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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2answers
71 views

If $U$ is an ultrafilter on $\mathbb{N}$, then $U$ limits exist.

This is rather silly, I expect Asaf will point out what I am missing immediately. Let $U$ be a filter on $\mathbb{N}$. If $\{a_n\}_{n=1}^\infty$ is a sequence of reals, we write $\lim_U a_n = a$ if ...
5
votes
1answer
86 views

A strengthening of delta system lemma

I would like to know if a strengthening of the delta system lemma is true. Suppose $\kappa$ is an infinite cardinal with $\kappa^{< \kappa} = \kappa$ (so $\kappa$ is regular). Suppose $S \subseteq ...
1
vote
0answers
68 views

Simple question of poset and names.

If $\mathbb{P}$ be a poset, $\dot{Q}$ a $\mathbb{P}$-name and $\mu$ an infinite cardinal such that $\Vdash 0<|\dot{Q}|\leq\mu$. $(a)$ Exist names $\langle \dot{q}_\alpha\rangle_{\alpha<\mu}$ ...
1
vote
0answers
37 views

A question regarding Silver's Theorem

I am reading Jech's book Introduction to Set Theory. As an introduction to Silver's theorem, it is stated that: If $\aleph_\alpha$ is a strong limit cardinal and if $k < \aleph_\alpha$ and ...
5
votes
3answers
317 views

pseudo numbers and surreal numbers

A surreal number $\{x_L\|x_R\} \in No$ is a number when for all $\xi\in x_L$ and all $\eta \in x_R$ we have $\eta > \xi$. All the things $\{x_L\|x_R\}$ which are not of that form are called ...
2
votes
1answer
79 views

What does an Ulam matrix look like?

I'm trying to visualize an Ulam matrix but I"m having trouble. So it has Aleph one rows and aleph null columns? What do elements of a Ulam matrix look like?
1
vote
1answer
50 views

Encoding countably many reals

Is there a way to encode a countable set of reals by a set theoretic formula with parameters a countable sequence of ordinals? By this I mean is there a formula $\varphi$ in the language of set ...
1
vote
1answer
46 views

Initial segments of trees

I am fairly sure this question will look quite trivial, but I have some trouble getting the reason behind the use of the symbole $\subset$ when we deal with sequences and initial segments of ...
0
votes
2answers
44 views

Can I use the independence of the (generalized) continuum hypothesis to prove that my set is countable?

I have a set $S$ which is countable. I have defined a subset $U \subset S$ that: cannot have a cardinality higher than $|S|$ (because it's a subset); is infinite (because I proved that). Now I'd ...
5
votes
1answer
39 views

How to prove $cf(\aleph_{\omega_1})=\omega_1$?

I am reading Jech's & Hrbacek's book "Introduction to set theory" and this is question 2.2 from chaper 9: Prove $cf(\aleph_{\omega_1})=\omega_1$ Isn't it clear from the fact that ...
2
votes
1answer
54 views

How to show that the Cohen forcing adding arbitrary many reals adds no dominating real

Let $\lambda$ be any infinite cardinal and let $Fn(\lambda, 2)$ be the set of finite partial functions from $\lambda$ into $2$. This is a forcing notion adding $\lambda$ many Cohen reals. It is a ...
1
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1answer
39 views

Converse to Steinhaus?

Does there exist a Lebesgue null set $A$ such that $\{ x-y : x,y \in A \}$ contains an interval? Under CH, yes. Let $\langle C_\alpha \rangle_{\alpha < \omega_1}$ list the closed nowhere dense ...
3
votes
1answer
86 views

How do I show the existence of a weakly inaccessible cardinal is not provable in ZFC?

So I know that if we assume the existence of inaccessible cardinals than we can show ZFC is consistent. How do I show that the existence of such cardinals is not provable in ZFC?
2
votes
1answer
43 views

Relaton between infinite order type $\Theta$ and $\omega$

I want to understand in case if $\Theta$ is an arbitrary infinite order type, why we have either $\omega \preceq \Theta$ or $\omega \preceq\Theta^*$. Where $\Theta^*$ is reverse of order type ...
4
votes
1answer
134 views

ZFC + not-CH, can a set with cardinality between $\aleph_0$ and $2^{\aleph_0}$ be defined?

Continuum hypothesis states, there is no set with cardinality between the integers and the reals. There is a milestone result, that CH is independent from ZFC. That means, both of ZFC + CH, and ZFC + ...
2
votes
1answer
54 views

A question about of $\mathbb{P}$-name

Let $ \mathbb{P}$ be poset and $\dot{Q}$ a $\mathbb{P}$-name. Question: we can find some cardinal $\mu$ in the ground model such that $ \Vdash |\dot{Q}| \leq |\mu|$.?
1
vote
1answer
45 views

Application of the Baire category theory

Definition: A set $M\subset X$ is called "of first category" if it is countable union of nowhere dense sets. Otherwise its called "of second category". I want to see whether the following sets are ...
2
votes
1answer
85 views

What are two disjoint stationary subsets of ω1?

I know if cf(μ)≥ ω2 then two disjoint stationary subsets of μ are {α less than μ : cf(α)=ω} and {α less than μ : cf(α)=ω1}. But I'm not sure what two disjoint stationary sets of ω1 are. Any help is ...
0
votes
1answer
27 views

Definition of Ordinal (w/ Axiom of Regularity) (problem 37, page 208, Enderton's Elements of Set Theory)

Given the definition of an ordinal to be well-ordered by $\in$ and transitive, I am interested with proving the following: I know and understand the following which easily proves it, but uses the ...
6
votes
4answers
265 views

Show that there are non-well-founded models of Zermelo Fraenkel set theory

I have been working on this problem for several hours, and my understanding just isn't there. Here's what I've gathered: Using downward Lowenheim-Skolem theorem, we know that any consistent set of ...
5
votes
1answer
76 views

Why we use $\mathrm{Fn}(\kappa\times\lambda,2,\lambda)$ to force $2^\lambda\ge\kappa$ instead of $\mathrm{Fn}(\kappa\times \lambda,2)$

I am reading Kunen's Set Theory and I learn that, $\operatorname{Fn}(\kappa\times\omega,2)$ forces $2^{\aleph_0}\ge|\kappa|$, where $$\operatorname{Fn}(I,J,\lambda)=\{p:p\text{ ...
0
votes
0answers
28 views

Simple question about $\mathbb{P}$-name [duplicate]

If $p \in \mathbb{P}$ and $\tau$ be a $\mathbb{P}$-name such that $p \Vdash \tau \in B$. Then there exist a nice name $\dot{b}$ for an object in $B$ such that $\Vdash \tau =\dot{b}$. Also, $\sigma$ ...
3
votes
1answer
165 views

Good book on foundations - axiomatic set theory

I'm currently planning on reading Suppes' Axiomatic Set Theory, because I'm interested in finding out what the currently accepted foundations of mathematic are. Is this a good book for doing so? What ...
1
vote
1answer
69 views

Exercise 14.12 of Jech's book

I'm dealing with the chapter about forcing in Jech's book and I need some help with the second part of exercise 14.12. How can I prove the following equality? $$\Vert (\forall y\in ...
2
votes
0answers
94 views

What is the name of proofs with (without) Axiom of Choice

In many contexts we distinguish between proofs using AC and proofs which do not use AC. (To phrase this somewhat differently: If there is a proof without AC, this proof is usually preferred.) I would ...
1
vote
2answers
93 views

The Recursion Theorem (Set Theory)

In the book 'Introduction to set theory' by Hrbacek and Jech, there is this theorem stated in the book: Then in the proof, there is this part: I don't understand the induction part. We are trying ...
1
vote
1answer
21 views

Closed subsets of Souslin orders

A total order without end-points is Souslin if it is complete, non-separable and ccc. Such orders may or may not exist but when they do, there can be a vast zoo of them. Can we get consistent ...
1
vote
1answer
61 views

Uncountable linearly independet family in $K^\mathbb{N}$

Let $K$ be a field. Consider the vector space $K^\Bbb{N}$ of $K$-sequences. Is there an uncountable linearly independent set of vectors in this vector space? If Yes, can you name it explicitely? Does ...
3
votes
0answers
63 views

Axiom of Pairing in T. Jech: “Set Theory”

I just opened up the book "Set Theory" by T. Jech and realised two things: The natural numbers are defined in Chapter 2 The Axiom of Pairing is used in Chapter 1 to define ordered n-tuples ...
1
vote
0answers
29 views

Dimension of space of linear maps between infinite dimensional vector spaces

Let $F$ be a field, and suppose $V$ and $W$ are vector spaces over $F$. What is the dimension (meaning cardinality of any basis) of the space of linear maps from $V$ to $W$? I hope there is an answer ...
0
votes
1answer
40 views

Generalization of name and nice names

Let $\mathbb{P}$ be poset. Let $D, B$ be sets. We say that a $\mathbb{P}$-name $\dot{z}$ is a nice name for a function from $D$ into $B$ if there is $\left<{A_{d},h_{d}}\right>_{d \in D}$ such ...
0
votes
0answers
67 views

Questions of $\mathbb{P}$-name for a set and functions

Let $\mathbb{P}$ be poset. Let $B$ be a set. We say that a $\mathbb{P}$-name $\dot{b}$ is a nice name for member of $B$ if there is a maximal antichain $A\subseteq\mathbb{P}$ and a function ...
82
votes
10answers
3k views

Different kinds of infinities?

Can someone explain to me how there can be different kinds of infinities? I was reading "The man who loved only numbers" by Paul Hoffman and came across the concept of countable and uncountable ...
4
votes
1answer
78 views

The topological product of path-connected spaces is path-connected $\Rightarrow\sf AC$?

There is a very natural proof that the product of path-connected spaces is path-connected: Let $X=\prod_{i\in I}X_i$ be a product of path-connected spaces $X_i$. Given $(x_i)_{i\in I},(y_i)_{i\in ...
-2
votes
1answer
35 views

Clubs in regular cardinal, give function [closed]

Given regular cardinal $\kappa > \omega$ and a closed unbounded set $C \subseteq \kappa$. Give a function $f : \kappa \rightarrow \kappa$ s.t. if $\forall \beta < \alpha [f(\beta) < \alpha]$, ...
3
votes
1answer
76 views

Question of $\Diamond$ in Generic Extension

Let $M$ is a transitive model of $ZFC$ and $G$ is filter which a countable transitive model. Assume $( \mathbb{P}$ is c.c.c and $|\mathbb{P}|\leq \omega_{1})^{M}$ and $\Diamond$ holds in $M$. I want ...
1
vote
1answer
68 views

Existence of Certain Names in Iterated Forcing

Suppose $\mathbb{P}$ is a forcing. Let $\dot{\mathbb{Q}}$, $\dot{<_\mathbb{Q}}$, and $\dot{1}_\mathbb{Q}$ be a name such that $1_\mathbb{P} \Vdash_\mathbb{P} ``\langle \dot{\mathbb{Q}}, ...
1
vote
1answer
62 views

Names and nice names

Let $\mathbb{P}$ be poset. Let $B$ be a set. We say that a $\mathbb{P}$-name is a nice name for member of $B$ if there is a maximal antichain $A\subseteq\mathbb{P}$ and a function $h:A\rightarrow{B}$ ...
7
votes
2answers
141 views

Which other fields of mathematics are relevant to modern set theory?

I am currently studying mathematics as an undergrad. Over the last year I discovered mathematical logic and set theory and took some courses in these subjects. Especially set theory is really ...
3
votes
1answer
54 views

Powers of $\mathfrak{c}^+$

Denote by $\mathfrak{c}^+$ the cardinal successor of continuum. Can we prove in $\mathsf{ZFC}$ that $(\mathfrak{c}^+)^{\aleph_0} = \mathfrak{c}^+$? I guess not. Of course this question is ...
4
votes
1answer
57 views

$\forall$ stationary $S \subset \omega_1$ $\forall$ $\alpha < \omega_1$ $\exists$ closed set of ordinals $A$ of length $\alpha$, s.t. $A \subset S$.

Edit: This is a question from Jech: Set Theory I have been trying for a few days my luck with the following question: For every stationary $S \subset \omega_1$ and every $\alpha < \omega_1$ there ...
4
votes
0answers
96 views

Non-Forcing and Independence

Do there exists sentences which are independent of ZFC, cannot be shown to be independent through some method of forcing, and do not increase the consistency strength of ZFC (e.g. so Large Cardinal ...
2
votes
0answers
69 views

Subtree finitely branching at limit nodes

I was reading Chapter 3 - Coherent Sequences - in Handbook of Set Theory and it says that every subtree of $\sigma \Bbb{Q}$ which is finitely branching at limit nodes is easily seen to be a special ...
1
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0answers
25 views

Is the statement that there is a bijection $A\cong A\coprod A$ for every infinite set equivalent to the axiom of choice? [duplicate]

To provide a little bit of context to the above question, it's not so hard to show that the statement is implied by the axiom of choice (first replace $A$ with an ordinal, then build a bijection by ...
1
vote
1answer
39 views

Splitting sets of cardinality greater than omega

Let $\omega \leq \kappa \leq 2^{\omega}$ and $cf(\kappa) > \omega$. Show that if $X \subseteq \mathbf{R}$ s.t. $|X| = \kappa$ then $\exists q \in \mathbf{Q}$ s.t. $|X \cap (-\infty , q)| = |X \cap ...
7
votes
1answer
218 views

What are disasters with Axiom of Determinacy?

It is well-known that Axiom of Choice has several consequences which might be viewed as counter-intuitive or undesirable. For example, existence of non-measurable sets or Banach-Tarski Paradox. H. ...
1
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0answers
28 views

Ordinals that are not cardinals [duplicate]

I am reading Jech's set theory and he defines a cardinal number as an ordinal $\alpha$ (a cardinal) if $|\alpha| \neq |\beta|$ for all $\beta < \alpha$, and he says that all infinite cardinals are ...
5
votes
1answer
97 views

A club-guessing exercise

I came across this club-guessing exercise on Cardinal Arithmetic by Abraham and Magidor in the Handbook of Set Theory. Let $\kappa, \lambda$ be regular cardinals $\kappa^{++}<\lambda$ and let ...
1
vote
1answer
60 views

I am confused about poset $\sigma$-centered.

Assume that $2 \leq |J| \leq \aleph_{0} $. Let $\mathbb{P}=\operatorname{Fn}(I,J)$ $\mathbb{P}=\operatorname{Fn}(I,J)$ is $\sigma$-centered iff $|I| \leq \mathcal{c}$ where ...
4
votes
1answer
189 views

Best Less-Famous Texts for Forcing

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