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
33 views

Cardinality of a set of subsets with given max cardinality [duplicate]

Let $S$ be a set of infinite cardinality $\kappa_1$. What is the cardinality $k$ of the set of subsets of $S$ with cardinality $\kappa\le\kappa_0<\kappa_1$? I understand that if $\kappa_0$ is ...
5
votes
1answer
56 views

Exercise on forcing

I got this homework in my forcing class: Let $G\subseteq P$ be generic over M. Show that there is a cardinal of M, $\lambda$ such for every set of ordinals $X\in M\left[G\right]$ there is a set ...
0
votes
1answer
100 views

Non WellFounded Set theories and Russell's Paradox

I am very puzzled by set theories which reject the axiom of regularity. If we reject the axiom of regularity, and allow a distinction to be drawn between wellfounded and non-wellfounded sets/classes, ...
4
votes
1answer
81 views

Ordinal exponentiation, is $3^\mu = \mu$?

I'm revising for my set theory final, and I've been asked to find an ordinal $\mu > \omega$ with $2^\mu = \mu$, then to answer whether $3^\mu = \mu$. The ordinal I picked as $\mu$ was the union ...
2
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1answer
52 views

“Every set is equinumerous to a well-founded set” - provable in $\sf ZF-Reg$?

What is the relationship of the following to other axioms of $\sf ZFC$? $\sf WB$: Every set $A$ is in bijection with a set well-founded by $\in$. Obviously, $\sf ZF$ implies $\sf WB$ (because ...
2
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2answers
125 views

What are some simple example of “forcing” in set theory?

Can someone illustrate the idea of "forcing" in set theory through some simple examples? The article on forcing on wikipedia goes straight to axiom of choice and continuum hypothesis, I wonder if ...
1
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1answer
36 views

Choosing a $x \in$ $X$, with $X$ an infinte set, $<$ a well-ordering on $X$, such that $x < x'$ for only finitly many $x' \in X$

Let $X$ be an (countable or not) infinte set, $<$ a well-ordering on this set. Lately I read a proof in which was explicity stated to choose an element $x \in X$ such that there are infintly many ...
9
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0answers
154 views

A consequence of zero sharp (successor cardinals having countable cofinality)

So the existence of $0^\sharp$ in set theory is really the assertion on the existence of indiscernibles for the constructible universe $L$ that also "generate" $L$ (see ...
8
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2answers
531 views

Is there a set theory that avoids Russel's paradox while still allowing one to define the set of all sets not containing themselves?

The main idea of Russel's paradox is that, in Naive Set Theory, if we define $R = \{x\ |\ x \not\in x \}$, then $R \in R \Leftrightarrow R \not \in R$. ZFC deals with this by making unrestricted set ...
3
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1answer
48 views

Cardinality of a Grothendieck Universe

Let us work in $ZFC+U$ where $U$ is the existence of a Grothendieck Universe. Let $\mathrm{On}(U) = \mathrm{On}\cap U$ denote the set of ordinals in $U$. How can I show that the cardinality of $U$ is ...
2
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1answer
62 views

stationary set,club,module theory,Auslander lemma,

Here http://www.ams.org/journals/tran/1990-322-02/S0002-9947-1990-0974514-8/S0002-9947-1990-0974514-8.pdf I do not understand the first two lines of the proof of lemma 9,on page 550:What and why is ...
0
votes
2answers
43 views

Axiom of foundation allows sets consisting of a descending sequences plus some 'atom'?

I am reading a text which describes how the Axiom of Foundation prevents sets that are built from a descending sequence such $$X=\{x_0, x_1,\ldots\},\text{ with } x_1\in x_0, x_2\in x_1,\ldots$$ ...
4
votes
1answer
83 views

Diamond and Suslin tree

I'm reading the proof (assuming $\Diamond$) of the existence of a Suslin tree in Nik Weaver's Forcing Mathematicians (Theorem 18.4 page 71) and I have difficulty seeing the use of $\Diamond$. Given a ...
3
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0answers
71 views

What do the ZFC axioms look like in terms of subset?

I'm reasonably familiar with the ZFC axioms. I know they can be formalized in many different ways. However, I usually see them presented in terms of the elementary "propositional calculus primitives" ...
3
votes
2answers
115 views

Splitting the Real Line

By definition a $\mathfrak c$-dense subset of $\mathbb{R}$ has $\mathfrak c$-sized intersection with every non-empty open set. Using transfinite recursion it is quite easy to prove that every ...
3
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1answer
78 views

Is there a set $A$ such that power set of $A $ has a bijection with $\mathbb{N}$? [duplicate]

Has this statement any relation with continuum hypothesis ?
4
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1answer
65 views

The Diamond Principle implies the Club Principle.

So the Diamond and the Club principles are both combinatorial principles in set theory. They are defined as follows (there are thinner definitions but I stick to this ones is $\omega_1$, as I am sure ...
6
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0answers
78 views

Elementary submodels in stationary logic.

In the paper "Stationary Logic" by Barwise, Kaufmann and Makkai the authors prove that stationary Logic L(aa) has Löwenheim number $\aleph_1$, i.e. every satisfiable set of sentences has a model of ...
3
votes
1answer
39 views

Splitting Stationary Sets

So by a theorem of Solovay we know that a stationary subset of a regular (uncountable) cardinal $\kappa$ can always be split into $\kappa$-many stationary subsets. Is the regularity assumption ...
0
votes
1answer
38 views

Ordinal arithmetic and limit ordinals

Suppose $1\leq\xi<\omega_1$ is a countable ordinal and that $1\leq\zeta<\omega^\xi$. Is it always true that $\zeta+\omega^\xi=\omega^\xi$? If so, why?
0
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1answer
79 views

Hanf Numbers and Decidability

Currently reading J.L. Bell's Models and Ultraproducts and at the end of Chapter 4 section 4 the authors comment that "In spite of the fact that most languages can easily be shown to possess Hanf ...
4
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1answer
61 views

Not meager in $V^{\mathbb{C}_I}$

Assume $A\subseteq 2^\omega $ is not meager in any non-empty open set, in The ground model $V $. Then is not meager in any non-empty open set, in $ V^{\mathbb{C}_I}$ where $\mathbb{C}_I$ is Cohen ...
1
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0answers
42 views

If $M\prec L_{\omega_1}$ then $M = L_\alpha$ for some $\alpha$ - we need a condition to prove it?

I try to prove the exercise 13.17 in Jech: If $M\prec (L_{\omega_1},\in)$, then $M=L_\alpha$ for some $\alpha.$ [Show that $M$ is transitive. Let $X\in M$. Let $f$ be the $<_L$-least ...
1
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3answers
68 views

Why do we have Axiom of Pairing but we don't have its generalisation, i.e a collection exists instead of pairing axiom?

For unions we have the generalised axiom, not just union for pairs. But for pairing we don't have generalisation, that a collection exists for any number of sets. If we have some axiom that says ...
1
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1answer
37 views

Can a sum of a nonprincipal ultrafilter and a principal ultrafilter be equal to the nonprincipal ultrafilter?

If $ \mathcal U$ is a nonprincipal ultrafilter and $\mathcal V$ is a principal ultrafilter, can $ \mathcal U \oplus \mathcal V$ be equal to $\mathcal U$ ?
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0answers
72 views

is axiom of powers required?

we can form any subset of a set using axiom of specification. then we can form collection of subsets using axiom of pairing repeatedly. we have for every condition S , (A={x belongs to X & S(x)} ...
9
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0answers
170 views

The ethics of Borel determinacy

I was speaking with a friend the other day, and I happened to say "morally, Borel determinacy is as strong as ZF." I was riffing on the well-known result of Harvey Friedman, that we need ...
7
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1answer
106 views

For an infinite cardinal $\kappa$, $\aleph_0 \leq 2^{2^\kappa}$

I'm trying to do a past paper question which states: $$ \text{For all infinite cardinals $\kappa$, we have } \aleph_0 \leq 2^{2^\kappa}. $$ I'm supposed to be able to do this without the axiom of ...
2
votes
1answer
48 views

Saturated models and $\kappa=\kappa^{<\kappa}$

Do not assume GCH. Can you characterize the cardinals $\kappa$ such that every theory $T$ with an infinite model has a saturated model of cardinality $\kappa$? I guess these are the cardinals such ...
2
votes
1answer
38 views

Clarifying some doubts on the definition of “extensional classes”

On page 68 of Jech's "Set Theory" 4th Edition there is the following definition : A class $\mathcal{M}$ is extensional if the relation $\in$ on $\mathcal{M}$ is extensional, i.e., if for any ...
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1answer
34 views

The set of ordinals $< \alpha$ of a given cofinality $\kappa< \text{cf}(\alpha)$ is stationary

I don't understand the last line of the following proof : I understand that we climbed up to $g(\kappa)$ in $\kappa$ steps (i.e. cf$\big(g(\kappa)\big) \leq \kappa$) but can we be sure that we did ...
5
votes
3answers
164 views

Lists of sets as objects of ZF axiomatics

I have a naive question about foundations of mathematics. A common opinion of most mathematicians is that the essential part of mathematics can be reduced to ZF(C) axioms. I do not quite understand ...
1
vote
4answers
301 views

What is the future of Set Theory if it is NOT the foundation of Mathematics?

Homotopy Type Theory has been receiving a lot of hype during the last couple of years and been hailed as the most effective foundation of all Mathematics compared with Set Theory. My question: If ...
2
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1answer
65 views

What do we call well-founded posets whose elements have a unique height?

What do we call those well-founded posets $P$ with the property that for every $x \in P$, all maximal chains in the lowerset generated by $x$ have the same length? Examples: The set of all finite ...
1
vote
1answer
85 views

Is this behavior of the continuum function consistent with ZFC.

I have an axiom, based on how finite cardinalities work. As we know, every ordinal can be written as the sum of a unique limit ordinal $L$ (where $0$ is a limit ordinal axiom for the purposes of this ...
2
votes
2answers
65 views

$|\alpha\times\alpha|=|\alpha|$ for infinite ordinal $\alpha$, definably

I am trying to prove Proposition 1.7 of http://math.bu.edu/people/aki/7.pdf: Proposition 1.7 (Halbeisen–Shelah). If $\aleph_0\le|X|$, then $|{\cal P}(X)|\not\le|{\rm Seq}(X)|$. In words, ...
5
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3answers
1k views

What's wrong with this proof of the inconsistency of the axiom of choice?

Let $\mathscr{T}$ be the (countable) collection of all theorems provable in ZFC. Define an equivalence relation on $\mathscr{T}$ by $\phi\sim\psi$ iff $(\phi \iff \psi)$. In other words, two theorems ...
1
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1answer
32 views

Justification for ordinal arithmetic at limit ordinals

Wikipedia tells me that: $$\alpha + \lambda := \bigcup_{\beta < \lambda} \left ( \alpha + \beta \right ) $$ for a limit ordinal $\lambda$. Multiplication and exponentiation, as Wikipedia says, ...
2
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0answers
54 views

Supplementing FOL with the Härtig quantifier

"The Härtig quantifier captures a certain fairly large fragment of second-order logic and contributes to understanding higher-order logic." (p. 1154 ...
1
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0answers
32 views

Condition for an Ultrafilter to be Ramsey.

I read in the english Ultrafilter article on Wikipedia that one can prove that a non-principal ultrafilter $D$ on $\omega$ is a Ramsey ultrafilter if and only if for every coloration $c:[\omega]^2 ...
1
vote
1answer
48 views

Indiscernibles and colorations.

Let $L$ be a language, and $(X,\leq)$ be a total order contained in an $L$-structure $\frak{A}$. Now if we denote by $[X]^n$ the set of $n$-sized sequences in $X$ and consider a set $\Gamma$ of ...
1
vote
1answer
48 views

If $|A|<|B|$ does $B$ surject onto $\aleph(A)$?

After reading Proving existence of a surjection $2^{\aleph_0} \to \aleph_1$ without AC I became curious if there is a generalization to arbitrary cardinals. That is, if $\frak m<n$, does it follow ...
1
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1answer
42 views

P-names are comprehensible in the ground model

If my understanding of $\mathbb{P}$-names $\tau$ is correct, they must be comprehensible by someone living in the ground model $\mathcal{M}$. However, the value $\tau[G]$ of $\tau$ need not be ...
1
vote
1answer
49 views

How to show countability of $\omega^\omega$ or $\epsilon_0$ in ZF?

I know that with choice, the countable union of countable sets is countable, making $\omega^\omega$ and $\epsilon_0$ both countable. Can we show this without choice? E.g. in the case that $\omega_1$ ...
3
votes
1answer
123 views

Must $\mathfrak{sd} = \mathfrak{d}$?

This should be fun. Let $\mathfrak{sd}$ be the least cardinal $\kappa$ such that there exists $\langle s_i : i < \kappa \rangle$ satisfying the following. (1) Each $s_i: \omega^{\omega} \to ...
4
votes
1answer
89 views

How to prove that $\sf CH$ implies $2^{\aleph_0}=\aleph_1$

Of course, most of you will, upon reading the title, exclaim "But isn't that the definition of the continuum hypothesis?" So I need to be a little more careful about the exact definitions. Let ...
3
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0answers
48 views

Does “$(\exists f:A\twoheadrightarrow B)\implies(\exists f:B\hookrightarrow A)$” implies the axiom of choice? [duplicate]

Let $P$ denotes the property that if there exists a surjection from set $A$ to set $B$, then there exists an injection from $B$ to $A$. It's apparent that $P$ can be proved in ZFC. My question is that ...
0
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0answers
35 views

Intersection with the empty set in Principia Mathematica

In Whitehead and Russell's Principia Mathematica, "arithmetical class-sum" is defined as I have tried my best to decipher the dated notation in the book. It seems that the sum is defined as the ...
5
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0answers
131 views

ZF and weakly inaccessible cardinals

This question should probably have been asked 3 years ago (perhaps it has and was removed for some reason?) In 2011, Alexander Kiselev claimed to have proved in ZF that there are no weakly ...
2
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1answer
35 views

Question about $\text{non}(\mathsf{nonstat}_\gamma)$

A lemma in Kunen's (2011) set theory states that if $\gamma$ is a limit ordinal with $\kappa=\text{cf}(\gamma)>\omega$, then $\text{add}(\mathsf{nonstat}_\gamma)$ $=$ ...