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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34 views

Why do we call “comprehension” and “regularity” to the axiom schemas in Set Theory?

I have several Set Theory books in my "shelve" but I have found the justification for the names in none. Usually all of them just state something like: "The axiom schema of ...
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2answers
21 views

Introducing a term in a first-order theory

Consider, in the first-order NGB theory of sets, the following axioms: $$\vdash\exists x\forall y(y\notin x)$$ and $$\vdash\forall y(y\notin\varnothing)$$ The second one defines the constant ...
4
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0answers
29 views

Can we associate values to elements in a poset? [duplicate]

My question comes from personal investigation. Suppose you have a poset $(X, \le_X)$. I would like to associate to all elements $x \in X$ a value $v(x) \in V$, where $(V, \le_V)$ is a totally ordered ...
3
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2answers
72 views

Countable ordinal

I haven't done ordinal algebra for a long time and I can't remember how to prove that $\omega + \omega$ is a countable ordinal. Precisely what is the bijection between $\omega$ and $\omega \cdot 2$ ? ...
2
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2answers
72 views

Isomorphisms of well ordered sets

Let $A$, $B$ be well ordered sets. If there exist order-preserving functions $f : A \to B$ and $g : B \to A$ (do not need to preserve initial segments), are $A$ and $B$ isomorphic? I know this is not ...
2
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1answer
38 views

Ordinal enumeration in ordered Mostowski model - does it not need the global choice?

In Jech's axiom of choice he proves following lemma (lemma 4.5(b) in his book): There is a injective mapping from $M$ to $\mathrm{Ord}\times \operatorname{fin}(A)$, where $A$ is a set of atoms and ...
2
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1answer
70 views

Different axiomatizations of set equality

I've seen two definitions (or axioms?) of set equality: $a=b \Leftrightarrow (\forall x : x \in a \Leftrightarrow x \in b)$ $a=b \Leftrightarrow (\forall x : a \in x \Leftrightarrow b \in x)$ That ...
3
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1answer
56 views

In what sense does forcing increase the width of a set-theoretic hierarchy?

I've often read that, whereas top extensions increase the height of the cumulative hierarchy, forcing extensions increase its width (see, e.g., the answer to this question where's it's said that ...
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2answers
94 views

uncountable well-ordered chain in $(\mathcal{P}(\mathbb{N}),\subseteq)$ without $AC$

If we assume $AC$ we can construct an uncountable well-ordered chain of subsets of $\mathbb{N}$ by well-ordering the reals and then using the same Dedekind's cut construction as in this question. ...
2
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2answers
61 views

Does the following condition implies full outer measure?

Let $X \subseteq 2^{\omega}$ be a set of positive Lebesgue measure. Suppose that for every $\eta, \nu \in 2^{<\omega}$ of the same length, the measure of $X$ above $\eta$ is the same as the measure ...
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5answers
778 views

Is the set of all topological spaces bigger than the set of all metric space?

I was wondering right that since the notion of a topology is much more general than that of a metric, and that "neighborhodness", if you will, and the concept of continuity, is generalized by the ...
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1answer
1k views

Which is the most powerful language, set theory or category theory? [closed]

As far as I know, mathematics is written based on a language which can be for example set theory or category theory. My concern is about the power of these languages. How can we realize which language ...
0
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1answer
27 views

If $R$ be a Union of zero measure sets , what is the cardinal of index set? [duplicate]

If $R$ be a Union of zero measure (lebesgue) sets , what can we say about the cardinal of index set? Does this question related to continuum hypothesis? Thanks.
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1answer
66 views

Does every diagonal intersection contain $0$?

This might even be a notational nuisance, but here it goes. Let $\kappa$ be a cardinal, $X_\alpha\subseteq\kappa$ for all $\alpha<\kappa$. As you know, the diagonal intersection of ...
1
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1answer
109 views

Subsets of the reals when the Continuum Hypothesis is assumed false

If one assumes that the continuum hypothesis is false then there are subsets of the reals of intermediate cardinality, uncountable but smaller than the continuum. What can be said about the necessary ...
0
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1answer
57 views

Ordinary Infinity and Dedekind Infinity

Does anyone know a good proof for the claim that one can add a finite number of elements to a set A, which is an ordinary infinity (take their union), and A will still be equipollent to this new set. ...
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0answers
30 views

Complete atomic boolean algebras as coalgebras of some endofunctor on Set

I was hoping to use the fact that CABAs are powersets with extra structure on the morphisms to find an endofunctor $F:\text{Set}\to\text{Set}$ with $\text{Set}^{op}\simeq\text{Coalg}F$. I started by ...
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0answers
162 views

An exercise in Fine Structure of constructible universe concerning projectum patterns

This question assumes some familiarity with Jensen's fine structure analysis of the constructible universe L (https://en.wikipedia.org/wiki/Jensen_hierarchy, ...
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1answer
76 views

How to simulate power sets in structural set theory (ETCS)?

How to simulate power sets in structural set theory (ETCS)? (nlab) It turns out that one of the primary attributes of a structural set theory is that the elements of a set have no “internal” ...
6
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1answer
103 views

Well-foundedness of cardinals and the axiom of choice

Without axiom of choice it is not generally true that the class of all cardinal (in this question we consider Scott cardinal rather than cardinals as ordinals) is not well-founded under the ordinary ...
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3answers
438 views

Why is the Axiom of Infinity necessary?

I am having trouble seeing why the Axiom of Infinity is necessary to construct an infinite set. According to a professor of who's mine teaching a class on "infinity," the Peano axioms are only ...
3
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1answer
104 views

Theorems not Formulable in Set Theory

Several sites I have been reading say that set theory is a good foundation for mathematics because virtually every theorem can be cast into a theorem in set theory. What is an example of a theorem ...
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0answers
54 views

Can a Banach measure be consistent with Friedman's Fubini-type theorem for non-measurable functions?

I read on Wikipedia that Harvey Friedman proved that the following is consistent with ZFC+¬CH: For all functions $f:[0,1]^2 \mapsto \mathbb{R}^+$ such that both $\int_0^1 \left ( \int_0^1 f(x,y) dy ...
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1answer
46 views

Recursive definitions in formal logic

A binary tree $T$ is either A single vertex, or A graph formed by taking two binary trees, adding a vertex, and adding an edge directed from the new vertex to the root of each binary tree. Suppose ...
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1answer
56 views

Algebraic closure with no nontrival automorphism

In Milne's notes on Galois theory, Chapter 7, p.91 he remarked that it is consistent without the axiom of choice that there exists an algebraic closure $L$ of $\mathbb{Q}$ with no nontrivial ...
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4answers
675 views

Continuum Hypothesis in formalized language. [closed]

The Continuum Hypothesis was advanced by Georg Cantor in 1878, before that Zermelo–Fraenkel set theory was stablished. "There is no set whose cardinality is strictly between that of the integers and ...
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1answer
43 views

Cofinality of ordinals

I have to calculate these cofinalities: $Cof({\omega_2}^{\omega_2})$=? $Cof({\omega_2}^{\omega_3})$=? $Cof({\omega_2}+{\omega_3})$=? I know that: $Cof({\omega_2}^{\omega_1})={\omega_1}$ ...
3
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1answer
61 views

How are 'primitive recursive' sets defined?

The following is taken from Inner Models and Large Cardinals by Martin Zeman (beginning of chapter 2). Let $\kappa$ be an ordinal. $U \subseteq \mathcal P(\kappa)$ is suitable iff $\kappa$ is ...
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0answers
59 views

Least $\alpha>\omega$ such that $V_{\alpha}$ satisfies certain ZF axioms

In my text i found this question: Which is the least $ \alpha > \omega$ such that $V_{\alpha}$ satisfies the axiom of power? which the least for satisfies the axiom of pairing? And the axiom of ...
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2answers
67 views

Cofinality of two ordinals

I would calculate cofinality of two ordinals: $Cof(\omega_2^{\omega_1})$ $Cof(\aleph_{\omega^\omega})$ I know that a $cof(X) < X$, but i cannot solve it..
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0answers
49 views

Inclusions between $H(\omega), V_\alpha$ and $L_\alpha$

Can you give me advices to understand the inclusions between set Universes $H(\omega), V_\alpha$ and $L_\alpha$? For example, i ve this question: $$ H(\omega) \subset L_\omega? $$ So, I know that ...
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1answer
76 views

Proof that the minimal inductive set does not contain any more elements than $\{\emptyset, s(\emptyset), s(s(\emptyset)),s(s(s(\emptyset))),…\}$?

I am studying axiomatic (ZF) set theory and in particular the construction of the natural numbers from axiomatic set theory. Let me start by giving some introduction to my question. The axiom of ...
0
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1answer
45 views

iterative conception of set --> axiom of regularity

In the book Mathematical Logic by Joseph R. Shoenfield, he describes the iterative conception of set/cumulative hierarchy. The explanation is posted below and marked with a *. The author explains ...
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2answers
92 views

Do fett ordinals exist?

Define a class $K$ of ordinals inductively as follows: $0=\emptyset\in K$. For all $\alpha\in K$, the succesor of $\alpha$ is also an element of $K$. For every function $f\colon \mathbb N\to K$, the ...
3
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1answer
49 views

Least fixed point of restricted function

Let $P$ be a poset with the property that every order-preserving map $f:P\to P$ has a least fixed point $\mu(f)$. Now for any $p\in P$, the poset $\downarrow(p)=\{x\in P|x\leq p\}$ must also have ...
7
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0answers
111 views

What can the reals of an inner model be?

This is probably a silly question. Call a set of reals $X$ a constructibility ideal (in analogy with a Turing ideal) if $X$ is closed under effective join $r\oplus s: n\mapsto 2^{r(n)}3^{s(n)}$ and ...
2
votes
1answer
102 views

Can a large $V_\alpha$ satisfy Comparability but not be well-orderable?

Say that a set satisfies Comparability if any two of its subsets are comparable: one is injectable into the other. Are there models of ZF containing ranks $V_\alpha$ which satisfy Comparability but ...
1
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2answers
98 views

Foundations of Mathematics [duplicate]

For a long time I've felt that the search for foundations in mathematics doesn't have an answer, that it is almost like physics where you can keep asking where things come from. However, is this not ...
6
votes
1answer
81 views

Are two generic filters in a common generic extension?

Let $M$ be a countable transitive set. Suppose $\mathbb{P}$ is a forcing in $M$. Let $G$ and $H$ be two generic filters for $\mathbb{P}$ over $M$. My questions are: Is there a forcing $\mathbb{Q}$ ...
3
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1answer
62 views

How can it be seen that the Von Neumann universe $V_{\omega+\omega}$ does not model the Fraenkel axiom

Consider the Von Neumann universe $V_{\omega+\omega}$. As mentioned on the Wikipedia page on Von Neumann universes, $(V_{\omega+\omega},\in)$ is a model for $\rm Z$, but not for the Fraenkel axiom of ...
3
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3answers
108 views

Why do we want the Axiom of the Power Set?

I'm just learning a bit about axiomatic set theory, and I'm kind of confused as to why we need/want this axiom? Does not accepting it imply that there exists some set which doesn't have a power set? ...
6
votes
1answer
81 views

How can one prove the axiom of collection in ZFC without using the axiom of foundation?

Say I want to prove the axiom(s) of collection from the axiom(s) of replacement. If you have the axiom of foundation, then you can use Scott's trick to do this. But suppose I'm working in a context ...
3
votes
1answer
84 views

Class models of $\mathsf{ZFC}$ and consistency results

First of all, I'm only starting to study independence results in set theory. And there is one obstacle that confuses me a lot. Probably such questions have already been asked, but I haven't found ...
8
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3answers
875 views

The smallest infinity and the axiom of choice

The short version of this question is: which (natural) axiom should be added to ZF so that the statement "$\aleph_{0}$ is the smallest infinity" becomes true? A set $A$ is called infinite if it can ...
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0answers
50 views

$|A^2|=|A|$ for every infinite $A$ iff Axiom of Choice holds. [duplicate]

I've seen this assertion in a few comments around the site, and I found the answer to the $\rightarrow$ implication here. Does anyone know a (hopefully simple) proof of the $\leftarrow$ implication?
2
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1answer
45 views

Assuming CH, can every tower be extended to a selective ultrafilter (or even a p-point)?

Assume CH. A tower is an almost decreasing family $(A_\alpha)_{\alpha\in\omega_1}$ with no pseudointersection. A selective (also called Ramsey) ultrafilter is one with the property that for every ...
1
vote
1answer
96 views

What is the first order formulation of Zorn's lemma in the language of set theory?

Very often in notes of courses in set theory you find the assertion that in ZF the Axiom of choice (AC) is equivalent to Zorn's Lemma (ZL) (which is equivalent to Well Ordering Principle which it ...
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0answers
49 views

Cardinality of the union and product of two sets without AC

We have the following results Let $A$ and $B$ be infinite sets s.t. $|A|=|B|$, then $|A\cup B|=|A|$. I was wondering if we can prove that without the Axiom of Choice or without using cardinal ...
8
votes
1answer
115 views

How can the Hadwiger–Nelson problem depend on the axioms of set theory?

The wikipedia page on the Hadwiger Nelson problem says the following two things: The correct value may actually depend on the choice of axioms for set theory. and the problem is equivalent ...
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0answers
53 views

Difficulty with exercise 24.12 in Jech

Exercise 24.12 in Jech's Set Theory (3d Millenium Edition): The lexicographical ordering $\omega \times {\omega}_1$ does not have true cofinality. True cofinality is defined (p.461) as the least ...