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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1answer
38 views

Constructing a Banach space of cardinality $\beth_{\omega+1}$

This is related to yesterday's question Constructing a vector space of dimension $\beth_\omega$; it's the next exercise (I.13.35 (a)) in Kunen's Set Theory. Let $B_0 = \ell^1$ and let $B_{n+1} = ...
2
votes
0answers
23 views

Exercise on preservation of cardinals

I'm trying to solve exercise 12.1 in Prof. Monk's lectures on set theory which asks me to show that $Fn(\omega_1, 2, \omega_1)$ preserves cardinals larger than $\omega_2$. Now, the only method in the ...
3
votes
2answers
82 views

Constructing a vector space of dimension $\beth_\omega$

I'm trying to solve Exercise I.13.34 of Kunen's Set Theory, which goes as follows (paraphrased): Let $F$ be a field with $|F| < \beth_\omega$, and $W_0$ a vector space over $F$ with $\aleph_0 ...
4
votes
1answer
83 views

cov(meager) strictly between $\aleph_1$ and $2^{\aleph_0}$

Is is consistent that $\aleph_1 < \text{cov(meager)} < 2^{\aleph_0}$? I can only seem to find references for results that assert it is consistent that it (or other cardinal characteristics) is ...
0
votes
1answer
29 views

Model of a system of set theory

I was trying to understand Easton's theorem's proof. But I am not able to understand a few terms- model of a system of set theory, ranked variables, free variables, abstraction term, set constant, ...
-1
votes
1answer
35 views

Functions with cardinals as domain

How do we actually define a function with cardinals as domain? For example, take domain of the function as $\aleph_1$. Do we define it for all cardinal numbers strictly less than $\aleph_1$?
2
votes
0answers
22 views

The notations $ M[X] $ and $ V[A] $ and intermediate models

I am confused about Jech's notations $ M[X] $ and $ V[A] $. Let us begin with exercise 13.34 (see [Jec02, p. 199]). Let $ \mathbf{M} $ be a transitive model of $ \mathsf{ZF} $ containing all ...
0
votes
1answer
21 views

Cardinal Arithmetic, Regular Cardinals, and Exponentiation

I cannot solve a simple cardinal exponentiation/regularity exercise. Let $\kappa$ be a regular cardinal. Why is the cardinal $\kappa^{\lt \kappa}=\sum_{\alpha<\kappa}\kappa^\alpha$ regular as ...
0
votes
1answer
17 views

Union of Dedekind-finite sets

$F$ is Dedekind-finite if for every $A\varsubsetneq F$ we have $A<_cF$. Need help to prove that if $F,G$ are Dedekind-finite sets, $F\cap G=\emptyset$ then $F\cup G$ is also Dedekind-finite. ...
3
votes
0answers
32 views

Complete embeddings and intermediate models

Let $ M $ be a countable, transitive model for $ \mathsf{ZFC}^* $, i.e. for a sufficiently large finite fragment of $ \mathsf{ZFC} $. Theorem. Suppose that $ \mathbb{P} := (P, {\leq_P}, ...
2
votes
1answer
53 views

In $\sf{ZF}$, is every finite set also hereditarily finite?

The following is an excerpt from Wikipedia's Hereditarily finite set page: A recursive definition of well-founded hereditarily finite sets goes as follows: Base case: The empty set is a ...
6
votes
1answer
67 views

Construction of Ultrafilters

I've been doing a lot of work with ultrapowers and saturation recently. In particular, I am reading chapter 6 of Chang and Keisler as well as Keisler's paper on "Ultraproducts which are not ...
2
votes
1answer
73 views

Definition in Kunen

In Kunen's second edition of set theory he gives the following definition Let $(\mathbb{Q},\leq_\mathbb{Q},\mathbb{1}_\mathbb{Q})$, and $(\mathbb{P},\leq_\mathbb{P},\mathbb{1}_\mathbb{P})$ be forcing ...
0
votes
2answers
98 views

Is the “Most Important Property a Set S has” Necessary and Sufficient to Define a Paradox-Free Notion of Set?

About a year and a half ago, while I was looking on the Web for papers regarding the Russell paradox, I chanced to find an interesting concept. This concept was contained in what (for want of a ...
1
vote
1answer
25 views

Is “constructible from” a transitive relation?

In Jech's Set Theory, exercise 13.27, it is hinted that $X \in L[Y]$ and $Y \in L[X]$ together imply $L[X]=L[Y]$. I tried to prove this fact without success, although I suspect the proof is simple. ...
2
votes
0answers
96 views

How large or small can the gap in Shelah's main gap theorem be, up to consistency?

Shelah's main gap theorem in model theory says: For each first order complete theory $T$ in a countable language if $I(T,\kappa)$ denotes the number of its models of size $\kappa$ then one of ...
1
vote
0answers
42 views

Banach Tarski paradox in Minkowski space

The Banach - Tarski paradox states: 'A three-dimensional Euclidean ball is equidecomposable with two copies of itself.' My simple question is: Does this paradox hold his validity also in the Minkowski ...
4
votes
5answers
514 views

What's so special with small categories?

Sometimes one encounter the requirement that the objects of a category needs to be a set. What if it was not, could you provide examples of what could go wrong? (One example per answer).
1
vote
0answers
29 views

Can one talk about equivalences classes in a class(which might not be a set). [duplicate]

equivalence relation on a set breaks it into disjoint sets. does 'disjointness' make sense in classes. specifically, given a category, isomorphism classes of objects makes sense or not. can it be so ...
1
vote
1answer
46 views

Why is $\alpha \mapsto L_{\alpha}[A]$ $\Delta_{1}$?

On page 187 of Jech's Set Theory, there is a proof sketch of the fact that $\alpha \mapsto L_{\alpha}$ is $\Delta_{1}$. As far as I can tell, Jech's argument only shows that this operation is ...
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votes
0answers
24 views

Does there exist a cardinal $\kappa$ such that $\aleph_{\kappa} = \kappa$? [duplicate]

Does there exist a cardinal $\kappa$ such that $\aleph_{\kappa} = \kappa$ ? Moreover is there one that is regular? Thanks in advance!
0
votes
0answers
37 views

Sets Constructible Relative To A Unary Predicate

The class $L$ of constructible sets is defined by recursion using the operation def$(M)=\{x \subset M: x$ is definable over $(M, \in) \}$. By adding a unary predicate, $P$, to our language, we can ...
0
votes
0answers
31 views

Tarski's theorem follows from choice [duplicate]

It is known that Tarski's theorem and axiom of choice are equivalent. Implication $\Rightarrow$ follows from considering bijection $(A+\aleph(A))^2\rightarrow(A+\aleph(A))$. Implication $\Leftarrow$ ...
3
votes
2answers
81 views

Distinguishing sets according to more fine-grained notions than cardinality.

I'm interested in distinguishing sets according to more fine-grained notions than cardinality. Now I don't know a thing about computability theory, but it seems to me that considering sets up to ...
-1
votes
1answer
92 views

Can't prove Continuum Hypothesis

My teacher said (and I checked on Wikipedia too), that there is a theory, the Continuum hypothesis, that can neither be proved nor disproved. However, the theory states that: There is no set ...
1
vote
1answer
55 views

Suppose $X$ is a Hausdorff Lindelöf scattered space. Is $\xi(X)$ a successor ordinal?

Let $K$ be a (Hausdorff) scattered topological space and for each ordinal $\alpha$ denote by $K^{(\alpha)}$ the $\alpha$th derivative of $K$ by the Cantor-Bendixson derivation (i.e., define ...
12
votes
2answers
620 views

Transfinite Induction and the Axiom of Choice

My question is essentially this: Why does the principle of transfinite induction not suffice to show the axiom of choice when the sets to be chosen from are indexed by a well ordered set? I have read ...
7
votes
2answers
183 views

Is the completeness theorem for first-order logic relative to one's choice of set theory?

By the completeness theorem for first-order logic, every consistent theory has a model. However, to even make sense of the word "model," I believe we're assuming a set theory. So is there a set theory ...
7
votes
6answers
1k views

Axiom of Choice and finite sets

So I am relatively familiar with the Axiom of Choice and a few of its equivalent forms (Zorn's Lemma, Surjective implies right invertible, etc.) but I have never actually taken a set theory course. I ...
6
votes
1answer
100 views

The continuum hypothesis (CH) and its equivalent

For a set $A \subseteq \mathbb{R}^2$ and $x,y \in \mathbb{R}$, we define $A^y=\{x \in \mathbb{R}\mid (x,y) \in A\}$ and $A_x=\{y \in \mathbb{R}\mid(x,y) \in A\}$. Proposition: The continuum ...
6
votes
1answer
170 views

What is semantics of “type”? Do “types” of “type theory” semantically differ from “set” of set theory?

"To be of a (certain type)" is a fundamental relationship for ontology and the computer science "ontologies" are in the core of Semantic Web (which is my interest). But I did not encounter a ...
3
votes
2answers
143 views

What is the meaning/purpose of finding the “foundations of mathematics”?

I've read in a lot of places how there was a "foundational crisis" in defining the "foundations of mathematics" in the 20th century. Now, I understand that mathematics was very different then, I ...
1
vote
1answer
62 views

Reference request, Descriptive set theory

I was wondering what a good text would be to learn descriptive set theory out of? Hopefully something more in the spirit of Kunen's text on the introduction to independence proofs.
4
votes
1answer
75 views

Is the class of countable posets well-quasi-ordered by embeddability?

The question is in the title. Here "$P$ embeds into $Q$" means there is a function $f : P\to Q$ such that for all $p,p'\in P$, $p \le_P p'$ if and only if $f(p) \le_Q f(p')$. A well quasi order $W$ ...
0
votes
3answers
340 views

Infinite paths that connect two vertices?

This is a follow-up to another question concerning infinite paths which was admittedly ill-posed. I hope this question is posed better. The graph $N$ with vertex set $V(N) = \mathbb{N}$ and $(x,y) ...
2
votes
1answer
102 views

Cardinalities of Collections of Models

Let $T$ be a complete theory in a countable language (with only infinite models). Recall the spectrum function: $I(\aleph_\alpha,T)=$ the number of non-isomorphic models of $T$ of cardinality ...
2
votes
1answer
45 views

Cardinal Arithmetic 2: $\beth_{\omega_1}$

This is question is simple to write but (I think) hard to solve. Does the following equality hold? $$\bigcup_{{\beta}<\beth_{\omega_1}}\beth_{\omega+1}(|\beta+\omega|) =\beth_{\omega_1} $$ Where ...
3
votes
2answers
58 views

Can a finite axiomatization of PA be expressed in a finitely axiomatizable first order set theory?

Peano Arithmetic has an infinite number of axioms because of its induction schema; Likewise $\sf ZFC$ has an infinite number of axioms because of its axiom schema of replacement. $\sf NBG$ however ...
2
votes
0answers
54 views

Proof of PFA from Supercompact

In Jech's proof (in chapter 31) that the consistency of a supercompact cardinal implies the consistency of PFA, he needs the following fact: Let $\mathbb{P}_\kappa$ be the countable support forcing ...
2
votes
0answers
49 views

Is $\mathcal{P}(\omega)$ bigger than $\omega$ in $NFU$ + infinity?

$NFU$ doesn't prove Cantor's theorem (that $\mathcal{P}(S)$ is cardinally greater than $S$) by a stratification dodge: the proof's critical step makes use of the unstratified formula $x \not\in f(x)$, ...
-2
votes
0answers
59 views

How can I make a set theory arbitrarily weak and finitely axiomatizable?

So the background is I'm trying to construct an automated theorem proving system based on first order resolution to formalize problems in mathematics and number theory. I'm familiar with the notion ...
4
votes
2answers
50 views

Example of a non-proper product of two proper forcing notions

I am looking for an example of a proper forcing notion $ \mathbb{P} $ such that $ \mathbb{P} \times \mathbb{P} $ is not proper. Maybe someone knows an obvious example or can give a reference to such ...
1
vote
0answers
34 views

Dual Constructions for Core Models

Roughly speaking, core models are inner models of ZFC which could contain some large cardinals. e.g. $L$ is the smallest core model and it is possible to have inaccessible, Mahlo, weakly compact, ...
1
vote
1answer
49 views

Closed subsets of $\beta \mathbb R$

Definitions. Suppose $X$ is a topological space. $w(X)=\min\{|\mathcal B|:\mathcal B$ is a base for $X\}+\omega$ $e(X)=\sup\{|D|:D\subseteq X$ is closed and discrete$\}+\omega$ $K(X)$ is the ...
18
votes
2answers
279 views

Covering $\mathbb R^2$ with function graphs

Suppose we have a countable family of function graphs (each function is $\mathbb R\to\mathbb R$, not necessary continuous). Obviously, they cannot cover the whole plane $\mathbb R^2$, because they ...
9
votes
1answer
98 views

Choosing elements of linear orders

Is it consistent with ZF that there can be a countable family of linear orders, each isomorphic to $\mathbb Z$ (that is, every element has a unique predecessor and successor, and any two elements have ...
3
votes
2answers
128 views

Kunen “Set Theory” 2011 versus 1980 edition - worth buying again?

What are the differences between the original edition (1980) of Kunen's famous book and the new edition (2011)? Is the updated version worth buying? (I hope this kind of question is allowed here. I ...
1
vote
1answer
69 views

Partitioning an infinite cardinal number

Can we partition an infinite cardinal greater than aleph null, into countable number of cofinal subsets? Can we have restriction on the cardinality of cofinal subsets? For example, suppose $\alpha$ ...
4
votes
1answer
62 views

Model of complete extension of Zermelo set theory

Chang and Keisler's Model theory gives the following exercise problem: Prove that there is a complete extension $T$ of Zermelo set theory which has arbitrary large natural models. (A model ...
4
votes
1answer
81 views

How to make an order isomorphism

Two linear orders $A$ and $B$ have starting points $a_0$ and $b_0$, and have cofinalities $\omega_1$. Let $(a_\alpha )_{\alpha<\omega_1}$ and $(b_\alpha )_{\alpha<\omega_1}$ be cofinal ...