Notice that ...
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The class of all functions between classes (NBG)

Is it possible in NBG (von Neumann-Bernays-Gödel set theory) to construct the class of all functions $X \to Y$ between two (proper) classes $X,Y$? I guess that this does not work. In the special case ...
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How should the relativized Kleene pointclass $\Sigma^1_1(A)$ be defined?

How should the relativized Kleene pointclass $\Sigma^1_1(A)$ be defined, when $A$ is a set of reals ($A \subset \omega^\omega$)? I assume that there is a standard definition, but I can't seem to find ...
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2answers
94 views

A Ramsey-type result for families of subsets

Let $S$ be a set of cardinality $\aleph_1$. Consider the directed family $\mathcal{C}$ (here directed means directed with respect to the inclusion) of all countably infinite subsets of $S$. Suppose ...
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1answer
48 views

There is no sequence $X_n$ such that $\forall n(\mathscr P(X_{n+1})\preceq X_n)$.

I'm working on the following exercise from Kunen: Define, in ZF without the axiom of regularity, $\aleph(\{\alpha: > \exists f \in \, ^\alpha X(f \text{ is } 1-1\})$. Show: ...
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1answer
31 views

Using ordinal arithmetic calculate the following ordinal numbers

(ω + 1) x ω (ω + 1) x 2 For Question #2, I can simplify to the point where I get (ω + (ω + 1)), but I'm not sure how to proceed from there
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1answer
69 views

Hilbert–Bernays provability conditions

Let "provability formula" ${\rm Prf}(x, y)$ written in the manner that provability operator $\square A$ defined as $\exists x\ {\rm Prf}(x, \overline A)$ satisfying Hilbert–Bernays axioms: If ZF ...
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1answer
52 views

Exercise (3) of Chapter III from Kunen's Set Theory: Intro to Independence Proofs

I'm a little stumped on the aforementioned question. It's statement is as follows: Let $M$ Be any class such that $\forall$x (x $\subset$ $M$ $\rightarrow$ x $\in$ $M$). Show that $WF$ $\subset$ ...
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2answers
73 views

Every II-finite set is III-finite

I need some help proving that if a set $X$ is II-finite then it is III-finite, i.e. if every non-empty family of subsets of $X$ which is linearly ordered by inclusion has a maximal element under ...
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$\phi(\mathbb{P}\ast \mathbb{\dot{Q}})$ has property $\mu$-linked and $\mu$-centered.

Let $\mathbb{P}$ be a poset and $\mathbb{\dot{Q}}$ be a $\mathbb{P}$-name of a poset. If $\phi(\mathbb{P})$ and $\Vdash_{\mathbb{P}} \phi(\mathbb{\dot{Q}})$, then $\phi(\mathbb{P}\ast ...
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1answer
40 views

forcing over a dense set.

I want to show that given a countable transitive model $M$ a notion of forcing $\mathbb P$ and a generic filter $G$, then forcing over a dense set of $\mathbb P$, $D$ is just as forcing over $\mathbb ...
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1answer
33 views

Function on a well-ordered set

Let $(W,<)$ be a well ordered set. Let $f : W\rightarrow W$ be a function such that $u < v$ implies $f(u) < f(v)$. Show that $\forall w \in W, w \leq f(w)$. I was thinking to consider $T=\{x ...
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1answer
40 views

A generalization of “any countable limit ordinal is the union of a sequence of increasing ordinal”

Using the fact that every countable ordinal is isomorphic to a closed subset of $\mathbb Q$, I find out that any countable limit ordinal is the union of a sequence of increasing ordinal. Now I'm ...
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1answer
45 views

The Amoeba poset has property Knaster.?

The Amoeba poset has property Knaster.? Amoeba poset $\mathbb{P}=\{p \subseteq \mathbb{R}: p$ $\text{is open} \wedge \mu(p)< \epsilon\}$ for $\epsilon>0$. Any suggestion. Thanks
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1answer
85 views

No Borel well-order of the reals?

I'm told there is no Borel well-order of the reals (in ZFC). I'm told, in fact, that this is because of Borel determinacy. However, this is usually a vague handwave of the form (a) take the usual ...
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Is $2^{\aleph_0} = \aleph_1$?

I was reading a thread on Examples of Common False Beliefs in Mathematics on MathOverflow, in which a user wrote: $$2^{\aleph_0} = \aleph_1$$ This is a pet peeve of mine, I'm always surprised ...
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2answers
128 views

Do we know if the consistency of $ \mathsf{ZFC} $ is sufficient to prove that $ \mathsf{ZFC} \nvdash \text{Con}(\mathsf{ZFC}) \to \mathsf{SM} $?

This is basically a sequel to this earlier post. There, I asked: If $ \mathsf{ZFC} $ is consistent, then is $ \mathsf{ZFC} \nvdash \neg \text{Con}(\mathsf{ZFC}) $, i.e., $ \neg ...
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1answer
35 views

uncountable repetitions

I have a question (or two) about recursive naming conventions. Consider the following recursive naming sequence: Base step: Let S be any nonempty set. Let x be any arbitrary element of S. Let S* be ...
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1answer
52 views

informal semantics regarding CH and AC

why is the assertion $\aleph_1=2^{\aleph_0}$ referred to as a hypothesis, whereas $$\forall \alpha( S_\alpha \ne \varnothing) \Rightarrow \prod_\alpha S_\alpha \ne \varnothing$$ is called an axiom? ...
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Dependence of axioms of Zermelo set theory

I'm reading a book about Zermelo–Fraenkel set theory, and I was told that there are a lot of dependence relation among the commonly used axiom system (in the question What axioms does ZF have, ...
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1answer
94 views

Reference request for the independence of $ \text{Con}(\mathsf{ZFC}) $.

Gödel’s Second Incompleteness Theorem says that if $ \mathsf{ZFC} $ is consistent, then $ \mathsf{ZFC} \nvdash \text{Con}(\mathsf{ZFC}) $, i.e., $ \text{Con}(\mathsf{ZFC}) $ is not provable in $ ...
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0answers
36 views

Surreal numbers in set theories other than ZFC

This isn't really a question rather than my thoughts on these things; I initially had questions but believe I managed to answer them. Regardless, here goes. Feel free to correct any mistakes, as I'm a ...
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1answer
78 views

Breaking the AC barrier using Kelley-Morse set theory

In her blog post Variants of Kelley-Morse set theory, Prof. Gitman proves that every model of the the common version of Kelley-Morse set theory ($\mathsf{KM}$) is a model of the Wikipedia version of ...
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1answer
32 views

Every set with more than point admits a permutation with no fixed point and the Axiom of Choice [duplicate]

Assuming axiom of choice , for any set $S$ with more than one point , there exist a bijection $f:S \to S$ such that $f(s) \ne s , \forall s \in S$ . Is the converse true , i.e. Does the statement " ...
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1answer
48 views

How to define “is a structure” by a first-order formula of ZFC

So, I've been trying to define a series of notions by first-order formulas in ZFC, and I'd like to know if I'm getting this right. In particular, I'd like to know if my formula for defining a ...
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Order-preserving map of regressive functions on $\omega_1$

I posted the following question a year ago on MO. It did receive some attention, but the answer there remains incomplete (as discussed in some detail in its comments). Meanwhile I got the impression ...
2
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1answer
53 views

Consequences of the negation of the Axiom of Dependent Choice

It seems to me that a proper reason to include The Axiom of Choice as a foundational axiom of set theory should be based on the observation that the negation of The Axiom of Choice has absurd ...
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26 views

Cardinal numbers greater than $\omega$ , ZF [duplicate]

Prove in $ZF$ that for every cardinal number exists a greater cardinal number. I managed to prove this fact in $ZFC$. But without using axiom of choice I can't well-order P(A), and can't build ...
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0answers
55 views

Pathologies involving non-generic filters

This is a question regarding exercise (IV.2.46) in Kunen's Set Theory (2011); it reads: "Assume that $M$ is a ctm for $ZFC$, and let $\mathbb{P}$ $=$ Fn$(\omega,2)$. Then there is a filter $G$ on ...
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0answers
73 views

Infinite palindromes in number of nonisomorphic posets is independent of $\mathsf{ZF}$

The following is an "exercise" in P. Stanley's book "Enumerative Combinatorics": Let $f(n)$ be the number of nonisomorphic $n$-element posets (...) let $\mathcal{P}$ denote the statement that ...
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1answer
88 views

Integer induction without infinity

In ZFC minus infinity (let us call this T), one can still define ordinals, and then define integers as ordinals all of whose members are zero or successor ordinals. I am looking for a formula $\psi$ ...
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1answer
80 views

Why does a nontrivial $V \to V$ have a critical point?

Let $V$ denote the von Neumann universe, and let $j: V \to V$ be a nontrivial (non-identity) elementary embedding. The critical point is the smallest $\kappa$ such that $j(\kappa) > \kappa$. The ...
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1answer
68 views

Is the cofinality function monotonic?

Is the cofinality function $\operatorname{cf}$ monotonic? I.e., if $\lambda \le \kappa$ for cardinals $\lambda$ and $\kappa$, does it then follow that $\operatorname{cf}(\lambda) \le ...
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2answers
52 views

Sets of “Isolated” Cardinals

Let $C\neq\emptyset$ be a set of infinite cardinals with the property that NO member of $C$ occurs as the supremum of strictly smaller members of $C$. So the cardinals in $C$ are sort of "isolated". ...
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3answers
102 views

Can I deduce ZFC Standard from “ZFC Dedekind”?

ZFC Standard: Infinity, Extensionality, Specification, Pairing, Union, Replacement, Power Set, Choice and Regularity. ZFC Dedekind: Infinity replaced bij Dedekind Inifinity, other 8 axioms the same as ...
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2answers
56 views

Increasing sequence of open sets in a separable metric space.

Suppose X is a separable metric space and ($U_α$ : α < γ) is an increasing sequence of open sets (i.e. $U_α$ ⊆ $U_β$ for α < β). Show that there is a countable $γ_0$ such that $U_α$ = $U_β$ for ...
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2answers
70 views

Fodor's Lemma for clubs

Fodor's (or Pressing Down) Lemma states that for every stationary subset $S$ of a regular cardinal and every regressive function $f:S\to \mathrm{Ord}$, there is an $\alpha$ such that $f^{-1}(\alpha)$ ...
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3answers
72 views

Decidability of the cardinality of a set given that the Continuum Hypothesis is independent from ZFC

I'm a Total Amateur (TM), please forgive me if this question makes no sense. The Continuum Hypothesis states that there are no sets with cardinality strictly between that of the integers and the ...
4
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1answer
44 views

Cardinality of sets of reals without choice

Assuming just ZF (no axiom of choice): Does $\aleph_n\leq|\mathbb{R}|$ for all $n<\omega$ imply $\aleph_\omega\leq|\mathbb{R}|$? (with $\kappa\leq|\mathbb{R}|$ meaning that there is a set of reals ...
2
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1answer
64 views

Why isn't second-order ZFC categorical?

This builds off of (and reuses some of the examples from) this question about failures of categoricity not related to size. Second-order $\mathsf{ZFC}$ isn't categorical, but it's almost-categorical, ...