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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56 views

Set partitioning in ZFC

Does $\sf ZFC$ allow the partitioning of a set by claiming that $a$ and $b$ are in the same subset if $f(a,b)$? Cause I've once seen this technique being used in a proof but I can't see how it is ...
2
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0answers
102 views

Cutting a Banach-Tarski Cake

I was reading a cake-cutting problem here (not really related, so I won't link to it), and for some reason, this variation occurred to me. I have no idea whether this problem is even well-formed: ...
2
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1answer
49 views

Why does a proof of $\exists f: X\to Y$ injection $\iff \exists g: Y \to X$ surjection requires the axiom of choice?

Why does a proof of $\exists f: X\to Y$ injection $\iff \exists g: Y \to X$ surjection requires the axiom of choice? This question is answered here: There exists an injection from $X$ to $Y$ if and ...
3
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1answer
31 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 ...
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2answers
50 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 ...
2
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1answer
37 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 ...
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0answers
31 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 ...
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3answers
193 views

Can set theory be inverted?

Has anybody investigated or constructed a set theory, call it $T$, which develops in a manner that is the reverse of how the majority of the standard set theories develop? Instead of starting with the ...
4
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1answer
100 views

Is there a non-trivial countably transitive linear order? [on hold]

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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1answer
49 views

Duals of filters, an explicit formula for meet?

Fix some set $U$. Recall that filters on $U$ are nonempty sets $F$ such that $A\cap B\in F \Leftrightarrow A\in F\land B\in F$. Replacing every element of $F$ with its complement and simultaneously ...
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1answer
103 views

Set with cardinality of Aleph 2

What is a simple or commonly known set that has a cardinality equal to Aleph 2 or greater?
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2answers
115 views

Good resources for studying independence proofs

I've finished most of Enderton's set theory. And I intend to spend some time studying independence proofs. I'm more interested in independence of axiom of choice not CH. From I know so far, there ...
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1answer
41 views

example of multiplication of ordinals with infinite cardinality with larger value on right where we dont' take the max?

I recall reading about a rule for multiplying ordinals where at least one is infinite, and where the cardinality of the multiplier (on right) is larger than the multiplicand (on left). If I recall ...
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2answers
55 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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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, ...
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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 ...
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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 ...
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 ...
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2answers
79 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?
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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 ...
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 $M \prec H(\theta)$, $M$ ...
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1answer
96 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 $\phi$?
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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 ...
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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 ...
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1answer
566 views

Strength of the statement “$\mathbb R$ has a Hamel basis over $\mathbb Q$”

I would like to know if there are "interesting" equivalences to the statement "$\mathbb R$ has a Hamel basis over $\mathbb Q$". I am not interested in more general statements, like "every vector space ...
8
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1answer
193 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" ...
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 ...
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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) ...
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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, ...
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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 $|L| + \aleph_0$ ...
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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 ...
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 ...
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4answers
301 views

What can I do with proper classes?

There are standard tricks, constructions and techniques in ZFC when working with proper classes; for instance one can form the cartesian product of a pair of classes without difficulty, or more ...
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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 ...
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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 ...
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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 ...
3
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2answers
77 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 ...
2
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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 ...
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4answers
1k views

Does learning logic and set theory before arithmetic, algebra, and geometry have an advantage?

I'd like to become conversant in a wide variety of serious mathematics, but i'm currently one of those students who did very poorly on mathematical subjects in school, never completing even basic ...
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0answers
154 views

A consequence of zero sharp (successor cardinals having countable cofinality)

So the existence of $0^\sharp$ in set theory is really the assertion on the existence of indiscernibles for the constructible universe $L$ that also "generate" $L$ (see ...
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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 ...
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!
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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 ...
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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 ...
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1answer
62 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 ...
6
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2answers
234 views

Boolean algebras without atoms

Why is the theory of Boolean algebras without atoms $\omega$-categoric?
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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 ...