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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5
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2answers
111 views

A construction of sigma-algebras - surely not new, right?

I know no descriptive set theory. I've stumbled on something that must be well known, being so simple. But it contradicts something I've been told by smart people; the question is whether it's well ...
4
votes
1answer
66 views

First Uncountable Ordinal Cofinality: Needs AC?

Say $\omega_1$ is the first uncountable ordinal. The reason I care about $\omega_1$ is Any countable subset of $\omega_1$ is bounded (or if you prefer, there is no countable cofinal subset). This ...
2
votes
1answer
64 views

Are there collections of sets that are neither a set nor a definable proper class?

Working in ZFC, are there collections of sets that are neither a set nor a definable proper class? I mean if some collection of sets is not a set can we necessarily conclude that it is a definable ...
4
votes
1answer
53 views

What conditions must be checked for that $c$ is Cohen over $V$.

$\textbf{Hechler forcing} $ Let $\mathbb{D}=\{(s,f): s \in \omega^{<\omega},f\in \omega^{\omega} \text{and} s \subseteq f\}$, ordered by $(t,g)\leq (s,f)$ if $s \subseteq t$, $g$ dominates $f$ ...
3
votes
0answers
58 views

References for Introductory Model Theory focusing on applications other than algebra

I would appreciate suggestions for references (books, lecture notes, articles etc...) on Model Theory (at an introductory level) that don't focus mostly on algebra when giving examples and/or applying ...
2
votes
1answer
44 views

ZF and the Existence of Finitely additive measure on $\mathcal{P}(\mathbb{R})$

My understanding is that Solovay (1970)'s relative consistency shows that if ZFC+I has a model then ZF+DC has a model in which every subset of the reals is Lebesgue measurable (and hence ...
5
votes
2answers
85 views

Proving that $\alpha\approx|\alpha|$ is not constructive

$\sf ZF$ tells us that for every ordinal $\alpha$ there is a bijection $f:\alpha\to|\alpha|$, more or less by the definition of $|\alpha|$, the cardinal of $\alpha$. What I would like to show is that ...
6
votes
2answers
122 views

The non-existence of non-principal ultrafilters in ZF

In Hrbacek and Jech (1999, p.205), they point out that "it is known that the theorem [the extension of any filter to an ultrafilter] cannot be proved in Zermelo-Fraenkel set theory alone." And in Jech ...
1
vote
3answers
102 views

Explanation of a “sentence” of Halmos's Naive Set Theory

While reading Halmos's Naive Set Theory I found that he has remarked in a place, If $\mathcal{C}$ be a collection of subsets of a set $E$ (that is, $\mathcal{C}$ is a subcollection of ...
3
votes
2answers
128 views

Naively addressing Russell's paradox

Russell's paradox prevents us from allowing any expression of the form $\{x \mid P(x)\}$ from being a set. His observation shook up the field of set theory, prompting formal axiomatizations of the ...
4
votes
1answer
64 views

Singularity of small cardinals under AD

It appears to be quite a folklore result that under AD we know which small well-orderable cardinals (for my purposes I mean below $\omega_\omega$) are regular, namely only $\omega,\omega_1,\omega_2$. ...
0
votes
2answers
58 views

Why is the axiom of pairing needed in Von Neumann-Godel-Bernays Set Theory?

Why do we need the axiom of pairing in Von Neumann-Godel-Bernays Set Theory? Doesn't the following prove it? Suppose $A$ and $B$ are sets. Using the axiom of class comprehension, we can form a ...
1
vote
1answer
60 views

Regarding chains and antichains in a partially ordered infinite space [duplicate]

I've been given this as an exercise. If P is a partially ordered infinite space, there exists an infinite subset S of P that is either chain or antichain. This exercise was given in the Axiom ...
4
votes
1answer
108 views

What is wrong with this “proof” that there is no $\omega$th inaccessible cardinal?

"Theorem": There is no $\omega$th inaccessible cardinal. "Proof": Assume ZFC. Let $\kappa_n$ be the $n$-th inaccessible cardinal; since $V_\kappa$ is a model of ZFC for inaccessible $\kappa$, ...
14
votes
2answers
641 views

Do we know that we can't define a well-ordering of the reals?

Folklore has it that it is impossible to define a well-ordering of the reals explicitly. There exist pointwise definable models of ZFC where every set is definable without parameters: it is the ...
5
votes
1answer
54 views

Ordinal arithmetic $(\omega+1) \cdot \omega$ and $\omega \cdot (\omega +1)$

Here is where I am so far: $(\omega+1) \cdot \omega = \sup\{(\omega +1) \cdot n, n \in \omega\} = \omega^2$ and $\omega \cdot (\omega +1) = \omega \cdot \omega + \omega = \omega^2 + \omega$ Hence ...
0
votes
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
54 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
75 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
80 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
votes
1answer
51 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 ...
1
vote
2answers
115 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
vote
1answer
35 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 ...
8
votes
0answers
137 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
votes
2answers
524 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
votes
1answer
45 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
votes
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
40 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
81 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
votes
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
votes
1answer
76 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
votes
1answer
62 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
votes
0answers
74 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
37 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
36 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
votes
1answer
77 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
votes
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
vote
0answers
41 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
vote
3answers
62 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
vote
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 $\mathcal U$ ?
1
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0answers
71 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
votes
0answers
161 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
votes
1answer
104 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 ...
1
vote
1answer
43 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
37 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 ...
1
vote
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
299 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
votes
1answer
63 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 ...