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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3
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0answers
20 views

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 ...
9
votes
3answers
222 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 ...
4
votes
2answers
34 views

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 ...
1
vote
3answers
119 views

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 ...
1
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0answers
69 views

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 ...
3
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0answers
38 views

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 ...
1
vote
2answers
37 views

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 ...
4
votes
2answers
85 views

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 ...
4
votes
2answers
88 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 ...
3
votes
1answer
43 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: ...
0
votes
1answer
30 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
2
votes
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 ...
1
vote
1answer
50 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 $WF$ $\subset$ ...
1
vote
2answers
71 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 ...
4
votes
0answers
53 views

$\phi(\mathbb{P}\ast \mathbb{\dot{Q}})$ 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 ...
2
votes
1answer
39 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 ...
0
votes
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 ...
3
votes
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 ...
2
votes
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
4
votes
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 ...
11
votes
4answers
195 views

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 ...
3
votes
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 ...
0
votes
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 ...
1
vote
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? ...
2
votes
0answers
52 views

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, ...
1
vote
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 $ ...
1
vote
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 ...
0
votes
1answer
77 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 ...
0
votes
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 " ...
2
votes
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 ...
7
votes
0answers
96 views

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
votes
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 ...
-2
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0answers
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 ...
1
vote
0answers
54 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 ...
7
votes
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 ...
4
votes
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 $\psi$ ...
2
votes
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 ...
0
votes
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 ...
1
vote
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". ...
4
votes
3answers
101 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 ...
1
vote
2answers
55 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 ...
1
vote
2answers
68 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 $f^{-1}(\alpha)$ ...
1
vote
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
votes
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
votes
1answer
63 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, ...
6
votes
0answers
87 views

$\sf ZF$ — Sets that can be proven to exist

There are only countably many formal proofs in $\sf ZF$. Thus, there are only countably many sets that can be proven to exist in $\sf ZF$. This collection of sets seems to satisfy $\sf ZF$'s axioms; ...
3
votes
0answers
115 views

How should one understand the foundation of set theory?

I have read the answer of Carl Mummert for the question on how to avoid circularity. I would like to ask further as I want to study models of set theory. As I understand, with say assuming the ...
0
votes
1answer
67 views

Number of “small” subsets to a “large” set

For the following we assume the axiom of choice. Let $X$ be a set of cardinality $l$ for some infinite cardinal number $l$, and let $p(X)$ be the number of subsets of $X$ that have cardinality ...
3
votes
1answer
92 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 ...
2
votes
1answer
49 views

Why this set is finite $E_n=\{l \in \omega: \exists{j<n}[r_j \Vdash \check{l} \leq \dot{h}(n)]\}$?

Lemma IV (4.9) (Kunen book). Let $M$ be a countable transitive model of $ZFC$ and fix $\mathbb{Q}\in M$ such that $(|\mathbb{Q}| \leq \aleph_{0})^{M}$. Let $G$ be $\mathbb{Q}$-generic over $M$. Then ...