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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How does ZFC describe addition?

Surprisingly, the Wikipedia article on addition doesn't contain the answer. I looked elsewhere online for it, but didn't find it. Intuitively, the cardinal of the union of two sets seemed ...
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1answer
36 views

Examples of subsets which cannot be specified using “a formula in the language of set theory”

In ZFC, the axiom schemas of specification and replacement refer to "formulae in the language of set theory." Are there any canonical examples of sets which cannot be specified in this way? Put ...
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112 views

Cantor-Schröder-Bernstein without elements

The Cantor-Schröder-Bernstein Theorem for the category of sets has several "analogues" for other categories as well, for example measurable spaces and operator algebras (but not for arbitrary ...
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3answers
193 views

How does ZFC define functions?

I found the following definition on Wikipedia. Is it the most common definition? How is the definition usually notated? A function f from X to Y is a subset of the Cartesian product X × Y ...
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2answers
161 views

About generically Knaster property

We say that a poset $\mathbb{P}$ is absolutely Knaster if, for every $ccc$ poset $\mathbb{Q}$, $1 \Vdash_{\mathbb{Q}} \text{``$\mathbb{P}$ is Kanster''}$. In general, we say that a poset $\mathbb{P}$ ...
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1answer
63 views

Countable intersection on an ultrafilter

Does there exist a nontrivial ultrafilter $a$ such that there are no sequence of sets $K_0,K_1,K_2\dots\in a$ such that $$K_0\cap K_1\cap K_2\dots = \emptyset?$$
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189 views

Problem with forcing with perfect trees

I've been reading the section on Forcing with Perfect Trees in Jech's Set Theory. The notion of forcing consists of all perfect trees $T\subseteq Seq(\{0,1\})$, where $p$ is stronger than $q$ if ...
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1answer
60 views

A well ordering on $\mathbb{R}$ and bigger sets

Consider the set of sequences $S = \{f:\mathbb{N}\to\mathbb{N}\}$, define an order on $S$ by the following: Based on the well-ordering of $\mathbb{N}$ and induction, either $f_1 = f_2$ or there is a ...
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2answers
158 views

What is the relation between axiomatic set theory and logical quantifiers?

On the one hand, the logical predicates $\forall$ and $\exists$ are defined using the concept of a Domain of Discourse, which itself is defined as a set (at least according to wikipedia). On the ...
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1answer
144 views

How large are measurable cardinals of higher orders?

For each ordinal $\alpha$ define the notions of $\alpha$ - measurable cardinals and $\alpha$ - normal measures as follows: A measure $\mu$ on a measurable cardinal $\kappa$ is a $0$-normal measure ...
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1answer
65 views

Order-preserving injections of ordinals into $[0,1]$

The usual "matchstick" representation of an ordinal number can be thought of as an order-preserving injection of that ordinal into the interval [0,1]. For example, here's a representation of ...
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1answer
105 views

Question regarding disjoint unions, sequential compactness, and Dedekind-finiteness

I have proved the following two results: $[\mathsf{ZF}]$ The disjoint union of a Dedekind-finite family of sequentially compact topological spaces is again sequentially compact. ...
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1answer
290 views

Why do we use groups and not GROUPS?

When a group is defined, we ask that the elements belong to a set. If we allow them to belong to a proper Class we get a GROUP. What is the advantage of working with groups? What properties do we ...
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1answer
796 views

Bijection between $2^{\mathbb R}$ and $\mathbb{ R ^ R}$

I'm well aware of the standard proof based on cardinality arithmetic to show that these two sets have the same cardinality and the question of defining a bijection between the two sets came up. I ...
3
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1answer
80 views

Non-well-founded models viewing well-founded models as non-well-founded.

I'm currently thinking about how different models of set theory view each other. In particular I'm looking at how well-foundedness behaves between different models. So we have the Axiom of ...
4
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1answer
73 views

How to show that $\mathfrak s \leq \mathfrak d$

I am trying to understand why $\mathfrak s \leq \mathfrak d$. Can anyone state a proof of it? I have a proof , which I don't understand yet. My question regarding that proof is here below: At the ...
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1answer
48 views

On Counted Languages

In my recent question on Godel Completeness I mentioned that there was a related question I wanted to ask, but would keep separate. I have been recently studying "non-well ordered sets" and Chapter 7 ...
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3answers
210 views

How to prove the Milner-Rado Paradox?

For every ordinal $\alpha<\kappa^+$ there are sets $X_n\subset\alpha$ $(n\in\Bbb{N})$ such that $\alpha=\bigcup_n X_n$ and for each $n$ the order-type of $X_n$ is $\le\kappa^n$. [By ...
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1answer
53 views

Initial Segment Order Isomorphic to the Ordinal Numbers

Prove that every well-ordered proper class has an initial segment order isomorphic to the ordinal numbers, ON. I have a plan to prove this but it uses a recursive definition and induction which I do ...
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2answers
138 views

$\kappa$-c.c. vs. $\kappa$-Knaster

For an infinite cardinal $\kappa$ and a partial order $\mathbb{P}$, we say: (a) $\mathbb{P}$ has the $\kappa$ chain condition ($\kappa$-c.c.) iff there is no subset of $\mathbb{P}$ of size $\kappa$ ...
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1answer
48 views

All models of $\mathsf{ZFC}$ between $V$ and $V[G]$ are generic extensions of $V$

I'm reading the proof of lemma 15.43 of Jech's Set Theory: Let $G$ be generic on a complete Boolean algebra $B$. If $M$ is a model of $\mathsf{ZFC}$ such that $V\subset M\subset V[G]$, then there ...
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1answer
30 views

Is there a difference between the order type of Q·ω and Q·Q?

From what I understand, the expression is "a countable amount of the order type of Q", which intuitively should be equal to the second expression. Is this true? How do I explain this formally? Thanks ...
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1answer
50 views

When do surjections split in ZF? Two surjections imply bijection?

We have that the Axiom of Choice is equivalent to the principle that every surjection has a right inverse. However, without the Axiom of Choice we can determine for some $X$ that $X\succeq Y\implies ...
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2answers
81 views

How do the ZFC axioms produce the ideas of order?

The notion of order (and cardinality, for that matter) seems so basic to me that I can't imagine how it could be derived from anything. In an answer to a previous question I learned that all the ...
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2answers
141 views

Which sets are well-orderable without Axiom of Choice?

I know that, assuming Axiom of Choice, every set is well-orderable. I know also that the assertion that $\mathbb{R}$ is NOT well-orderable is consistent with ZF. How can I find other sets such that, ...
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1answer
49 views

Consistency result vs. True in every model of Axiom X

Suppose a forcing extension of ZFC has been found which satisfies statement $A$. For example, say the extension is formed by Cohen or Laver forcing, so that the model satisfies $\neg$CH. At this ...
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1answer
42 views

How to prove by induction that the set of all natural numbers is an ordinal

I have seen alternative methods of this proof, with one being: let $n$ be the set of all natural numbers. Then (1) $\omega$ is an ordinal, (2) If $\alpha$ is an ordinal and $\beta \in \alpha$, then ...
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0answers
87 views

Zero sharp and Kunen inconsistence.

There is some (deep/shallow/interesting/trivial) relation between: (1) $\exists 0^\sharp$ iff $\exists j:L\longrightarrow L$ nontrivial elementary embedding. (2) The Kunen inconsistency $\not\exists ...
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1answer
115 views

A Question Regarding an Accessibility Relation on the Class of All Models of ZFC

Consider the class of all models of ZFC and consider the following accessibility relation on the class--that each model in the class can know what are the cardinals and ordinals of every other member ...
6
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1answer
108 views

Do second-order categoricity proofs require a background concept of set?

In his article "The Set-Theoretic Multiverse", Joel David Hamkins (as part of his reply to Donald Martin's argument that the set-theoretic universe is unique, found in "Multiple Universes of Sets and ...
4
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1answer
97 views

Do Ramsey idempotent ultrafilters exist?

I was studying idempotent ultrafilters when I saw that no principal ultrafilter could ever be idempotent, because $\left\langle n \right\rangle \oplus \left\langle n \right\rangle = \left\langle 2n ...
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0answers
139 views

Large Cardinal Inequalities

Solovay showed that the existence of $0^\dagger$ follows from the existence of two measurable cardinals. We know existence of a measurable cardinals is weaker than existence of $0^\dagger$ so we ...
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1answer
89 views

Consistency strength of Turing measurability

This is probably well-known to recursion theorists, but as google didn't help me, I'll ask it here. Convention: All sets of reals in the following discussion are assumed to be closed under Turing ...
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2answers
47 views

Reference Request for GB-set theory

Could anyone give me a reference for a book which has an introduction to set theory from the GB axioms as opposed to ZFC, everything I read seems to just look at things from ZFC (Jech...) Thanks for ...
2
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0answers
71 views

$\mathfrak b \leq \mathfrak r$. Is this proof correct?

I am trying to prove that $\mathfrak b \leq \mathfrak r$ where $\mathfrak r$ is the minimal cardinality of a reaping family of sets. A family $\mathcal R \subseteq [\omega]^\omega$ of sets $\mathcal ...
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1answer
66 views

Difference between $V^P$ and $V[G]$

This may be a basic question. I am studying forcing at Kunen's book. However, in several other papers that I am reading, they use that something is true in $V^P$ instead of $V[G]$. I know that if ...
2
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1answer
64 views

Transfinite recursion and Replacement

I have a question about the use of Replacement when proving the transfinite recursion theorem. It seems that the crucial use of Replacement is made in the step involving the set of all partial ...
2
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1answer
54 views

On proving that $\mathcal{P}(\omega)/Finite$ is atomless

As I mentioned elsewhere, I'm working on Schimmerling's A Course on Set Theory. One of the nice features of the book (for me, anyway) is the addition of some interesting exercises on Boolean algebras. ...
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2answers
78 views

How to show that a free ultrafilter cannot have an infinite pseudointersection?

The following text is a quote from p.180 of Halbeisen's book Combinatorial Set Theory. This book is also available on website of a course taught by the author. (As mentioned in Asaf's comment, it is ...
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3answers
129 views

Identity of indiscernibles

It follows from the axioms of identity alone that $x = y \Rightarrow \big((\forall z) x \in z \equiv y \in z\big)$ and $x = y \Rightarrow \big((\forall z) z \in x \equiv z \in y\big)$. One of the ...
4
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1answer
77 views

How to prove that $\omega_1 \leq \mathfrak p$

I am trying to understand the stated above Theorem. $\mathfrak p$ is the smallest cardinality of any family $\mathcal F \subseteq [\omega]^\omega$, which has the strong finite intersection property, ...
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1answer
59 views

Prove that $\omega + \omega_1 = \omega \cdot \omega_1 = \omega^{\omega_1} = \omega_1$

I am assuming already that a) the union of countably many countable sets is countable and b) $\omega_1$ is the least uncountable ordinal, so $x < \omega_1$ if and only if $x$ is a countable ...
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0answers
89 views

What axioms are necessary to prove Mostowski Collapse?

I've been reading Schimmerling's A Course on Set Theory, and have been enjoying it a lot so far. However, some times he's less clear than I think he could be. For example, during a proof of Mostowski ...
1
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1answer
70 views

Is any subset of the open ordinal space $[0,\Omega)$ $G_\delta$?

Consider the open ordinal space $[0,\Omega)$, where $\Omega$ is the first uncountable ordinal. Can I say that every subset of $[0,\Omega)$ is $G_\delta$? If yes, does this imply that $[0,\Omega)$ is ...
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3answers
53 views

What is the cardinality of all limit ordinals $\alpha$ s.t. $\alpha < 2^\mathfrak c$

Let $\Omega$ be the first ordinal with cardinality $2^\mathfrak c$. Take now the set of all ordinals $\alpha < \Omega$ which are limit ordinals. Is the cardinality of this set countable or is it ...
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3answers
140 views

Why are irrational numbers uncountable and rationals contable?

Question 1: Why are irrational numbers uncountable and rationals contable? I really struggle to understand this. I initially thought it had something to with the fact that between any two numbers ...
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0answers
77 views

Follow up question for an exam question

Some time ago I got an answer to this old exam question. Now I found the following question, which strikes me as very similar: Let $\kappa$ measurable cardinal, and $\preceq$ a partial ordering on ...
2
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2answers
116 views

Natural numbers in Set Theory

We seem to accept the fact $(\omega,+,\times,<,0,1)^{V}$, where $V:=x=x$ is the set theoretic universe, properly reflects what is intuitively understood to be the set of natural numbers, i.e. we ...
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5answers
249 views

Statement that is provable in $ZFC+CH$ yet unprovable in $ZFC+\lnot CH$

My understanding of logic is really basic, and I ask this question out of curiosity. Is there an explicit example of a statement whose proof uses the continuum hypothesis and is unprovable in $ZFC + ...
5
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2answers
108 views

When can we switch the order of forcing iteration

I am interested in when two forcing iterations are isomorphic (or at least add the same reals) when the order of the forcings is switched. I know that each forcing does not properly exist in the ...