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 ...

learn more… | top users | synonyms

0
votes
0answers
53 views

Consequences of the principle of dependent choices (DC)

It is known that if we assume the axiom of determinacy every set of real numbers is lebesgue measurable. In order to study this, I'm following Jech's Set Theory book. There, Jech says that apart from ...
6
votes
1answer
99 views

Consequence of $V=L$

Assume $V=L$. Define $\langle A_\alpha\mid\alpha<\omega_1\rangle$ as follows: Let $A_\alpha$ be the $<_L$-least $A\subseteq\alpha$ such that $(\forall\beta<\alpha)A\cap\beta\not=A_\beta$ if ...
1
vote
1answer
40 views

Models of Comprehension Schema

Let $M_\alpha$ for $\alpha\in ON$ be transitive sets and let $M=\bigcup_{\alpha\in ON}M_\alpha$. Suppose that (i) for every $\alpha<\beta$, we have $M_\alpha\in M_\beta$ and (ii) for every limit $\...
1
vote
1answer
49 views

Principle equivalent to $\diamondsuit_\kappa$

Let $\kappa>\omega$ be regular. The principle $\diamondsuit_\kappa$ is as follows: There exists a sequence $\langle X_\alpha\mid \alpha<\kappa\rangle$ such that each $X_\alpha\subseteq\alpha$ ...
0
votes
1answer
26 views

Help with exercise from Kunen (II. Ex. 51)

Show that the following are equivalent: (1) $\diamondsuit$ (2) The existence of $B_\alpha\subseteq\alpha\times\alpha\ (\alpha<\omega_1)$ so that the set $\{\alpha<\omega_1\mid B\cap (\...
2
votes
0answers
62 views

Intuitive motivation for the $\diamondsuit$-principle?

I'm having difficulty internalizing the $\diamondsuit$-principle. I understand what it says, and can see why some of the constructions that I've seen that use it work. But I don't think I understand ...
8
votes
1answer
85 views

If ZFC has a model then the union of ZFC with the negation of inaccessible cardinals has a model

I would appreciate some help with proving the statement in the title. I'm new to model theory, so I won't understand technical terms or symbols that well. Hence a sketch of the proof will suffice. ...
2
votes
2answers
56 views

$\Pi_2$ formula true for $L_{{\omega_1}^L}$ but not for $L$

I tried the formula [${\forall x}{ \exists} {f} $$ {x}$ is an ordinal ${\land} {f}$is an injective function ${\land}$ $dom f =x$ ${\land}$ $ran f$ ${\subseteq} {\omega}$$]{ \lor}$ $x$ is not an ...
2
votes
1answer
48 views

Amalgam of trees

Definition A tree is a partially ordered set $(T, <)$ such that for each $t \in T$, the set $\{s \in T : s < t\}$ is well-ordered by the relation $<$. For trees $(T,<_T)$, $(S,<_S)$, $...
5
votes
1answer
73 views

Weakly Mahlo cardinals and weakly inaccessible cardinals

A cardinal $\kappa$ is weakly Mahlo if it's weakly inaccessible and the set $\{\lambda\in\kappa:\,\lambda\,\text{weakly inaccessible}\}$ is stationary in $\kappa$. Let's define $E_0=\{\lambda:\,\...
1
vote
2answers
110 views

Neighborhood chain in $\mathbb{R}^\mathbb{R}$

Edited after SamM's comment: Consider the topological space $\mathbb{R}^\mathbb{R}$, with the usual topology. Pick a point $x \in \mathbb{R}^\mathbb{R}$ and a neighborhood $V = V_0$ of $x$. I wish to ...
5
votes
2answers
190 views

How many sorts are there in Terry Tao's set theory?

In his 2010 post, A computational perspective on set theory, Terry Tao writes: The standard modern foundation of mathematics is constructed using set theory. With these foundations, the ...
2
votes
1answer
47 views

$L$ and $L_{\omega_2}$ agree on all $\Sigma_1$ formulae?

I remember seeing this mentioned somewhere in a textbook I was looking at a while ago, but I can't seem to find the proof. Is anyone aware of the reference I'm looking for? Or can someone provide a ...
4
votes
0answers
64 views

Help with a consequence of $V=L$

Assume $V=L$. Prove that there is a function $f:\omega_1\rightarrow\omega_1$, $\Delta_1$ definable over $L_{\omega_1}$, so that for every $A\subseteq\omega_1$ and every $\Sigma_1$ formula $\varphi$, ...
1
vote
1answer
34 views

Show that there exists a fixed point for this (set theoretic) class function

I see that this question might be trivial but I can't seem to figure it out myself: Suppose that $F:ON\to ON$ is a class function: that is, for every ordinal $\alpha$ there is unique ordinal $F(\...
0
votes
1answer
57 views

Is there a key difference between the principle of transfinite induction and the principle of transfinite induction for ordinals?

In a recent question I was asked to prove the principle of transfinite induction for ordinals but I mistakenly proved the principle of transfinite induction, since I have a only a vague understanding ...
1
vote
0answers
18 views

Uncountable sequence of functions which differ on finitely many values [duplicate]

I've been struggling with this question I've been assigned for a couple weeks now. The question is to show from ZFC that there is an uncountable sequence $〈f_α:α→ω:α<ω_1〉$ of functions such that ...
1
vote
1answer
65 views

What do propositional function in ZFC mean?

I know that a propositional function is a WFF which can be either true or false depending on the value of at least one variable. The axiom schema of specification (subsets) says that for every ...
3
votes
1answer
55 views

Question about a countable family of infinite subsets of $\mathbb{N}$

Assume CH. Show that given a countable family {$A_n|n\in \mathbb{N}$} of infinite subsets of $\mathbb{N}$, show that there is a subset $S$ of $\mathbb{N}$ such that $S$ intersects each $A_n$, but ...
5
votes
0answers
47 views

The negation of the continuum hypothesis as an axiom [duplicate]

Suppose that one considers the formal theory "ZFC+~CH", or the set of all axioms of ZFC plus the negation of the continuum hypothesis. Then we have that there exists at least one set S whose ...
1
vote
1answer
82 views

Set theory with multiple countable infinities [closed]

In set theory, all sets that are countably infinite are generally considered to have the same size since there is a bijection between them. Has anyone tried formalising set theory in a way which ...
3
votes
1answer
69 views

The Meta-Mathematics of Multiple Forcing

In forcing we have the forcing theorem (also called the truth and definability theorem). It guarantees that forcing works. What are the similar theorems for multiple forcing? To elaborate: Kunen, in ...
1
vote
1answer
90 views

Is it circular to define the Von Neumann universe using “sets”?

I was just reading the Wikipedia page on the Von Neumann universe, where it is stated that this universe "is often used to provide an interpretation or motivation of the axioms of ZFC." However, later ...
3
votes
2answers
87 views

Explicit example of countable transitive model of $\sf ZF$

Do we know any explicit example of a countable transitive model for $\sf ZF$ or $\sf ZFC$?
3
votes
1answer
125 views

How can I construct a sequence of injections $\langle f_\alpha\colon\alpha\to\omega\mid\alpha<\omega_1\rangle$ with a particular coherence property?

Assuming the axiom of choice, show that there exists a sequence of function $\langle f_\alpha\mid\alpha<\omega_1\rangle$ such that: (1) each $f_\alpha:\alpha\rightarrow\omega$ is injective and (2) ...
1
vote
1answer
34 views

Smaller, well-behaved subsets of the function space of a successor cardinal

Suppose $\kappa$ is an infinite cardinal. I'm trying to show that there's a subset $X\subseteq \kappa^{+^{\kappa^+}}$ with $|X|=\kappa$ and for any $\alpha<\beta$, we have $f\in X$ with $f(\alpha)=\...
2
votes
0answers
49 views

Quick question about $\diamondsuit_\kappa$

I've run into the claim that $\diamondsuit_\kappa$ is equivalent to the (weaker looking) principle $\diamondsuit_\kappa^*$: That there's a sequence of $X_\alpha$ ($\alpha<\kappa$) with $X_\alpha\...
2
votes
1answer
62 views

Equivalents of Jensen's diamond principle

Let $\kappa>\omega$ be regular. The principle $\diamondsuit_\kappa$ is as follows: There exists a sequence $\langle X_\alpha\mid \alpha<\kappa\rangle$ such that each $X_\alpha\subseteq\alpha$ ...
3
votes
1answer
82 views

Help with a problem about consequences of the continuum hypothesis

Suppose that the continuum hypothesis holds. I'm trying to prove that there is a set $T\subseteq\omega_1\times\omega$ such that every set $S$ with $S=A\times B\subseteq\omega_1\times\omega$ with $A$ ...
2
votes
1answer
73 views

Independent families of sets

I'm having a difficulty understanding some exercises related to independent families of sets. Recall that $ \mathcal{A} $ is $\lambda$-independent if for any disjoint $ P, Q \in \mathcal{A} : |P|, |Q| ...
3
votes
1answer
40 views

Existence of a particular element of an ultrafilter

I'm getting to know some ultrafilter theory. I'm stuck on the following exercise: Suppose $ \mathcal{U} $ is an ultrafilter on $ \omega $. Prove that there exists $ A \in \mathcal{U} $ such that $$\...
4
votes
1answer
50 views

Consistency of ZFC + “for every function there exists a class inaccessible to it”

Is ZFC + the following statement consistent (and if so, is it equiconsistent to some known large cardinal): For every function $f:ORD \rightarrow ORD$ such that: $f(\alpha)\geq \alpha$, $\alpha >...
1
vote
0answers
102 views

Maximal antichains in a forcing which adds surjections

Let $P$ be a separative partial order such that $\left| P \right| \leq \left| \alpha \right|$ and $$\Vdash_P\exists f(f\colon\omega\to\alpha\text{ is surjective}\land f\notin\check V).$$ I want to ...
1
vote
1answer
78 views

The generalized continuum hypothesis can't fail first at $\omega_{\omega_{1}}$

I am willing to prove that the GCH cannot first fail at a singular cardinal. For this purpouse I am following the strategy outlined by Kunen in his 2013 book (see Exercises III.6.16-6.17-6.18-6.19). I ...
2
votes
1answer
81 views

Why wasn't Bertrand Russell surprised by the set of all sets that contain themselves?

Russell's paradox deals with the question: "Does the set of all sets that do not contain themselves, contain itself?" What about the question: "Does the set of all sets that contain themselves, ...
2
votes
1answer
32 views

If $a\sqcup b$ and $a\times b$ biject, then $b$ either injects or surjects in-/onto $a$

Let $a$ and $b$ be sets such that there is a bijection $a\sqcup b\to a\times b$. Show, without assuming the Axiom of Choice, that there is either a surjection $b\to a$ or an injection $b\to a$. ...
0
votes
1answer
52 views

Does the MK axiom of infinity and the Power set axiom hold in the cumulative hierarchy?

Question: Define the cumulative hierarchy inductively as: $$V_0 = \emptyset$$ For each ordinal $\alpha$: $$V_{\alpha+1} = \mathcal{P}(V_\alpha)$$ ie the power set of $\alpha$ and for any nonzero ...
1
vote
1answer
38 views

Does there exist a metric space of cardinality aleph-two that has a countable epsilon cover?

Does there exist a metric space $(M,d)$ such that $|M|=\aleph_2$ but there exists a countable $\epsilon$-cover of $M$?
3
votes
1answer
64 views

Well-pruned $\kappa$-Suslin trees have the $\kappa$-Baire property

I'm having troubles seeing that a well-pruned $\kappa$-Suslin tree, with $\kappa$ uncountable and regular, has the $\kappa$-Baire property. I saw in Kunen's Set Theory that it can be seen by proving ...
1
vote
1answer
28 views

Order-preserving maps on a directed-complete poset

Definitons. A poset $P$ is directed if every finite subset has an upper bound in $P$. It is directed-complete if every directed subset has a least upper bound in $P$. For any poset $P$ let $[P\to P]$ ...
1
vote
1answer
35 views

A proof of maximality of an antichain in a complete Boolean algebra

Suppose $(\mathbf{B},\leqslant)$ is a complete Boolean algebra and let $|\mathbf{B}|=|\gamma|$. Let $C=\langle c_\alpha\mid \alpha<\gamma\rangle$ be a maximal descending chain without a lower bound:...
1
vote
1answer
49 views

Reference: Mahlo cardinals remain Mahlo in L

The following is stated on Wikipedia for Mahlo cardinals. Unfortunately, it's not sourced. Where can I find details? I wasn't able to google any articles dealing with Mahlo cardinals in L. Since ...
1
vote
0answers
51 views

A classification of the types of poset extensions that raise local-global principle is equivelent to the axiom of choice?

EDIT: Completely overhauled the question, remove category theory from it (stated in plain set theory): The original question Adding relations to a partial order Let $X$ be a set and let $P: Pow(X) \...
7
votes
1answer
117 views

Is ZFC ω-consistent over ZF?

Gödel proved that if ZF is consistent, then ZFC is also consistent. He did this by showing that his constructible universe is a model of ZFC. Gödel also introduced the notion of an $\omega$-consistent ...
33
votes
7answers
2k views

Do the axioms of set theory actually define the notion of a set?

In Henning Makholm's answer to the question, When does the set enter set theory?, he states: In axiomatic set theory, the axioms themselves are the definition of the notion of a set: A set is ...
4
votes
1answer
64 views

Show forcing condition compatibility by induction

Say that we have a countable support forcing iteration $ \mathbb{ P}_{ \alpha }$ ( using Jech's definition ) where $ \alpha $ is a limit ordinal, and consider two conditions $ f , g \in \mathbb{ P}_{ ...
0
votes
0answers
27 views

Let $A \subseteq \mathbb{R}$ and $|A|=\omega$, prove $|\mathbb{R}\setminus A| = 2^\omega$. [duplicate]

Let $A \subseteq \mathbb{R}$ and $|A|=\omega$, prove $|\mathbb{R}\setminus A| = 2^\omega$. Hint: $\mathbb{R}\sim \{0,1\}^\mathbb{N}$ I'm somewhat stuck trying to prove this, could someone give ...
0
votes
1answer
37 views

If $ \bigcup N_\alpha $ is stationary, then $ \{ \min(N_\alpha) \}$ is stationary

It's the last set-theoretic question for tonight, I promise. I'm trying to figure out why the following holds true: Suppose we have a regular, uncountable cardinal $ \kappa $ and a disjoint family ...
2
votes
2answers
26 views

Unboundedness of the set of subgroups of a cardinal

I'm trying to figure out why the following is true: Let $ \kappa $ be an uncountable, regular cardinal. Suppose we turn it into a group (i.e. there are operations $ (\cdot, ^{-1}, e) $ with which $ \...
2
votes
1answer
49 views

Checking whether some sets are clubs in $ \aleph_2 $ and $ \aleph_1 $

I'm getting acquainted with the stationary sets and clubs. I don't yet quite get everything well, so I'd appreciate some help with this question: which of the following sets are clubs, contain a club,...