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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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 ...
7
votes
1answer
122 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} = ...
4
votes
2answers
112 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 ...
0
votes
1answer
39 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, ...
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1answer
40 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
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0answers
45 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
39 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
38 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. ...
2
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1answer
76 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 ...
5
votes
1answer
144 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 ...
1
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1answer
27 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. ...
6
votes
1answer
104 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 ...
1
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0answers
55 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 ...
1
vote
0answers
33 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 ...
2
votes
1answer
67 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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0answers
39 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
35 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
86 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
164 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 ...
0
votes
2answers
120 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 ...
8
votes
5answers
636 views

Do we need Axiom of Choice to make infinite choices from a set?

According to the answers to this question, we do not need choice to pick from a finite product of nonempty sets, even if each of the sets is infinite. The axiom of choice is required to ensure that a ...
3
votes
2answers
175 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
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1answer
95 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.
2
votes
1answer
48 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 ...
2
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0answers
71 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 ...
3
votes
2answers
80 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
59 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 $x \not\in f(x)$, ...
4
votes
2answers
67 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
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0answers
51 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, ...
2
votes
1answer
64 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 ...
7
votes
1answer
191 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 ...
1
vote
1answer
50 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 ...
3
votes
1answer
92 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 ...
1
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1answer
78 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 $\alpha$ ...
10
votes
1answer
102 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 ...
4
votes
1answer
75 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 ...
5
votes
1answer
70 views

Cardinal Arithmetic: $\beth_\omega$

Is the following equality independent of ZFC: $$\bigcup_{\aleph_{\beta}<\beth_{\omega}}2^{\aleph_{\beta}}=\beth_\omega $$ Consistent: Now that the equality is consistent with ZFC since it holds ...
4
votes
1answer
84 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 ...
0
votes
2answers
61 views

Ordinal existence

Is there any ordinal $\alpha$ such that $\omega ^ {\omega ^ \alpha} = \alpha$? Could you please suggest me how to even try to solve this?
1
vote
0answers
108 views

Does the Math-tea Argument Have Any Relevance to the Method of Forcing?

The Math-tea Argument (i.e. the argument that, for example, there must be real numbers that we cannot describe or define, because there are only countably many definitions, but uncountably many reals) ...
2
votes
2answers
89 views

An exercise in Kunen

The following exercise appears in Kunen; In $M$, let $\mathbb{P}=Fn(\kappa,\lambda)$ (i.e. the finite partial functions), where $\aleph_0\leq\kappa<\lambda$. Then $\lambda$ is countable in $M[G]$, ...
1
vote
1answer
27 views

Class Sizes and Ordering

It is independent of ZFC that every proper class has the same "size". (See Size Of Proper Classes). However, is it necessary that the proper class sizes must be linearly ordered if then are not the ...
2
votes
1answer
96 views

Are All Nonstandard Models of PA Ill-Founded?

We often use sets to represent natural numbers, but we can also use natural numbers to represent sets. For example, we can use the binary expansion of a natural number to represent a set. The number ...
2
votes
1answer
132 views

Best Less-Famous Texts for Forcing

There are many books, papers and lecture notes which give an introduction to forcing (e.g. Jech or Kunen's books) but here I am looking for some possibly less-famous useful comprehensive texts for ...
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0answers
63 views

Large Cardinals with Elementary Extension Characterization

Question: Which large cardinals $\kappa$ have a characterization in the following form: $\kappa$ is large if and only if for all cardinals $\lambda>\kappa$, $\langle W_{\kappa},\in\rangle\prec ...
6
votes
1answer
111 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 ...
0
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1answer
60 views

Newbie approach to understand generalized continuum hypothesis

There is this theorem that size of power set constructed from infinite set is "more" infinite than the previous set: $$ \begin{eqnarray*} \aleph_0 &= |\mathbb{N}| \\ \aleph_{n+1} &= ...
2
votes
1answer
107 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 ...
0
votes
1answer
96 views

Representative elements in the symmetric difference metric

The symmetric difference is a natural way to quantify the distance between measurable sets: $$d(S,T)=measure([S\setminus T]\cup[T\setminus S])$$ This is a pseudo-metric because there may be ...
2
votes
1answer
37 views

A problem in transitivity of set

Let $D$ be a transitive set with the following property : $$\forall a\in D:\quad a\subseteq B\,\Longrightarrow a\in B$$ Prove that $D\subseteq B$. I think there will be needed axiom of ...