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Questions tagged [set-theory]

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 theory, definability, infinite combinatorics, transfinite hierarchies; etc. More elementary questions should use the "elementary-set-theory" tag instead.

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Study for Ternary and N-ary relations: books and practice material recommendations?

I have learnt the topic binary relations and its types and i am curious about ternary and N-ary relations and i want to learn about there properties. So please can I be recommended books and online ...
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28 views

Is completeness inherited upwardly?

Language: first order logic with equality and membership and the omega set rule (see below). Define: $set(x) \iff \exists y (x \in y)$ Axioms: $\sf ID$ axioms + Extensionality: $\forall z (z \in x \...
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1answer
21 views

Difference between Compactness Proof Structure Creation and Creation of a Forcing Extension Structure

Having just started to learn Forcing and having looked though the large number of Forcing questions on StackExchange, I apologize if I overlooked the following 'overview' novice question: Enderton ...
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2answers
60 views

If an extension of ZF is not consistent with $V=HOD$, does this imply that it must prove existence of large cardinals?

Suppose we have a first order theory $\text{T}$ such that $\text{T} \supset \text{ZF}$ $\forall M [(M\models \text{T}) \to \neg (M\models \forall x (x \in HOD))]$ Does that mean that $\text{T}$ must ...
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2answers
65 views

Cardinal exponentiation

My understanding of exponentiation of cardinals leads to the conclusion that if $2 \leq \kappa \leq \lambda$, then $2^\lambda = \kappa^\lambda$ , because: $2^\lambda \leq \kappa^\lambda \leq (2^\...
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1answer
57 views

Can we use metatheory of set theory for proof statements in set theory?

I wanna know if is possible or have examples of theorems in set theory, for example $\beta$, that have a demonstration of forme $Cons(ZFC)\Rightarrow \beta$ but $\beta$ is independent of axioms of $...
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2answers
90 views

Does every uncountable Borel subset of $\mathbb R$ contains a perfect subset?

This question came from (London Mathematical Society Student Texts) Krzysztof Ciesielski-Set Theory for the Working Mathematician-Cambridge University Press. Chapter 6.2 Exercise 5. I have thought ...
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1answer
43 views

Is the class of all hereditarily transitive definable sets a model of ZFC?

Is it the case that every transitive definable set is also ordinal definable? Formal definition of the former would be: $TD^M=\{u\in M: \exists \tau_1,...,\tau_n\in Trs^M, \varphi\in Formula (M\...
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0answers
24 views

Ordinals which satisfy $\beta \cdot \alpha=\alpha$

Let $\beta>1$ be a fixed ordinal. I want to find a nice characterization of the ordinals $\alpha>1$ which satisfy $\beta\cdot\alpha=\alpha$. I have already seen that $\beta+\alpha=\alpha$ if and ...
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3answers
295 views

How do the consequences of Russell's paradox extend beyond universal comprehension principle as far as the set of all sets problem? [duplicate]

I think I understand the way in which Russell's paradox shows that the following principle is wrong: " for every predicate, there is a set having as elements all the objects that satisfy this ...
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0answers
21 views

Finite sets of a infinite set A is in bijection with A [duplicate]

Suppose $A$ is a infinite set. Let $P_f(A)$ be the collection of all finite subsets of $A$. Now we want to show that $P_f(A)$ is in bejection with $A$. I can do this for the case that $A$ is countable,...
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1answer
36 views

Let $α$ be an ordinal and let $n\in\omega$. If $α\geq ω$ and $γ<\log (α)$, then $\log(\alpha +(\omega^{\gamma}\cdot n))=\log(\alpha)$

Let $\alpha$ be an ordinal and let $n\in \omega$ ($\omega$ is the least infinite ordinal). Suppose that $\alpha\geq\omega$ and $\gamma<\log\left(\alpha\right)$. I want to prove that $$\log\left(\...
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1answer
58 views

Adding a new unary function to Morse–Kelley set theory [on hold]

Suppose we take MK and add a new unary function symbol α, with a countable set of extra axioms α(1) ∈ α(0) α(2) ∈ α(1) α(3) ∈ α(2) (i) Show that the resulting theory is consistent.For all n ∈ N $\...
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0answers
31 views

Can this re-definition of proxi-finiteness of theories help guiding extending ZFC?

A formula $\varphi^n$ is to be labeled as graded formula only if it is in prenex normal form and every variable in it is of the form $x_i$ where $i$ is a natural number smaller than $n$, and all ...
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2answers
38 views

“Normality” condition on an ordinal operator.

Call a rule $\Phi:\Bbb{ON}\rightarrow\Bbb{ON}$ normal if for any ordinal $\alpha$, $\Phi(\alpha)<\Phi(\alpha^+)$, and for limit ordinals $\lambda$, $\Phi(\lambda)=\bigcup_{\alpha<\lambda}\Phi(\...
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1answer
57 views

Developing new theories by transfinitely iterating the Godel sentence construction

In Turing's Ph. D thesis "Systems of Logic Based on Ordinals", he writes of a simple way to use Gödel's incompleteness theorem to devise a transfinite sequence of new theories. The sequence proceeds ...
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0answers
67 views

Is Isles' theorem 2.6 correct?

In "Regular Ordinals and Normal Forms", Theorem 2.6, David Isles claims that Bachmann proves that the sequence of normal functions generated by a Bachmann collection has property (6). Firstly, ...
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0answers
49 views

Can power set axiom be proved in a class theory of well ordered hereditarily accessible sets?

Working in a pure class theory, where sets are defined as elements of classes. That is: Define: $set(x) \iff \exists y (x \in y)$ Let's have the following known three axioms from $\text{MK}$ ...
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2answers
41 views

Cardinality of the set of all total orders on $\Bbb{N}$

I need to compute the cardinality of the set of all total orders on $\Bbb{N}$. Now, by definition there is an inclusion of this set into $\mathcal{P}(\Bbb{N}\times\Bbb{N})$, and so has cardinality $\...
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1answer
41 views

Cardinality of all infinite subsets with an infinite complement.

Working in $\text{ZF}$... Let $X$ be an infinite set with a given well-ordering relation $\le$. Define $\tag 1 \mathcal B(X) = \{ S \in \mathcal P(X) \, | \, S \text{ is infinite } \text{ and } X \...
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2answers
63 views

$\aleph_0\times 2^{\aleph_0} = 2^{\aleph_0}$ requires the Axiom of Choice?

In a model solution it is stated that the cardinality of a set which is the countable union of sets of cardinality $2^{\aleph_0}$ is $\aleph_0\times 2^{\aleph_0}$, and, using the Axiom of Choice, $\...
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76 views

Classifying the chains of orderable sets' power sets up to isomorphism

Recently, while trying to understand another result, I began to wonder about the following question: Given some orderable set $A,$ what (if anything) can we conclude about the order type or ...
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1answer
38 views

Comparing the size of ordinals involving sums of $\omega$

I had come across this question when revising for an upcoming exam in Set Theory: Put these ordinals in ascending order: $\omega^3 + \omega^2 + \omega, \\ \omega + \omega^2 + \omega^3, \\\omega^3 + \...
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4answers
69 views

If $X$ is infinite, then there is no surjective map $f : X \rightarrow WO(X)$

I had come across this question when revising an upcoming exam in Set Theory. Here we are assuming Axiom of Choice, and $WO(X)$ denotes the set of well-orders on $X$, which was already established ...
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1answer
43 views

What is the consistency strength of $ZC+\neg CH\ \forall x (|x|>1)$?

What is the consistency strength of "ZC + failure of $\text{CH}$ for all many membered sets"? I know that for the case of ZFC the failure of $\text{CH}$ for every many membered set is too strong, ...
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35 views

Equivalence of surjectivity and existence of right inverse of any function imply the axiom of choice. [duplicate]

Actually this is question in pinter's set theory. (Equivalence of surjectivity and existence of right inversw)Let $A$ be a set and let $f : A \rightarrow B$ be a function; $f : A \rightarrow B$ is ...
2
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2answers
121 views

How early do the second-order definable subsets of $\mathbb{N}$ occur in the Constructible Universe?

$ZFC+V=L$ implies that $P(\mathbb{N})$ is a subset of $L_{\omega_1}$. But I’m wondering what layer of the constructible Universe contains a smaller set. My question is, what is the smallest ordinal $...
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0answers
89 views

Why not consider the model of $ZFC_2 + 0-inaccessibility$ as the standard model of ZFC?

Let's take $ZFC_2 + 0$-inaccessibility; That is, ZFC written in full second order logic and add to it absence of existence of any inaccessible set. So this defines a unique model of all hereditarily ...
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1answer
38 views

Is $\forall x(x\in L\land\mathrm{rank}(x)=\alpha\rightarrow x\in L_{\alpha+1})$ consistent with $\mathsf{ZFC}$?

The assertion $$\varphi=\forall x(x\in L\land\mathrm{rank}(x)=\alpha\rightarrow x\in L_{\alpha+1})$$ is not provable in $\mathsf{ZFC}$: under $V=L$ there's uncountably many sets of rank $\omega$ in $L$...
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1answer
48 views

Prove that $A=\{\{\Phi(b\restriction n) : n \in\omega\}\mid b\colon\omega\to 2\}$ is not a mad family.

Let $\Phi\colon{}^{<\omega}\omega\to\omega$ be a bijection and let $A=\{\{\Phi(b\restriction n) : n \in\omega\}\mid b\colon\omega\to 2\}$ Prove that A is not a mad family. I'm not sure how to ...
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0answers
27 views

How to compare values of Buchholz functions?

https://en.m.wikipedia.org/wiki/Buchholz_psi_functions Is there a simple method to compare values of Buchholz functions? Assuming that we have two ordinals represented using addition and $\psi_\alpha$...
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1answer
59 views

Can we describe any subsets of $\mathbb{N}$ occurring in a late layer of the Constructible Universe?

There is a certain large countable ordinal referred to in the literature as $\beta_0$. It was first discovered by Paul Cohen, and here are some equivalent characterizations of it: The smallest ...
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1answer
60 views

Is there a subsystem of ZFC which constrains the Universe to $L$?

The Axiom of Constructibility states that $V$, the Universe of all sets, is equal to $L$, the Constructible Universe. When added to $ZFC$ does not place a constraint on what sets exist, instead what ...
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1answer
55 views

Cantor-Bendixson Theorem Proof (existence of an ordinal)

Let $A$ be a topological space, denote the set of accumulation points of $A$ by $A'$. If $\alpha$ is an ordinal, we define $A^{(\alpha)}$ by transfinite induction: $A^{(0)} = A$, $\, \,A^{(\alpha+...
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1answer
37 views

Finding an order isomorphism from $\text{On}\times\text{On}$ to $\text{On}$

Let $\text{On}$ be the class of all ordinals and let $\leq_{\text{c}}$ be the canonical well-ordering on $\text{On}\times\text{On}$. More specifically, $\preceq$ is defined as follows. Let $\left(\...
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0answers
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Given a poset with a join such that any pair has a meet, find a total order with a special property.

Let $(A,\geq)$ be a partially ordered set such that there exists the join $\bigvee A$, i.e. $a\in A$ such that $a\geq b$ for any $b\in A$; for any pair $(b,c)\in A\times A$ there exists the meet $b \...
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1answer
56 views

What is the lowest layer of the Constructible Universe which is a model of $ZFC-P$?

This answer says that the smallest ordinal $\lambda$ such that $L_\lambda$ is a model of $ZFC$ isn’t easy to describe, other than to say that $\omega_1^{CK}<\lambda<\omega_1$. But my question ...
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1answer
53 views

Which stage in the Neumann hierarchy do powers of the reals fit in?

To be more specific than the short title, I try to gauge the size of some "normal" function spaces as e.g. found in functional analysis against set universe sizes at certain stages. For the sake of ...
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1answer
47 views

Uses of Axiom of Choice [duplicate]

I am a first-year maths student but I occasionally drift away from our taught material. Some years ago I saw the ZFC axioms for the first time, but now that I am in college, and although the stuff I'...
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2answers
31 views

Show that there exists non-zero countable ordinal $\delta$ such that $\omega\delta=\delta$

I am stuck on the following problem that says: Show that there exists non-zero countable ordinal $\delta$ such that $\omega\delta=\delta$ My Attempt: Previously I solved a question similar to ...
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0answers
21 views

Finite axiomatisation of set theory

I am looking for suggestions of "closest" set theories to ZF that admit a finite axiomatisation -- both theories with have independent interest, and theories perhaps devised specifically with this ...
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0answers
98 views

Baire space $\mathbb{N}^\mathbb{N}$ written as $\mathbb{R}$ [duplicate]

I'm writing my bachelor thesis on various topics from set theory and descriptive set theory (mainly topological games), and I just read a paper in which the Baire Space $\mathbb{N}^\mathbb{N}$ is ...
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1answer
46 views

Is “PA+ω-rule” and “Zermelo-infinity+every set is finite + ω-set-rule” equi-interpretable?

We know that "PA" and "Zermelo-infinity+every set is finite" are equi-interpretable. Now is "PA+$\omega$-rule" and "Zermelo-infinity+every set is finite + $\omega$-set-rule" equi-interpretable? ...
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1answer
41 views

Can $ZFC + \neg Con(ZFC)$ be interpretable in $PA + \omega$- rule?

Suppose ZFC is consistent. Then can $ZFC + \neg Con(ZFC)$ be interpretable in $PA + \omega$- rule? The idea is that "interpretability" doesn't preserve truth, so even if we hold ZFC + CH to be true, ...
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1answer
79 views

Difference between class $\{x\mid x\in Y\}$ and $Y$ with $Y$ being a set?

Consider $X,Y$ 2 sets. One wants to define $X\cap Y$. There is no guarantee that $X\cap Y$ will be a set on the formal level. Consider class $C=\{x\mid x\in Y\}$ where $x\in Y$ is treated as ...
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1answer
49 views

Size issues in the reduction of Colimits to a Coequalizer of Coproducts.

This is in regards to a proof in Emily Riehl's Category Theory in Context (Available free here), on page 97. This is what I understand of the proof: For a locally small category $C$, we wish to ...
2
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1answer
60 views

Understanding how to construct a tall narrow tree

I am trying to read the following presentation by Hamkins: http://jdh.hamkins.org/wp-content/uploads/2017/01/Bonn-Logic-Seminar-2017.pdf At page 30(34) there is a lemma called "Uniform covering ...
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0answers
79 views

Show that continuous function from omega_1 to R is eventually constant [duplicate]

Let $f: \omega_1 \to \mathbb{R}$ be a contionuous function. Prove that $f$ is eventually constant. I was trying to prove it by contradiction but I did not have any idea.
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0answers
18 views

Domain of the neat reduct of a cylindric set algebra

The following is the definition of the neat reduct of a cylindric algebra: For ordinals $\alpha < \beta$, If $\mathfrak{A}$ is a $\beta$-dimensional cylindric algebra then the neat-$\alpha$ reduct ...
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0answers
42 views

A consistent ZFC implies that ZFC has a model that is not well-founded [duplicate]

I've come across a problem in a book about the Zermelo–Fraenkel set theory I'm reading and am kind of stuck: Assume ZFC to be consistent. Show that ZFC has a model that is not well-founded/regular. ...