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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64 views

How do I show that Lκ = Vκ?

I'm trying to show that Lκ is a model of ZFC if κ is weakly inaccessible. How do I show that Lκ = Vκ? Since we know Vκ is a model of ZFC I believe this is all I have left to show that Lκ is a model.
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2answers
60 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 ...
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0answers
65 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 $\langle \dot{q}_\alpha\rangle_{\alpha<\mu}$ ...
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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
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1answer
148 views

Is category theory constructive?

Roughly: This question concerns the process and the constructive nature of formalizing and proving category theoretic statements within $\textsf{ZFC}$. $\textsf{ZFC}$ can only talk about sets, ...
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1answer
58 views

Suppes' Axiom of Cardinal Numbers

In Suppes' book, $\textit{Axiomatic Set Theory}$ he introduces an axiom concerning cardinal numbers,before introducing them, namely that each set is associated with an object known as a cardinal ...
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1answer
71 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?
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1answer
85 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 ...
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1answer
47 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 ...
4
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1answer
37 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 ...
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1answer
83 views

How to define a nice name?

Let $\mathbb{P}$ be a poset and $B,D$ be sets. Let $p \in \mathbb{P}$ and $\sigma$ be a $\mathbb{P}$-name such that $p \Vdash \sigma \in B$. Then there exist a nice name $\tau$ for an object in $B$ ...
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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 ...
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1answer
80 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?
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1answer
46 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 ...
2
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1answer
42 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 ...
2
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1answer
49 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|$.?
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1answer
40 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 ...
5
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1answer
73 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{ ...
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1answer
26 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 ...
4
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1answer
132 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 + ...
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0answers
26 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, $\sigma$ ...
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1answer
67 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 ...
3
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1answer
143 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 ...
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0answers
90 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 ...
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0answers
61 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 ...
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0answers
66 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 ...
1
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1answer
45 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 ...
4
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1answer
74 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 ...
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1answer
33 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 $\forall \beta < \alpha [f(\beta) < \alpha]$, ...
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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 ...
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1answer
60 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 $h:A\rightarrow{B}$ ...
0
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2answers
41 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 ...
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1answer
57 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
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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 ...
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0answers
28 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 ...
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0answers
92 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 ...
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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 ...
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0answers
68 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 ...
4
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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 ...
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2answers
132 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 ...
1
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1answer
38 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 ...
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4answers
252 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 ...
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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 ...
1
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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 ...
3
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1answer
75 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 ...
3
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0answers
25 views

For which $\alpha > \kappa$ is “X has cardinality $\kappa$” absolute in $L_\alpha$?

This is a much more general question than a question that is relevant while doing exercises 13.7 and 13.8 in Jech, namely "Is countability absolute in $L_\kappa$" for $\kappa$ regular? Countability is ...
5
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1answer
94 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 ...
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3answers
177 views

What is the meaning of “uncountably infinite” within countable models of set theory?

So, I don't know much about countable models of set theory, other than that they exist. To me, their existence is a very weird thing (and a reason to move away from first-order formulations). Here is ...
2
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1answer
44 views

How to find centered subsets in the forcing Hechler

Let Hechler forcing $\mathbb{D}$. Define $\mathbb{D}=\omega^{<\omega }\times{}\omega^{\omega }$( not cardinal arithmetic) ordered as $(t,g)$ iff $s \subseteq t$ and $f \leq g$ (that is, ...
3
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1answer
71 views

If $V$ is the von Neumann constructible universe, is $\langle V,\in \rangle\models V=L$ true?

My question is essentially in the title. I'm reading some notes about the proof of consistency of ZFC assuming the consistency of ZF. The author first assumes theres is a model for ZF-Foundation and ...