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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If $\alpha$ and $\beta$ are ordinals, prove that $\alpha ^ \beta$ is a countable ordinal.

In this question I am supposing that both $\alpha$ and $\beta$ are ordinals. My definition of an ordinal is that: $x$ is an ordinal if $x$ is well-ordered by $\in$ and $x$ is $\in$-transitive. So ...
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1answer
76 views

Axiom of choice and an example of a Well-ordered $\Bbb R$

From the axiom of choice we get that every set can be ordered in a way that will make it a well ordered set, including $\Bbb R$. However, since the ordinal of such a well-ordered set of $\Bbb R$ will ...
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2answers
47 views

Prove $\epsilon_0$ < $\omega_1$ [duplicate]

This is a question in ordinal arithmetic. (If anyone has read 'Classic Set Theory' by Derek Goldrei, this question comes from page 252.) $\epsilon_0$ = sup {$\omega$, $\omega^\omega$, ... } and ...
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1answer
42 views

How to pronounce the partition relation

The partition relation $$ \kappa \to (\alpha)^m_\lambda $$ says that for any $f:[\kappa]^m \to \lambda$, there is a $X\subseteq \kappa$ such that $f$ is constant on $[X]^m$ and the order type of $X$ ...
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1answer
55 views

Prove $\omega + \omega_1 = \omega_1$ [duplicate]

I am assuming that $\omega_1$ is the first uncountable ordinal and I'm using ordinal arithmetic. I have so far that if $\alpha$ and $\beta$ are ordinals, then $\alpha + \beta$ = sup{$\alpha + ...
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0answers
117 views

Levy collapse gone bad

Let $\kappa$ be strongly inaccessible, and let $\mu<\kappa$ be regular. What is the effect on $\kappa$ of forcing with the following? (1) The product of $Col(\mu,\alpha)$ for $\alpha<\kappa$, ...
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3answers
79 views

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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194 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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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 ...
2
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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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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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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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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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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
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 ...
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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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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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187 views

A truth definition, wrong, but where

Consider the theory ZFC of language with connectives $\neg$, $\rightarrow$, predicative symbol $\in$ and quantifier $\forall$. Assume that we are working in ZFC. Let $\mathcal{V}$ be the class of ...
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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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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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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
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
89 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
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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141 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 ...
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1answer
178 views

Problem 24 from Chapter 1 of Kunen's Set Theory: An Introduction to Independence Proofs

Just want to make sure I'm tracking Kunen here, and hopefully the proof I have is correct. Comments / Suggestions welcome. Thanks! Problem 24. Let T be any consistent set of axioms extending ZF. ...
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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 ...
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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 ...
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1answer
69 views

$n$th-power of ccc posets

We know that it is relatively consistent with $\textbf{ZFC}$ that there is a ccc poset $\mathbb{P}$ such that its cartesian square $\mathbb{P} \times \mathbb{P}$ is not ccc. Indeed, if $\mathbb{P}=T$ ...
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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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1answer
60 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 ...
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2answers
92 views

What is the cardinality of the open ordinal space $[0,\Omega)$ if we remove it's limit ordinals?

Let $\Omega$ be the first uncountable ordinal. and for any limit ordinal $\lambda < \Omega$ Let $U_\lambda$ an open neighborhood of $\lambda$ with the order topology. what is the cardinality of the ...
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1answer
71 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
141 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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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 ...
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5answers
250 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 + ...
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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 ...
2
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2answers
117 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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1answer
66 views

Similar Proofs using ZF

I have been given two question. Both require the use of the Zermelo-Fraenkel axioms. One asks to prove that there is no set a such that $a= \{a \}$. The other asks to prove that there is no set $b$ ...
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88 views

Old exam question about well ordering on $\mathbb{R}$

I am not sure how to start tackling this question and would love a hint: Let $<$ the regular order relation on $\mathbb{R}$, and $<_w$ well ordering on $\mathbb{R}$. We define a coloring ...