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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 ...

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Distinguishing between set membership functions and proper class membership functions

If a set is defined by its membership function, then we can see that there exist membership functions that will not define a set but a proper class. If we try to define the class of set membership ...
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2answers
11 views

Consider a two-point set $M = \{a,b\}$ whose topology consists of the two sets, $M$ and the empty set.

Consider a two-point set $M = \{a,b\}$ whose topology consists of the two sets, $M$ and the empty set. Why does this topology not arise from a metric on M? May someone please clarify this question (I'...
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1answer
27 views

Loeb Measure construction in axiomatic approach to NSA

Hello dear StackExchange, in my upcoming bachelor's thesis, I plan to present an overview of some topics in nonstandard analysis, including some higher applications, like Loeb Measures (among other ...
3
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2answers
29 views

Proving that if $\kappa$ is a limit ordinal, then $\alpha+\kappa=\bigcup_{\gamma\in\kappa}\left(\alpha+\gamma\right)$

Let $\alpha$ be an ordinal and let $\kappa$ be a limit ordinal. I want to prove that $$\alpha+\kappa=\bigcup_{\gamma\in\kappa}\left(\alpha+\gamma\right).$$ Here I define the sum of ordinals not using ...
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1answer
35 views

Applying transfinite recursion

Let $f\colon\alpha\to\beta$ be a function between ordinals $\alpha,\beta$. I want to define a function $g\colon\alpha\to\gamma$ for some ordinal $\gamma$ so that $(\forall \eta < \alpha)(g(\eta) =...
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1answer
103 views

Could Hypercomputer solve undecidable problems? [on hold]

firstly lets assume that a human brain is no more powerfull then a Turing Machine. A question that I would like to ask is whether a hypercomputer capable of doing uncountably many computational steps ...
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1answer
66 views

Huge vs. compact cardinal

How is it possible and how does one prove that the least huge cardinal is less then the least compact cardinal (if both exist) but at the same time huge has higher consistency strength then the ...
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1answer
93 views

Higher-order Busy Beaver functions and the language of first-order set theory

I have a question, but before asking this question, it is required to ask the preliminary question: is it possible to define a particular higher-order Busy Beaver function by a formula (of finite ...
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1answer
152 views

Where can I find a proof of ($\aleph_1 \leq 2^{\aleph_0}$ is independent of ZF)?

In "A tutorial on countable ordinals" [1], in page 25, Forster uses the fact that $\aleph_1 \leq 2^{\aleph_0}$ is independent of ZF to prove that there is no definable family of fundamental sequences ...
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2answers
71 views

Set-Theoretic Omniscience and Set Definition

It occurred to me the other day that a set can, in some sense, know more than I do. This is because a set does not contain duplicate objects, and yet if the same object is defined in two different ...
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1answer
44 views

Question on $[\mathbb{N}]^{\omega}$

I'm reading Fremlin's article about $\mathfrak{p} = \mathfrak{t}$, and in 4B proposition, he shows that $\Vdash_{\mathbb{P}}\,\mathcal{P}(\mathbb{N}) = \mathcal{P}(\mathbb{N})\check{}$, where $\mathbb{...
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1answer
31 views

Absoluteness of $\Delta_0$ formulas for Boolean-Valued models.

I am stuck on the following lemma from Jech's forcing chapter and need a little bit of help. Lemma. If $\varphi(x_1, \dots, x_n)$ is a $\Delta_0$ formula, then $$\varphi(x_1, \dots, x_n) \hspace{...
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0answers
37 views

Showing $\kappa^{cf(\kappa)}$ determines the map $\kappa\mapsto 2^\kappa$

Let $\kappa$ be a limit cardinal. I want to show 1) If $2^\alpha$ is eventually constant as $\alpha\rightarrow\kappa$, then $2^\kappa=2^{<\kappa}\cdot\kappa^{cf(\kappa)}$. 2) If $2^\alpha$ is not ...
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1answer
18 views

Sup and Well-ordered

Let X be well-ordered. Let S be a bounded subset of X. Then clearly by well-ordering there exists a sup of S (the set of upper bounds of S is a subset of X and so by well-ordering it has a minimal ...
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0answers
21 views

Showing $2^\lambda=\prod_{\alpha<\text{cf}(\lambda)}2^{\lambda_\alpha}$ for $\lambda$ singular cardinal [on hold]

Let $\lambda$ be a singular cardinal, so we can write $\lambda=\sum_{\alpha<\text{cf}(\lambda)}\lambda_\alpha$ with $\lambda_\alpha<\lambda$ . Then why does $2^\lambda=\prod_{\alpha<\text{cf}(...
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1answer
55 views

ZF with “double powersets”, is there a set with the same size as ℝ?

If you replace the axiom of the powerset, with one guaranteeing the existence of double powersets only ... roughly speaking, what changes? Here's the axiom of power set in ZF**, written out in first ...
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1answer
29 views

Cardinality of reduced products

Suppose $\mathcal{A}_i (i \in I)$ is a family of $L$-structures and consider the reduced product $\mathcal{A}$ of this family by a filter $F \subset \mathcal{P}(I)$. Is there a way to determine the ...
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1answer
33 views

Transfer principles in reduced products and the axiom of choice

Let $\mathcal{A}_i (i \in I)$ be a family of $L$-structures, $F \subset \mathcal{P}(I)$ a filter and denote by $\mathcal{A}$ the reduced product of this family by $F$. I'll denote the domain of $\...
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1answer
29 views

How much infinite sums and products of cardinal numbers depend on the index set?

Generalizing the usual defining or addition and multiplication of two cardinal numbers, one can define the sum and the product of the entire indexed family $(\kappa_i)_{i \in I}$ of cardinals: $\sum_{...
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2answers
34 views

Definition of function application in ZF-like theories

What is the definition of function application in material set theories like ZF? I know how functions are represented using functional relations (optionally paired with the codomain). I'm aware of ...
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0answers
73 views

How to prove a generalized version of Gödel's Second Incompleteness Theorem?

Let's start from Gödel's Second Incompleteness Theorem (GST) in the form $\not \vdash_\mathsf{PA} \mathsf{Con}_\mathsf{PA}$, where $\mathsf{PA}$ is first-order Peano Arithmetics, assumed to be ...
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1answer
46 views

existence of uncountable well-ordered set admitting a special type of function

Does there exist an uncountable Well-ordered set $(X,<)$ and a function $f: X \times X \to \mathbb R$ such that $f(x,y)<f(x,z),\forall x<y<z$ ? Note that if we want $X$ to be countable, ...
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0answers
59 views

Why Does Closure Under $κ$-sequences Imply an Ultrapower Embedding Preserves $\mathcal{L}_{κ,κ}$-sentences?

This post was moved from MathOverflow because it is not quite appropriate within the context of research mathematics. My previous question on MathOverflow, "A weakening of cardinal compactness - is ...
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1answer
30 views

Existence of a name which has elements with given truth values

Let $\mathcal{A}$ be a complete boolean algebra. Let $ D $ be a set of $ \mathcal{A}-names $, and $ f:D \to \mathcal{A} $ any function (itself a name). I wonder if it's possible to construct $ \...
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2answers
68 views

Is there a maximal universe of sets?

In asking a question on this site about the unprovability on the Continuum Hypothesis, many people explained to me that for a given set of axioms, there are many different models that satisfy that set ...
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0answers
36 views

Ultrapower of V by D

Let $D$ be a nonprincipal $\kappa$-complete ultrafilter on $\kappa$. The following are equivalent: (i) $D$ in normal (ii) In the ultrapower $Ult_D(V)$, $\kappa=[d]$ where $d$ is the diagonal ...
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1answer
31 views

Proper initial segment $X$ of a von Neumann ordinal $Y$ has different order-isomorphism class from $Y$

I mean for von Neumann ordinal a $\in$-transitive set well-ordered by $\in$, I proved that the class of such ordinals is well ordered by $\in$ (or equally by inclusion), and so different von Neumann ...
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1answer
50 views

A question on metamathematics of forcing.

This question might be a duplicate but i have found no other questions related to this so here we go. In kunens (An introduction to independence proofs) and Nik weavers (forcing for mathematicians) ...
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1answer
81 views

2-huge cardinal, the most usual definition [closed]

How are 2-huge cardinals usually defined? Is the definition contained in Jech's set theory book (the millennium edition)? Which page?
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1answer
53 views

Are there countably many numbers than can be described?

I can explain to you every natural number (in theory) in the sense, that I could describe it and you would know exactly which number I'm talking about, e.g. by writing it down, this can be done in a ...
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1answer
222 views

How do set theorists view this issue?

Since the question has changed significantly over the course of last few days, it was suggested to modify it in that light. I have removed those parts which aren't directly related to the main ...
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1answer
51 views

Absent choice, if $A$ and $B$ have a bijection, must every surjective function from $A$ to $B$ have a right inverse?

This question seems elementary and there's probably a really obvious argument for this that I'm not thinking of but I can't find it anywhere. For equinumerous $A$, $B$, must every surjection $f:A \...
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1answer
91 views

$x$ is a limit point of $X$ if and only if there exists a sequence of distinct elements which converges to it.

Let me preface that this is the second time I am asking this question because it was incorrectly marked as a duplicate of Proving there is a sequence convergent to a limit point of a set without axiom ...
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0answers
126 views

What is the strongest large cardinal property that can be argued for by Grothendieck-Zermelo potentialism? [closed]

The question is stated in the title. I am asking this question because of my interest in the Grothendieck-Zermelo potentialist system $\mathcal Z$ ={$V_{\kappa}$| $\kappa$ strongly inaccessible} ...
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0answers
32 views

$x$ is a limit point of $A$ is and only if there exists a sequence of distinct elements which converges to it. [duplicate]

I have probably seen and done this problem a hundred times, but thinking back on it, I realized something which I hadn't before. Here, I am only concerned with the forward direction of the proof, and ...
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0answers
48 views

Is there a way to prove the Banach-Tarski theorem from just the Partition Principle?

The Banach-Tarksi theorem is one of the most notorious "counterintuitive consequences" of choice (and in fact in all of mathematics), making heavy use of non-measurable sets. However, full choice is ...
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1answer
95 views

How can axiomatic set theory handle real-life objects? [closed]

For what I know the answer seems to be "It can't handle them" because the axiomatic set theory (at the least in its classical variation) allows only one type of objects: sets. So, for example, we can'...
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1answer
106 views

Is it possible to base the slow-growing hierarchy on the ordinal defined as “the height of the minimal model of ZFC (assuming it exists)”?

Let $\alpha$ denote the ordinal described in section 2.24 of the book “A zoo of ordinals” [David A. Madore]: 2.24. The smallest ordinal $\alpha$ such that $L_{\alpha} \models {\text{ZFC}}$ (...
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0answers
33 views

Show the cardinality of the infinite union of sets is less than the cardinality of the Cartesian product of those sets.

Let $X_0, X_1, X_2, \dotsc$ be sets such that $X_0 \neq \emptyset$ and $$Card(X_0) < Card(X_1) < Card(X_2) < \dotsc$$ Prove that $Card(\bigcup\limits_{i=0}^{\infty}X_i) < Card(\prod\...
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3answers
102 views

If $\subseteq$ is defined set-theoretically, it doesn't exist

I came across the following statement: If the subset relation $\subseteq $ is defined as a set in ZFC, then it doesn't exist because $\mathrm{dom(}R\mathrm{)}$ is guaranteed to exist for any ...
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0answers
70 views

Does the axiom of choice allow a set well ordered both ways to have cardinality greater than countable? [closed]

Does the axiom of choice allow a set well ordered both ways to have cardinality greater than countable?
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2answers
27 views

Regularity Enforces Trivial $\in$-Isomorphisms

Theorem 6.7. in Jech's Set Theory: Assume Regularity. Let $T_1, T_2$ be transitive classes and let the bijection $\pi \colon T_1 \to T_2$ satisfy $u \in v \implies \pi u \in \pi v$. Then $\pi u = u$...
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1answer
47 views

Von Neumann integers definition

I am learning the ZF (C) set theory considering all the 6 standard axioms, here is how I defined the set of natural integers $\mathbb{N}$. Is it correct and how do I prove the Lemma 2 ? Axiom of ...
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0answers
93 views

What are well-documented instances or applications of a truncation function on the Cantor normal form of ordinals in $\omega^{<\omega}$? [on hold]

What are well-documented instances or applications of a truncate & shift operation on the Cantor normal form of ordinals in $\omega^{<\omega}$? The function truncates the finite term of the ...
2
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1answer
29 views

Sets definable by $\Sigma_2$ parameters

Say that a formula $\phi$ defines a set $x$ from parameters $a_1, \dots, a_n$ if $\phi(a_1, \dots, a_n, x)$ holds (in $V$) but for $y \neq x$ $\phi(a_1, \dots, a_n, y)$ does not hold. Is it true that:...
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1answer
47 views

What does a hyperreal version of the Cantor Set look like?

I would like to construct a hyperreal version of the Cantor set. Let $X_0$ be the interval $[0,1]$ in the hyperreal line, and for any $n$, let and let $X_{n+1}$ be the set of hyperreal numbers ...
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1answer
88 views

Existence of Natural Numbers as an Axiom

By natural numbers $\mathbb{N}$ I understand any set satisfying Peano axioms: $0 \in \mathbb{N}$ $\sigma : \mathbb{N} \to \mathbb{N}$ $\forall n \in \mathbb{N} \; . \; \sigma(n) \neq 0$ $\forall n,m \...
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0answers
23 views

Proving the Pressing down lemma [duplicate]

I'm having troubles in finding how to use the first part to prove the second Here is the link to my question about part 1 that was already answered Prove that if: $g: \omega_1\rightarrow\omega_1$ ...
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2answers
177 views

A question about the adequacy of first-order logic for ZF to formalize most (or all) of mathematics

Zermelo-Fraenkel set theory (ZF) is a set of axioms about sets (“set” is an undefined term in ZF.) To complete the formalization of mathematics using ZF (or ZFC, see below), axioms and rules of ...
2
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1answer
44 views

Iterated forcing and non-existance of a model $N$ such that $\langle G_n : n \in \omega \rangle \in N$ and $\text{o}(N) = \text{o}(M)$.

I have encountered the following exercise in kunen's forcing chapter and i have only partially solved it. Any hint or sketch would be very helpful. Let $\mathbb{P} \in M$ be non-atomic. Let $$M = ...