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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2answers
144 views

equivalence between formal and informal proof

I'm reading Cohen's book on the independence of the continuum hypothesis, and I see that all the proofs that he gives when he's defining the basic notions of set theory (ordinals, cardinals, ...
3
votes
1answer
82 views

Help with intuition on Cardinal Arithmetic Problems

It happens a lot to me that when I find an intuitive model (picture) of a mathematical entity, the proofs left as exercises in books are very easy to solve. For example when dealing with filters and ...
3
votes
1answer
86 views

Finitely-additive measure over $\Bbb{N}$

On the set of natural number, we can consider the finitely-additive measure defined as: $$\mu(A) = \lim_{n\to\infty}\frac{\#(A\cap [1,n])}{n}.$$ However, there is a definable (by PA, or some ...
3
votes
1answer
86 views

Is there any set theory without something like the Axiom Schema of Separation?

I appreciate any insight to this question, including suggestions for other terms to learn about first. I am self-taught with regards to set theory and not a mathematician, so my question may not be ...
6
votes
1answer
324 views

Is there any known uncountable set with an explicit well-order?

There is no known well-order for the reals. Is there a known well-order for any uncountable set? If not, is it known whether or not an axiom stating that only countable sets can be well-ordered is ...
3
votes
1answer
65 views

Why can fix $W_0 \subseteq \kappa$ and select $W$ such that $W_0\subseteq W \subseteq \kappa$?

Lemma V$.2.19$ (book Kunen) In $M$, let $\mathbb{P}=Fn(\kappa,\omega)$, where $\kappa$ is any cardinal. Let $K$ be $\mathbb{P}$-generic over $M$. Then $M[K]\models \mathfrak{d} \geq \kappa$ Proof: ...
0
votes
1answer
98 views

Infinity of every ZF model

Let's define $S(x) = x \cup \{x\}$. Prove that axioms of ZF (semantically) imply that all sets $\emptyset, S(\emptyset), S(S(\emptyset)), \dots$ are pairwise distinct. Prove (without axiom of ...
3
votes
2answers
138 views

Equivalence classes of real sequences, an interesting concept of closeness

Consider $\mathbb R^{\mathbb N}$, the set of infinite sequences of reals. Two such sequences are equivalent if and only if they eventualy coincide. That is, if $x_1,x_2,\dots$ is one of the sequences, ...
4
votes
1answer
89 views

A conceptual link between trees and Polish spaces

Could somebody explain to me why trees are so relevant for the study of Polish spaces and descriptive set theory? I still do not get the proper connection (...and when I think I got it – see the ...
3
votes
1answer
59 views

Replacing an ordinal with its cardinality in a partition relation

In The Higher Infinite, Kanamori claims that if $\alpha$ is a cardinal, and $\beta \to (\alpha)^\gamma_\delta$ for some $\beta$, then the least such $\beta$ is a cardinal. I can't seem to think of a ...
0
votes
1answer
95 views

versions of diamond,$\Diamond^*_S$

For an infinite cardinal $\lambda$ and stationary subset $S\subseteq\lambda^+$, why does $\Diamond^*_S\Rightarrow \Diamond_S$? We use the notation from page $127$ of Assaf Rinot, Jensen’s diamond ...
6
votes
1answer
170 views

Jech's Set Theory logic prerequisites

I have read some of the books suggested in What are the prerequisites to Jech's Set theory text?, so I have some beginning experience with transfinite recursion, ordinals, cardinals, order types, ...
1
vote
3answers
104 views

How do sets of language used to formulate ZFC axioms escape Russell's paradox?

We formulate sets using ZFC. Though, to write its axioms we already use the notion of sets. For instance, in formulating the Axiom of Extensionality, we write the following concatenation of symbols: ...
2
votes
1answer
38 views

Countable union of sets of cardinality $c$ has cardinality $c$

The book Theory of Functions of a Real Variable by I. P. Natanson, proves that a denumerable or finite union of pairwise disjoint sets of cardinality $c$ has cardinality $c$. The proofs given in the ...
2
votes
1answer
42 views

Question related to ordinal number without using Axiom of Choice.

Can we proof this result without using Axiom of Choice :- $$A\cap \alpha=\emptyset \,\,\,\, \mbox{and}\, \, \, A\times \alpha \sim A\cup \alpha$$ then there is an $A^{'} \subset A$ such that $\alpha ...
1
vote
1answer
68 views

The Definition of Definition by Recursion

The following is presented as the Transfinite Recursion on well-founded relations in Kenneth Kunen's book. Assume that $R$ is set like and well founded on $A$ and ...
1
vote
0answers
21 views

Resources for studying different set theories [duplicate]

Ok so lately I have been fascinated about general structure of mathematics and I have read some books on set theory. I have gone through the Endertons introductory book on set theory which operates ...
1
vote
1answer
157 views

Construction of a model of Peano Arithmetic

I'm studying the axioms of Zermelo-Frankel Set Theory at the moment. I already know the following six axioms: The axiom of empty set The axiom of extensionality The axiom of pairing The axiom of ...
2
votes
2answers
96 views

In ZFC, which axioms of set are not required to class?

In Set Theory , Thomas Jech says Classes Although we work in ZFC which, unlike alternative axiomatic set theories, has only one type of object, namely sets, we introduce the informal ...
4
votes
1answer
63 views

Does $\lambda^2 \leq \kappa^2 \Rightarrow \lambda \leq \kappa$ imply the axiom of choice?

I'm aware that the statement "for all cardinals $\kappa$, $\kappa^2 = \kappa$" is equivalent to the axiom of choice (I believe this was proved by Tarski). More generally, does anyone know if the ...
2
votes
1answer
97 views

Existence of an uncountable set of sequence

I'm back with a question really close to this one : Does it always exist an infinite subset of sequences that satisfy this property? Now I'm asking myself : does it exist an uncountable set that is ...
0
votes
1answer
89 views

Enderton's Elements of Set Theory Scott's Trick Exercise (page 207 problem 31)

31. Define kard $A$ to be the collection of all sets $B$ such that (i) $A$ is equinumerous to $B$, and (ii) nothing of rank less than rank $B$ is equinumerous to $B$. (a) Show that kard $A$ is ...
2
votes
1answer
36 views

Generalization of binomial coefficients

Let $X$ be a set. Write $S(X)$ for the set of all bijections $X\longrightarrow X$. One can easily see that $S(X\sqcup\{\operatorname{pt}\})\cong (X\sqcup\{\operatorname{pt}\})\times S(X)$, where ...
5
votes
1answer
77 views

There is no infinite sequence $x_1 \ni x_2 \ni x_3 \ni …$

We take the "usual" axioms of Zermelo-Fraenkel set theory (axiom of extensionality, axiom of the unordered pair, axiom of the sum set, axiom of the power set, axiom of the empty set, axiom of choice ...
1
vote
0answers
65 views

A universally valid second-order sentence only if CH holds.

I'm looking for a second-order sentence that is universally valid only if CH holds. I'm thinking a surjection between all not enumerable sets onto $\mathbb{R}$ but I don't know how to write it. Thank ...
0
votes
1answer
30 views

Given a collection of functions $f_i$ with the same domain, how to replace with values (w/o axiom replacement)

I know from a collection of ordered pairs we can project onto the first coordinate. I'm interested if there's a way (without using the axiom of replacement) to "project" a collection of functions onto ...
2
votes
1answer
65 views

Quasi-disjoint subsets of an infinite cardinal

Let $\kappa$ be an infinite cardinal and let $S$ be a collection of subsets of $\kappa$ such that for $s\neq t\in S$ we have $|s\cap t| \leq 1$. Is it possible that $|S|>\kappa$?
8
votes
3answers
122 views

What is a $P$-name in forcing theory

I am having troubles understanding what is a $P$-name is forcing theory and what is the purpose of this term in the forcing tecnique. Is there any simple way to explain this term. If there was I ...
2
votes
2answers
75 views

Halmos' Naive Set Theory Cardinal Arithmetic Exercise

On page 95 of Halmos' Naive Set Theory, we get the exercise If $\{a_i\}$ and $\{b_i\}$ are families of cardinal numbers such that $a_i< b_i$, then $$\sum_i a_i<\prod_ib_i$$ I know that we ...
9
votes
1answer
132 views

Can anyone explain what is the intuition behind the following definition of $p \Vdash^* \phi $?

Can anyone explain what is the intuition behind the following definition? Definition 4.25 Let $\Bbb P$ be a poset. Let $\phi(x_1,\ldots,x_n)$ be a formula, $p\in\Bbb P$, and let ...
1
vote
1answer
42 views

If $S$ is an Infinite Set, and $f(S)\subseteq{G}$ is a Set that Generates a Group $G$, then the Cardinality of $G$ is $\le$ Cardinality of $S$?

As the header says, if we map an infinite set $S$ into $G$ by $f$, such that $f(S)$ generates $G$, then is $|G|\le|S|$? I know that this should be true in the case where $S$ does not include into $G$. ...
7
votes
1answer
107 views

how to collapse $\omega_2$ to a smaller cardinal

Let $M$ be a model of ZFC and take the forcing notion $P(\omega,\omega_2)$ where: $P(\omega,\omega_2)=\{p|p \space is \space a \space function \space and \space \exists n \space s.t. (dom(p)=n) \space ...
3
votes
1answer
52 views

Function $f\in M[G]$, $f:\kappa\to M$ is in the ground model implies $\kappa^+$-Baire

Let $M$ be countable transitive model of ZFC, $P\in M$ be poset and $\kappa$ be a cardinal in $M$. In addition, for every $P$-generic filter $G$ over $M$, if a function from $\kappa$ to $M$ is in ...
2
votes
0answers
48 views

$Gal(\bar{\mathbb Q}/\mathbb Q)$ without choice, and constructive Galois theory

By this question: Algebraic closure for $\mathbb{Q}$ or $\mathbb{F}_p$ without Choice? We have that over ZF, algebraic closures of $\mathbb Q$ aren't unique. Are their Galois groups as extensions over ...
1
vote
1answer
76 views

Explanation of $\mathrm{ZFC} + \lnot \mathrm{Con(ZFC)}$

I read in JDH's answer to this mathoverflow question that $\mathrm{ZFC}$ is equiconsistent with $\mathrm{ZFC} + \lnot \mathrm{Con(ZFC)}$. But this second statement is a bit weird. What does ...
2
votes
0answers
17 views

Cardinalities of different sets [duplicate]

I want to show that given infinite cardinals $\kappa\leq{\lambda}$, we have that $\vert\{X : X\subseteq{\kappa} \text{ and } \vert{X}\vert ={\lambda}\}\vert=\kappa^{\lambda}$. I wanted to use ...
4
votes
1answer
71 views

Simple question about of $\Vdash \varphi$ [closed]

Let $\mathbb{P}$ a poset. The following are equivalent. $(1)$ $p\Vdash \varphi$. $(2)$ $\forall r\leq p(r\Vdash \varphi)$. $(3)$ $\{r: r\Vdash \varphi\}$ is dense below $p$. I am confused when ...
1
vote
2answers
66 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.
0
votes
2answers
72 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 ...
2
votes
0answers
71 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
40 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
1answer
174 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, ...
0
votes
1answer
66 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 ...
2
votes
1answer
86 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?
5
votes
1answer
90 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
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 ...
5
votes
1answer
43 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
90 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 $B$ ...
1
vote
1answer
42 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
91 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?