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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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20 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
32 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
57 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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2answers
31 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
53 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
64 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
48 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
63 views

Is there a dual to term “vacuously true” for a universal set?

For an empty set, any statement that claims "for all ... is true/false" are considered "vacuously true". So, can we construct a universal set in which any statement that claims "there exists ... is ...
3
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2answers
37 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 $\...
2
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1answer
40 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
60 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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0answers
68 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
36 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
65 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
40 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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0answers
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
117 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
88 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
37 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
46 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 $\psi_\alpha$...
3
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1answer
57 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
53 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+...
2
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1answer
34 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
52 views

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
55 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
46 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
29 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
97 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 ...
0
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1answer
33 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
78 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
48 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
59 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
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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. ...
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1answer
92 views

What is the strongest known proxi-finite theory?

[NEWEST EDIT] This a try to salvage this method, addressing the two objections that was raised by Noah. So I'll re-exposite this approach: EXPOSITION A theory $T$ is to be labeled as "proxi-finite" ...
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1answer
29 views

If $X$ and $Y$ are $\in$–isomorphic transitive sets, then $X = Y$ and the identity on $X$ is the unique $\in$–isomorphism $f\colon (X,\in)\to(Y,\in)$

Prove that if $X$ and $Y$ are $\in$–isomorphic transitive sets, then $X = Y$ and the identity on $X$ is the unique $\in$–isomorphism $f\colon (X, \in) \to (Y, \in)$. I've started this question, and ...
3
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1answer
40 views

Are there uncountable cardinals $\kappa$ such that $|\kappa\cap\mathsf{Card}| = \kappa$?

All the cardinals $\kappa\leq\aleph_0$ have the property that there are precicely $\kappa$ cardinals less than $\kappa$. Of course, $\aleph_1$ lacks this property since there are only $\aleph_0 +1= \...
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0answers
40 views

Recursive way of calculating cofinality of ordinals

I was trying to understand why A regular ordinal is always an initial ordinal. and in the course of this, came to the following hypothesis. For any ordinals $\alpha$ and $\beta$ such that $$\...
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2answers
93 views

What is the analogue to this cardinal arithmetic theorem for infinite products?

$\begin{array}{l}{\text { 1.3 Theorem Let } \lambda \text { be an infinite cardinal, let } \kappa_{\alpha}(\alpha<\lambda) \text { be nonzero cardinal}} \\ {\text {numbers, and let } \kappa=\sup \...
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1answer
65 views

Redundant axioms IN ZFC

I am studying enderton elements of set theory and trying to figure out what combinations of axioms are redundant. Given: extensionality, empty set, pair set, union, power set, SUBSETS, AOC, ...
2
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1answer
38 views

Weakening of the perfect set property

The perfect set property says that every uncountable set of reals contains a perfect subset. Now consider the following statement: P: For every $X\subset\mathbb{R}$, either $X$ or $\mathbb{R}\...
2
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0answers
83 views

Applications of Zorn's lemma with orderings other than set inclusion

In my set theory lecture notes, there is the following paragraph, after proving Axiom of Choice from Zorn's lemma: The demonstration of this theorem is a typical example of the application of Zorn'...
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1answer
34 views

Why every small set is a moderate set, but not conversely (as universe itself)?

I read here a subset of the universe, which may be small or large. (every small set is moderate, but not conversely; again, the universe itself is the standard counterexample.) But I don't ...
5
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1answer
99 views

Shooting a club is Baire

I'm attempting this problem from Kunen: I'm trying to do it by a direct combinatorial argument. Namely, let $C$ be the set of countable ordinals for which there exist such an $\omega$-chain. I want ...