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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1answer
9 views

Proving that $(\omega_n)^\omega=\omega_n$ providing CH but not GCH

This is an exercise from a book from Kunen - SET THEORY, An Introduction to Independence Proofs Assume CH but don't assume GCH. Show that $(\omega_n)^\omega=\omega_n$ for $1 \le n < \omega$. I ...
1
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1answer
17 views

Number of subsets of $\mathfrak c$ that are different no matter how high you go?

I am interested in collections $\mathcal X \subseteq \mathcal P(\mathfrak c)$ such that for any distinct $X,Y\in\mathcal X$ and $\alpha<\mathfrak c$ we have $X\cap [\alpha,\mathfrak c]\neq Y\cap ...
2
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0answers
32 views

Is the following a legitimate proxy for the Axiom of Replacement?

I'm working on an interface between set theory and plural logic. Here's my question: If one were to endow set theory with the expressive resources of plural quantification, could the following count ...
3
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0answers
28 views

Over ZF does “every non-seperable Hilbert space has an orthonormal basis” imply “there exists a non-Lebesgue measurable set”?

I know from this question that it's an open problem whether or not the existence of a dense orthonomral basis for every real or complex Hilbert space $(\text{B}_\text{orth})$ implies the axiom of ...
4
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1answer
90 views

Is there a non-trivial countably transitive linear order?

A linear order $<$ on a set S is countably transitive iff, whenever A and B are order-isomorphic countable subsets of S, there is an order-automorphism of S which maps A onto B. Does such an order ...
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2answers
74 views

What application/deeper meaning do countable and uncountable infinities have? [on hold]

Georg Cantor proved that there are two different infinities but what application does this proof have? Is this result used in some other more useful theorem?
4
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2answers
64 views

Intuition behind constrution of the Hyperreals

Just want to attempt to check if my understanding/intuition for the construction of the Hyperreal numbers via an ultraproduct is correct. Appreciate any corrections or help. So Hyperreals are ...
5
votes
1answer
72 views

Existence of stationary subset of $\omega_2$ with certain properties

How can I show the existence of a stationary subset $X\subset\omega_2$ with the properties $x\in X$ implies $cof(x)=\omega$ For every $\alpha<\omega_2$ the set $\{x\in X\mid x<\alpha\}$ is not ...
2
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1answer
103 views

Undergraduate set theory research [closed]

What are some topics for undergraduate Set Theory research? And what are the prerequisites for such topics? I'm graduating next year, and I'm interested in Set Theory. I have been always interested ...
4
votes
1answer
95 views

If $\phi$ holds for all standard models of ZF and ZF proves this, then does ZF prove $\phi$?

I apologize if this is a nonsensical question. Suppose $\phi$ holds in all standard models of ZF. Suppose further that ZF proves this. Then does ZF prove $\phi$?
2
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1answer
68 views

(totally) (M,P)-generic forcing condition

We say a cardinal $\theta$ is sufficiently large for a forcing $Q$ if $\mathcal{P}(\mathcal{P}(Q)) \in H(\theta)$. And a set $M$ is a suitable model for $Q$ if $Q \in M$ and $M \prec H(\theta)$, $M$ ...
2
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0answers
59 views

Is this a typo in Jech's Set Theory?

In Jech's Set Theory, p. 603 in the chapter about Proper Forcing, the proof of Theorem 31.7. In the second but last paragraph, the proof says By Theorem 8.27 (Menas), $\lbrace M \cap \lambda ...
8
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1answer
192 views

Objects Too Big To Care About?

I was wondering if in certain fields of math (denoted by some set of axioms describing some class of objects), that there is a cap on size beyond which the existence of larger objects is "irrelevant" ...
0
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1answer
29 views

$\mathbb{P}_{\kappa}$ forces $\text{non}(\mathcal{M})\leq \kappa$ and $\text{cov}(\mathcal{M})\leq \kappa$

Let $\mathbb{D}$ Hechler forcing. Let be $\kappa$ an uncountable regular cardinal. Consider the finite support iteration $(\langle \mathbb{P} \rangle _{\alpha < \kappa}, \langle \dot{\mathbb{Q}} ...
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0answers
62 views

Looking for this theorem by Devlin and Shelah

This is a theorem of Devlin and Shelah which I am looking for more details and also proof: $2^{\aleph_0}=2^{\aleph_1}$ is equivalent to the following statement: There is an $F:H(\aleph_1) ...
1
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1answer
44 views

About the cardinality of the set of all terms/formulas when $|L| > \aleph_0$

This may be a silly question; it was occasioned by exercise 17 of Doets Basic Model Theory book, in which he asks us to prove that, for an arbitrary language $L$, there are at most $|L| + \aleph_0$ ...
1
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2answers
59 views

Cardinal Exponentiation Inequality

Let $\lambda, \kappa$ be infinite cardinals with $\lambda<\kappa$, what is known about $\kappa^\lambda$? specially in the case either $\kappa$ is regular. Or is there very little that can be ...
2
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0answers
62 views

Coding of a function f (relative to a ladder system $\overrightarrow{C}$)

Let $\overrightarrow{C}=\langle C_\delta \colon \delta \in \mbox{Lim}(\omega_1)\rangle$ be a ladder system on $\omega_1$. Let N $\subseteq M$ be countable subsets of $\gamma$ and $ \lbrace\omega_1, ...
1
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4answers
69 views

Why do we define an inner forcing relation?

Studying forcing I came across different definitions of the forcing relation $\Vdash$: the outer forcing relation $\Vdash^M$ where we define $p \Vdash^M \varphi(\tau_0,\dotsc,\tau_n)$ to hold if for ...
2
votes
2answers
64 views

Is there much of difference between set models and class models?

When we talk of class models and set models, is there a need to talk about them separately? What would be an example? What I can think of is that a class cannot technically be talked inside a model ...
0
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2answers
33 views

Splitting condition of forcing posets

I was looking at Wikipedia for brief reminders of what I learned in my elementary set theory class, and discovered the forcing page (which I did not learn): A forcing poset is an ordered triple, ...
8
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3answers
419 views

Definition of Category

In Spanier's book of algebraic topology, there is a definition of "categories" which entails "a class of objects". I realize that the vagueness of the concept of "class of objects" is exactly used ...
3
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2answers
63 views

Proving that $\langle \omega+1, \leq \rangle$ and $\langle \omega + \omega^*, \leq \rangle$ are not elementarily equivalent

This is an exercise from Kees Doet's Basic Model Theory. The basic idea is to show that an ordering of type $\omega + 1$ is not isomorphic to an ordering of type of $\omega + \omega^*$, where ...
10
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1answer
135 views

Decidability of equality of two set-theoretical terms constructed without replacement or specification

Define the set of NS-terms (NS is for "no schemes") to be the smallest set of terms satisfying the following rules : $\emptyset,\omega$ are NS-terms. if $x$ and $y$ are NS-terms, then so are $x\cup ...
1
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1answer
79 views

Differences between set theory and second order logic?

In first order logic only $x$ in $p(x)$ is quantified but in second order logic it is also possible to express quantified predicates. Set theory is defined in first order logic, as far as I ...
2
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0answers
63 views

Generalizing beyond proper classes

I noted that issues such as Russell's Paradox involving the set of all sets that don't contain themselves can be resolved by stating that the object that is all set that don't contain themselves is a ...
2
votes
1answer
37 views

Example of a Well-Ordered Class that is not Proper

I am currently studying well-ordered classes in the context of NBG set theory and I am trying to find a well-ordered class that is not proper. Here is the relevant definition of the terminology ...
5
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0answers
97 views

A question about HOD

Suppose $\phi$ is a sentence in the language of ZFC such that ZFC proves $\phi^{HOD}$. I need to show that ZFC already proves $\phi$. Could you give any hints? Thank you!
2
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1answer
47 views

Find a uniquely defined element in this $\aleph_1$-indexed Cartesian product

Denote by $A$ the set of all ordinals with cardinality exactly equal to ${\aleph}_0$, and for $\alpha\in A$ let $B_{\alpha}$ denote the set of all bijections between $\alpha$ and $\omega$ ; finally ...
3
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2answers
76 views

Cardinality of universal set?

I read that there are some non-standard versions of set theory that allow for the existence of a universal set. My first question is: what (if anything) can be said about the cardinality of the ...
6
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2answers
133 views

Troubles with continuum hypothesis

Axiom of choice is discussed very often, because it should be a lot of paradoxes (Banach-Tarski paradox, for example) and in general it is considered by many non-obvious (for uncountable case, of ...
2
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0answers
38 views

Is the set of all of possible uncountable cardinalities uncountably or countably infinite? [duplicate]

I assume that the number of unique possible cardinalities that are uncountably infinite is either uncountable or countable because it is possible to take the powerset of each set, resulting in an ...
1
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1answer
61 views

Ultrapower construction of the Hyperreal numbers

I've been learning about non-principal ultrafilters with the overall aim of understanding the ultrapower construction of the Hyperreals. Couple of things I'm confused on: Firstly the Ultrafilter ...
1
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1answer
76 views

Quotient of Cohen forcing

How do we know that the quotient of the Boolean algebra associated with Cohen forcing by a generic filter is either atomic or isomorphic to the Cohen forcing? I know that Cohen forcing is the unique ...
4
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1answer
50 views

Any two countable atomless Boolean algebras are isomorphic

How to prove that Any two countable atomless Boolean algebras are isomorphic. This is an exercise of Jech - Set theory book for which I have some difficulty.
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2answers
117 views

Where is the axiom of choice used in Rudin's proof of “the countable union of countable sets is countable”?

Baby Rudin proves that the countable union of countable sets is countable. From reading other proofs online, the axiom of choice has to be invoked; however, I'm not seeing immediately where that is ...
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0answers
50 views

Enumerating the reals using a definability hierarchy

(edit : for those perplexed with the meaning of "truth" in the following, let us say we believe in the consistency of ZFC, use the completeness theorem and reason in a fixed model $M$ of ZFC. The ...
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0answers
111 views

What well-orders are definable over $V$?

Let $V$ be a (the) universe of sets, and $On^V$ denote the ordinals of $V$. It is well known that there are formulae that seem to define orderings `longer than' $On^V$. For example: $\alpha < ...
1
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2answers
119 views

Which should I study first: Logic or set theory?

I'm an undergraduate student in a college of sciences and technics studying maths, physics, computing and some chimestry so we studied elementary materials in logic and set theory. As I am interested ...
24
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2answers
844 views

W. Mückenheim claims a severe inconsistency of transfinite set theory; true? [closed]

The abstract for a paper on arxiv.org (http://arxiv.org/pdf/math/0408089v3.pdf) reads (with my emphasis): "Transfinite set theory including the axiom of choice supplies the following basic theorems: ...
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0answers
62 views

Importance of (the number of) ultrafilters

I have seen a proof about the numbers of ultrafilters in topology, namely a theorem of Pospíšil stating that there are $2^{2^\kappa}$ ultrafilters on every infinite set $X$ with $\kappa = |X|$. The ...
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0answers
93 views

Application of the reflection theorem in ZFC

In a proof that a certain theory $T$ in conservative over $\textsf{ZFC}$, the author makes the following step: (Here $\Delta$ is a finite set of formulas such that $\textsf{ZFC}\vdash\Delta$.) ...
0
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0answers
55 views

Prove the Axiom of Replacement implies the Axiom of Specification. [duplicate]

The Axiom of Specification says that: For a set $ A$ and objects $x\in A$,for any object $y$ there exists a set: $$ y \in \{x: P(x) \wedge x \in A\} \implies y \in A \text{ and } P(y)$$ Where $P(x)$ ...
1
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2answers
53 views

A road-map through “Combinatorial Set theory: With gentle intro to independence proofs”

I'm going to study independence proofs form Halbeisen's book. It seems that some material is not needed to study independence proofs, so it seems that the book contains more material than my needs. ...
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1answer
122 views

Books for Ordinals and Cardinals

I am looking for a nice introductory book to read to learn and master ordinals and cardinals. Please help me!
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0answers
77 views

Many $(\kappa+\alpha)$-strong cardinals below a $(\kappa+\beta)$-strong one for $\alpha < \beta < \kappa$

Let $\lambda$ be an ordinal. A cardinal $\kappa$ is $\lambda$-strong iff there is some inner model $M$ and an elementary embedding $$ j \colon V \rightarrow M $$ s.t. $crit(j) = \kappa$ and $V_\lambda ...
3
votes
1answer
80 views

$\operatorname{Fn}(\lambda,2,\lambda)$ collapses $\lambda^+$ to $\operatorname{cf}\lambda$ if $\lambda$ is singular?

It is an exercise problem in Kunen (VII G5). I shall show that $\operatorname{Fn}(\lambda, 2, \lambda)$ adds a map from $\theta = \operatorname{cf}\lambda$ onto $\lambda^+$ for singular $\lambda$. ...
2
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0answers
37 views

Help regarding a proof about Dedekind finiteness

I got this one as an exercise. If $F$ is Dedekind finite and $t \notin F$ then prove that $F \cup \{t\}$ is also Dedekind finite. I gave this as an answer: If $F\cup \{t\}$ is Dedekind infinite then ...
1
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2answers
103 views

Are Cantor's set theory and the axiom of choice well accepted nowadays? [closed]

Is there a "more consistent" theory ? Do mathematicians agree about the validity of the results obbteined by this theory ?
2
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1answer
33 views

limit and infinite ordinals: same thing? [duplicate]

I'd like to clarify my understanding re: limit and infinite ordinals. This post says: An initial infinite ordinal is a limit ordinal. Is it true the other way around? That is if I have a limit ...