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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1answer
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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
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1answer
43 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 ...
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0answers
40 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 ...
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1answer
73 views

Set theory with multiple countable infinities [on hold]

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 ...
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1answer
82 views

Examples of provably${}^n$ unprovable statements

Given any statement $A$ and a classical theory $T$ which we assume is at least as strong as Peano Arithmetic ($\sf PA$), we have that $T\vdash A$ implies $T\vdash T\vdash A$ (that is, if a statement ...
3
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1answer
61 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 ...
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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 ...
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1answer
75 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 ...
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2answers
71 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 $\sf ZFC$?
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1answer
44 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
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3answers
235 views

Why is this contradiction using axiom of constructibility incorrect?

Today I thought of this paradox and I'm trying to find the wrong assumption that causes it. Does anyone know what is wrong in the following argument: let $$A=\{x\in \mathbb{R}|\exists\phi:\forall ...
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1answer
30 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 ...
3
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1answer
76 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 ...
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0answers
40 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 ...
3
votes
1answer
68 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
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1answer
65 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
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1answer
34 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
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1answer
46 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 ...
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0answers
87 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 ...
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1answer
60 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
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1answer
73 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, ...
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2answers
194 views

Sets of Cardinals without choice

I have a theory that the axiom of choice is equivalent to the statement that sets of distinct cardinals are well ordered by cardinality. I can prove that the axiom of choice implies this. However I am ...
2
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1answer
29 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 $b\to a$. ...
3
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1answer
59 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 ...
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1answer
51 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 ...
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1answer
45 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 ...
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1answer
34 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 ...
1
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1answer
35 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 $M$?
1
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1answer
27 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 $[P\to P]$ ...
3
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1answer
107 views

Justification of ZFC without using Con(ZFC)?

I saw this post by Carl Mummert, which describes how one can 'see' the reasonableness of ZFC, via iterating the power-set operation starting from the empty set. However, this iterations proceed in ...
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0answers
50 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) ...
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1answer
111 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 ...
2
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1answer
72 views

Does PA prove that Con(PA) implies Con(ZF-I) and Con(NFU)?

I read from many sources that PA and ZF-I (a suitable axiomatization of ZF minus Infinity plus its negation) are bi-interpretable, but is PA enough to prove that they are equiconsistent? Specifically ...
4
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1answer
63 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}_{ ...
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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 ...
0
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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
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2answers
25 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
44 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 ...
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2answers
105 views

Proof that the minimal inductive set does not contain any more elements than $\{\emptyset, s(\emptyset), s(s(\emptyset)),s(s(s(\emptyset))),…\}$?

I am studying axiomatic (ZF) set theory and in particular the construction of the natural numbers from axiomatic set theory. Let me start by giving some introduction to my question. The axiom of ...
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1answer
29 views

Relation between limit ordinals and alephs. [duplicate]

I was wondering what the relation is between a limit ordinal and the alephs. Are all limit ordinals alephs and if so can it be proven.
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1answer
27 views

Is the set of all limit ordinals a set? [closed]

It could be a limit ordinal too, then would be a class, but I'm not sure about is that an ordinal number
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1answer
31 views

Proof involving well ordering

Let $A,B$ be well ordered sets with corresponding well orderings $\leqslant $ and $\leqslant '$. If $A$ is order isomorphic with $B$ initial segment $B '$ and $B$ is order isomorphic with $A$ initial ...
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1answer
39 views

Comparing infinite cardinals [closed]

I have a question concerning infinite cardinals which I found on an old exam paper: Let $c=2^{\aleph_0}$, $x=2^c, y=2^{2^c}, z=2^{2^{2^c}}$. Put $x^{y^z}, x^{z^y}, y^{z^x}$ in ascending order. I'm ...
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1answer
31 views

Cardinality of the set of all Hamel bases.

If we look at $\mathbb{R}$ as a vector space over $\mathbb{Q}$, then a Hamel basis $B$ has cardinality $\mathcal{c}$ (assuming the continuum hypothesis). What about the cardinality of the set of ...
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1answer
44 views

Diagonal union of non-stationary sets

I have a family of non-stationary sets $A_{\alpha}$ for $\alpha < \kappa,\ A_{\alpha}\subset\kappa$. The exercise is to show that $\triangledown A_{\alpha}$ is also non-stationary. I've been ...
6
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3answers
241 views

Why is establishing absolute consistency of ZFC impossible?

Why is establishing the absolute consistency of ZFC impossible? What are the fundamental limitations that prohibit us with coming up with a proof? EDIT: This post seems to make the most sense. In ...
2
votes
1answer
112 views

Prove $\bigcup \omega_1 = \omega_1$

This is a question regarding ordinals and probably requires some background knowledge of ordinal arithmetic. I will list what I hope is the relevant information for this question: $\omega_1$ is ...
3
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0answers
80 views

Existence of $\lambda^+$ Aronszajn trees when $\lambda$ is regular and $2^{<\lambda}=\lambda$

While I was dealing with Aronszajn trees I found the following exercise from Kunen's old book. If $\kappa=\lambda^+$ and $\lambda$ is a regular uncountable cardinal and $2^{<\lambda}=\lambda$ ...
3
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2answers
87 views

A class that contains itself as an element

Can a class be defined which contains itself as an element? I know its forbidden for a set to contain itself, and I have seen arguments that suggest it's possible for a class to do so but none of ...
2
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2answers
74 views

Can number 2 be defined as a formula in set theory?

Natural numbers can be represented as sets, however there are more than one representation of natural numbers in set theory (for example von Neumann's and Zermelo's). But all the representations of ...