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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28 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
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 ...
2
votes
3answers
185 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
votes
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
votes
1answer
77 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 ...
1
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0answers
40 views

Class of forcings with an approximation property to subsets of \omega_1

Is there a class of forcing notions which has the following property? For every $A \subseteq \omega_1 \cap V[G]$, there exists $A' \subseteq \omega_1 \cap V$ with $|A \triangle A'| \leq \omega$? That ...
4
votes
1answer
43 views

If two transitive models of ZFC have the same sets of ordinals, they are equal

Let M and N be transitive models of ZFC such that for every set x of ordinals, $x \in M \iff x \in N$. I want to show M=N. It will suffice to show $M \subset N$ by symmetry. I was thinking of assuming ...
1
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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 ...
1
vote
1answer
80 views

Is there a “strong” version of the axiom of choice?

First: My apologies for the probably imprecise way I am phrasing this question, to the point where I expect I will be told it is actually meaningless as presented here. The axiom of choice can be ...
1
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1answer
46 views

Question on Komjáth's “three clouds may cover the plane”

I am reading a wonderful paper by Komjáth, "Three clouds may cover the plane," and am having difficulty proving that certain sets are countable. Assume CH (the continuum hypothesis) holds. Let ...
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 ...
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0answers
127 views

Choice-less Set Theory for Dummies

In almost every graduate set theory text there are some parts about equivalences of $AC$, its consequences, some axioms like $AD$ which imply $\neg AC$, some well-known axiomatic systems which $AC$ ...
0
votes
1answer
41 views

$\mathcal P(\omega)\in L_\alpha$

Assuming the V=L axiom, what's the smallest $\alpha$ such that $\mathcal P(\omega)∈L_\alpha$? (Notation from here.) (Remember that V=L implies the continuum hypothesis.) Also, if anyone can explain ...
2
votes
1answer
61 views

Is $L_{\omega+1}$ uncountable?

I'm trying to understand the constructible universe, but I'm having trouble understanding what $\operatorname{Def}(X)$ means (especially since I don't really understand the $\models$ symbol). Is ...
4
votes
1answer
43 views

How do we know that $\omega_1$ exists in ZF? [duplicate]

In ZF, not all "collections of objects" are sets. For example, there is no set of all sets, and there is no set of all ordinals. So, how do we know that there is a set of all countable ordinals? In ...
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0answers
59 views

Von Neumann universe implies Foundation

Suppose we are working in $\mathsf{ZF-Foundation}$. Additionally, let us assume that $V = \bigcup\{V_{\alpha} | \alpha \in \mathsf{Ord}\}$ holds. Now I want to show that this implies Foundation. I am ...
2
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0answers
34 views

Questions of complete embedding and dense embedding

Let $\kappa$ a regular uncountable cardinal and $\mathbb{P}$ and $\mathbb{Q}$ poset. $\mathbb{P}$ has the $\kappa$-Knaster property iff for every $A \subseteq \mathbb{P}$ of size $\kappa$ there is $B ...
2
votes
1answer
59 views

regular open (Boolean) algebra is complete

To prove that regular open (Boolean) algebra is complete, I tried to show following claim, but I couldn't. I saw this statement in Kunen's 'Set Theory' p.64 but in other books what I checked, ...
2
votes
1answer
74 views

Relative Consistency Lemma with finitistic proof

in set theory, one uses the following Lemma in order to provide relative consistency proofs. I have question concerning the proof of this lemma. First, here is the statement: Suppose that $S$ and $T$ ...
2
votes
0answers
70 views

Examples of forcings which add no “definable” Aronjain tree

Maybe a bit board question but: Fixing a regular cardinal $\kappa$ in the ground model, I am looking for examples of set forcing notions which preserve regularity of $\kappa$ and add no new $\kappa$ ...
1
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1answer
55 views

Consistent choice of elements in inverse directed system

Is there some theory related to this... Let $(\{X_\alpha\}_\alpha,f_{\alpha\beta}:X_\beta\rightarrow X_\alpha\}_{\alpha\preceq\beta})$ be an inverse directed system of non-empty sets resp. surjective ...
4
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0answers
94 views

Incompleteness theorem

Correct me if I am wrong at any point! Godel's incompleteness theorem allows us to express "PA is consistent" in the language of Peano arithmetic, and shows that this is not provable in PA. Let's ...
2
votes
1answer
45 views

Is this presentation of the Random real forcing separative and $\sigma$-linked?

Random real forcing is the poset formed by the closed subsets of $[0,1]$ that are non-null (with respect to the Lebesgue measure), ordered by $\subseteq$. Is the Random real forcing $\sigma$-linked? ...
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1answer
34 views

Is strict cardinality ordering preserved by power sets? [duplicate]

Let $A$ and $B$ be sets. Suppose that $\#A<\#B$. That is, $B$ has strictly greater cardinality than $A$. Is it necessarily true that $\#(2^A)<\#(2^B)$? In particular, is this question ...
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1answer
62 views

Proving statements , that require Zorn's lemma , for countable case directly by well-ordering principle of natural numbers

We know that for countable sets , the existence of a choice function is a consequence of the well-ordering principle ; and it is also known that the results like "every vector space has a maximal ...
7
votes
1answer
212 views

Natural Numbers Object and the Axiom of Infinity

It is well known (if you're a topos-theorist, you will call it the definition), that the natural numbers $\mathbb{N}$ together with the zero constant $0$ and the successor function $1\xrightarrow{\ 0\ ...
2
votes
0answers
49 views

Set theory, motivating factors [duplicate]

Set theory seems to pop up in many different fields of mathematics. As someone with a CS degree, I've only encountered very basic set theory; dealing with non-specific sets, and their intersections, ...
4
votes
1answer
53 views

If $\phi$ is a $\Sigma_2$ sentence and $H_{\kappa} \models \phi$, then $V \models \phi$?

In the title question, $\kappa$ is any infinite cardinal. It's easy to see that the result is true if $\phi$ is $\Delta_0$ or $\Sigma_1$. I first tried proving the result for $\Pi_1$, but I don't see ...
0
votes
2answers
72 views

How much foundation do we get for free in $\sf ZFC$?

In $\sf ZFC$ we have the axiom of infinity and thus can define the natural numbers $$\mathbb N \equiv \bigcap\{X:\emptyset\in X\land \forall n(n\in X\implies n\cup\{n\}\in X)\}.$$ From this it's not ...
2
votes
1answer
107 views

Questions of $n$-linked in poset

Let $2\leq n \leq \omega$ and $F$ be a set of size $\leq n$ . Let $\mathbb{P}$ be a poset and $Q \subseteq \mathbb{P}$ an $n$-linked subset. Questions: if $\dot{a} $ is a name for a menber of ...
1
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1answer
117 views

Definable real numbers

Reading this Wikipedia page I found this definition: A real number $a$ is first-order definable in the language of set theory, without parameters, if there is a formula $\phi$ in the language ...
3
votes
1answer
65 views

About $\kappa$-Knaster and $\kappa$-linked

For an infinite cardinal $\kappa$ and a partial order $\mathbb{P}$, we say: (1) $\mathbb{P}$ has the $\kappa$-linked is a union of $\kappa$-many linked subsets. (2) $\mathbb{P}$ has the ...
4
votes
2answers
92 views

$\omega_2$ is not a countable union of countable sets [duplicate]

Without using axiom of choice, can we show that $\omega_2$ is not a countable union of countable sets? I know this cannot be done for $\omega_1$.
3
votes
1answer
82 views

Understanding the condition of $\diamondsuit^+$

I try to understand $\diamondsuit^+$, a variant of the diamond principle. It says that there is a sequence $\langle \mathcal{A}_\alpha\rangle_{\alpha<\omega_1}$ of collection of subsets which ...
1
vote
1answer
30 views

No generic is definable in a perfect notion of forcing of a model of Peano Arithmetic

I would like to prove Lemma 6.1.2.2 from The Structure of Models of Peano Arithmetic by Kossak and Schmerl. Let $\mathcal{M}$ be a countable model of Peano Arithmetic and $\mathbb{P}=\langle P, \le ...
3
votes
0answers
42 views

When a $\mathbb{P}$ - generic filter is $\kappa$ - complete?

By definition a $\mathbb{P}$ - generic filter $G$ over a ground model $M$ is $\aleph_0$ - complete because for any finite set of conditions in $G$ there is a condition $p\in G$ such that $p$ is ...
9
votes
3answers
387 views

Is $\text{Hom}(\prod_p \Bbb Z/p\Bbb Z, \Bbb Q) = 0$ possible without choice?

That divisible abelian groups are precisely the injective groups is equivalent to choice; indeed, there are some models of ZF with no injective groups at all. Now, given that $\Bbb Q$ is injective, ...
2
votes
1answer
70 views

Non analytic ideals on $\omega$

I would like to gather examples of NON analytic ideals on $\omega$. However, I have found nothing in the books and papers I have consulted. Could anyone tell me some reference/s?
0
votes
1answer
130 views

Proof that the finite product of nonempty sets is nonempty without axiom of choice from ZF

How do you prove that for $X_{i} \neq \emptyset$, $i \in \{1,...,n\}$ that $\prod_{i=1}^{n} X_{i} \neq \emptyset$ only using the ZF axioms but not the Axiom of Choice? I would like to see a rigorous ...
1
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1answer
30 views

Proving that if $A\leq_c B$ then $\chi(A)\leq_o \chi(B)$

By Hartog's Theorem we knoe that for every set $A$ There is a definite operation $\chi(A)$ which associates with each set $A$, a well ordered set $\chi(A) = (h(A),{\leq_{\chi(A)}}),$ such that $h(A) ...
0
votes
0answers
41 views

Connection beween closure property in forcing and preservation of $H_{\kappa}$

I would like to know about the relation between closure property of forcing notions and preservation of hierarchy of hereditary small sets, $\langle H_\lambda \mid \lambda\in \mathrm{Card}\rangle$, at ...
4
votes
2answers
176 views

Understanding proof of Hartogs’ Theorem on Set Theory

I'm trying to understand the proof of the Hartogs’ Theorem on page 100 of this book. My especific question is: If we have for each set $A$ $$\mathrm{WO}(A)=\{ (U,\leq_{U}) \, | \, U\subseteq A \, ...
0
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1answer
46 views

Condition to be a prewellordering.

I'm trying to do the problem 7.17 on th book Notes of Set Theory of Moschovakis. First I will define what is a prewellordering: A prewellordering on a set $A$ is any relation $(\lesssim) ⊆ A×A$ ...
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0answers
47 views

Infinite connected graph such that every vertex has finite degree

Let $G=(V,E)$ be an connected graph with $|V| \geq \aleph_0$ such that $\text{deg}(v)$ is finite for all $v\in V$. Does this imply that $|V|=\aleph_0$?
1
vote
2answers
56 views

Injections from all ordinals into a set $X$

We are working in $\mathsf{ZF}$. Let $X$ be a set. Let $A$ be the class of all injections $f: \alpha \to X$ for arbitrary ordinals $\alpha$. I am quite sure that, in fact, $A$ is a set, since if ...
6
votes
3answers
267 views

Does $A\times A\cong B\times B$ imply $A\cong B$?

This is similar to What does it take to divide by $2$? about $(A\sqcup A\cong B\sqcup B)\Rightarrow A\cong B$ which is valid in $\textsf{ZFC}$ by using cardinalities and also in $\textsf{ZF}$ by some ...
1
vote
1answer
72 views

Is this the real definition of an ordinal number?

I have to prove that if $\alpha$ is ordinal and $\beta \in \alpha$ then $\beta$ is oridinal. To do this I wanted to prove that for every $x\in\beta$, $x\in\alpha$. This seems like it should be true, ...
0
votes
0answers
98 views

Question about the foundation of mathematics [duplicate]

I have studied mathematical logic and set theory as an undergraduate. I studied mathematical logic (propositional and predicate logics) before set theory. When I studied mathematical logic, I was a ...
5
votes
2answers
97 views

Determining a charge through subsets

A charge is a finitely additive set function $c: \mathcal{P}(\mathbb{N}) \to [0, 1]$ such that $c(\mathbb{N}) = 1$ and $c(\{n\}) = 0$ for every $n \in N$. Here $\mathcal{P}(\mathbb{N})$ is the set of ...
2
votes
2answers
137 views

Prove that every totally ordered set has a well-ordered cofinal subset

I thought this was easy until I realized that for a set to be well-ordered, EACH of it's non-empty subsets must have a least element. Let $A$ be a non-empty totally ordered set. Let $x_0$ be some ...