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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What is forcing isomorphism? [closed]

This question is from Kunen's set theory book. My questions are: What is the definition of isomorphism between forcing notions? When do we say that two forcing notions $\mathbb{P},\mathbb{Q}$ ...
4
votes
1answer
65 views

Countable cartesian product and Axiom of Choice

In the A taste of Topology book, when talking about Cartesian product $\prod\{S:S\in\mathcal{S}\}$, the author writes the following: It is straightforward that ...
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0answers
77 views

Minimal Possible Ordinal for a Cardinal

For a given $\aleph_\alpha$, what is the minimal ordinal possible such that $|\beta| = \aleph_\alpha$? More precisely, assume we have a model $N$ of ZFC. $N$ could be an extension of many different ...
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1answer
56 views

If $M[G] \subseteq M[H]$ are forcing extensions, why is $M[H]$ a generic extension of $M[G]$?

I know that, wlog, we may assume $G = H \cap A$ for some complete subalgebra $A$ of the complete Boolean algebra $B$ over which $H$ is $M$-generic.
4
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1answer
55 views

Can $V$ only have well-orderings definable with respect to a parameter?

In this answer, Professor Hamkins gives a proof that for models $M$ of ZF, $M$ being a model of $\text{ZFC} + V = \text{HOD}$ is equivalent to there being a definable well-ordering of the universe: ...
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0answers
59 views

Can this set exists $\{\{\{\ldots\}\}\}$? [duplicate]

I know from the axiom of regularity that the set $\{\{\{\ldots\}\}\}$ cannot exist in ZFC. Is there another set theory where it's possible ? If yes, how would we construct it ?
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1answer
50 views

In NBG, can a set contain a proper class?

For example, I would want to consider the set of all classes Xi (for i a natural number), where Xi is the class of all sets of cardinality i.
5
votes
1answer
113 views

On the order of natural functions {f:N→N}

Define a partial order on natural-valued functions (or sequences, depends on how you see it): $f<g$ iff $\exists x:\forall n(n>x\rightarrow f(n)<g(n))$. Intuitively, $f<g$ if $g$ ...
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2answers
90 views

Definition of “set” in HoTT

In the homotopy type theory book (https://hott.github.io/book/nightly/hott-online-1007-ga1d0d9d.pdf) "set" is defined as follows (see chapter 3): Definition 3.1.1. A type A is a set if for all x, y : ...
2
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2answers
106 views

Can every non-empty set satisfying the axioms of $\sf{ZF}$ be totally ordered?

Let us first propose the following axiom, Axiom of Ordering $(\sf{AO})$. If $S$ be a non-empty set. Let $a,b\in S$. Then the set $\{a,b\}$ can be totally ordered. Now let us consider $\sf{ZFO}$ ...
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1answer
52 views

Grothendieck Universes: Atoms

I'm very much beginner in set theory; my apologize for mistakes. Grothendieck universe: $$\forall X(X\in \mathcal{U}\Rightarrow\mathcal{P}(X)\in U\}$$ $$\forall X\forall x(x\in ...
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2answers
177 views

Can every set be expressed as the union of a chain of sets of lesser cardinality?

If a set $S$ has countably many elements $\{x_n\}$, it can be expressed as a union of a chain of finite sets $$ \{x_1\} \subset \{x_1,x_2\} \subset $$ But what about a set of arbitrary cardinality ...
0
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1answer
52 views

Is union of a countable set of Tarski-finite sets countable in ZF? Reference request.

This is a reference request on the consistency, or not, with ZF, of "There exists a countable set $S$ of Tarski-finite sets such that $\cup S$ is not countable." Only this....NOT about AC, Dependent ...
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vote
2answers
91 views

Are there axioms that imply ZFC?

The question is simple: do we know if there are non-trivial axiom collections stronger than (imply but are not implied by) $ZFC$? To clarify what I mean: Do we know a way of replacing the axioms in ...
3
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1answer
95 views

Where do models of set theory live?

When we are studying independence proofs we are dealing with statements of the form $Cons(T)\rightarrow Cons(T')$ where $T$ and $T'$ are first order theories; commonly $T$ and $T'$ are subtheories of ...
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2answers
91 views

Troubles with a proof from Kunen's book

I'm studying Godel's constructible universe and those statements that are true inside it: specifically $L\vDash\lozenge$. In this regard, I need to understand the following Lemma from Kunnen's book ...
3
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2answers
60 views

What's the definition of a “collection”? [duplicate]

I cannot seem to find a formal definition for the following. What's a "collection" in the context of set theory?
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0answers
47 views

Notation for inhabited sets

A set $X$ is called inhabited if it has some element. In classical mathematics, this means that it is not the empty set, so that one usually writes $X \neq \emptyset$. However, in intuitionistic ...
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0answers
50 views

Proof: Every lattice has a maximal filter iff AC

I'm working through a proof of Herrlich's book Axiom of Choice, p.58 (Google books): Equivalent are Every lattice has a maximal filter. Axiom of Choice. In this book, a lattice is ...
5
votes
1answer
86 views

There is no Baire bijection between $\mathbb R$ and the set of functions $\mathbb Z\to\mathbb R$ modulo shifts

Let $X$ denote the set $\mathbb{R}^\mathbb{Z}$ (the set of all functions from integers to reals), and $\sim$ the equivalence relation on $X$ defined by: $f \sim g$ iff there is a $z \in \mathbb{Z}$ ...
1
vote
1answer
54 views

Does there exist a $\in$-decreaaing sequence of sets?

Axiom of infinity asserts that there exists a sequence of sets $\{a_n\}$ such that $a_n\in a_{n+1}$. An example is the set $\omega$. I'm curious whether there exist a sequence of sets $\{a_n\}$ such ...
3
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1answer
75 views

Definable non-computable number which contain no information

We have three types of numbers AFAIK: a) Computable b) Definable and non-computable, but contains information about Halting of some turing machines, extractable in a computable way if you were given ...
6
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1answer
73 views

Question about collapsing cardinals

Suppose, in $M$, $\kappa$ regular, $\lambda>\kappa$ regular. Is there a generic extension of $M$ in which $\kappa^+ = \lambda$ and in which cardinals $\leq \kappa$ and $\geq \lambda$ are preserved? ...
4
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3answers
80 views

Is there is a real $r$ and a countable transitive model $M$ such that $r$ is not in any forcing extension of $M$?

It is a theorem that if $M[G]$ is a generic extension $M$, then for every model $N$ of ZFC with $M \subset N \subset M[G]$, $\ N$ is some generic extension of $M$ (and is, in fact, $M[G\cap D]$ for ...
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2answers
271 views

Where is the flaw in my Continuum Hypothesis Proof?

I am not a mathematician, but rather a computer engineer with a curious mind. The continuum hypothesis (CH) has gripped my attention today, and I even asked a question about it earlier today. ...
2
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1answer
83 views

Is this interpretation of the continuum hypothesis correct?

I am not a mathematician, but rather a computer engineer with a curious mind. The continuum hypothesis states (I believe) that there does not exist a set $S$ such that $\aleph_0 < |S| < ...
2
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2answers
86 views

For transitive model $M$, is $L^M$ really a model of ZFC?

There are two concepts of "models" in set theory: one says that for each axiom $\phi$ of ZFC, $\phi^M$ holds. Remaining one just says that $M\models \mathsf{ZFC}$. First one is a schema rather a ...
2
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1answer
61 views

Definition of the forcing relation in class-forcing

Jech really glosses over class forcing. I cannot find a good reference online. I have two questions about it. 1) Jech says, "As for the forcing relation in general, we cannot generally define ...
2
votes
3answers
65 views

Does ZF $\vdash$ Con(ZF) $\rightarrow$ Con(ZF+$\lnot$AC)?

Certainly ZFC $\vdash$ Con(ZF) $\rightarrow$ Con(ZF+$\lnot$AC), but the usual forcing argument to construct a model of ZF+$\lnot$AC seems to require choice to find a generic filter.
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3answers
196 views

Existence of mathematical objets constructed using the axiom of choice

Let consider the Vitali set $V \subset \mathbb R$, which is constructed using the axiom of choice. (I could take any other mathematical "object" that can be constructed using the axiom of choice, but ...
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2answers
633 views

The set that only contains itself

Ignoring the axiom of regularity (and therefore the implication of "no set can contain itself"), would it be correct to state that the set that contains only itself is unique? My argument is that if ...
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0answers
82 views

Can you define a sensible probability measure on the set of countable transitive models of ZFC?

It is well known that the set of hereditarily countable sets $H(\omega_1)$ —or, if you prefer, $H_{\omega_1}$— has cardinality $2^{\aleph_0}$, and I understand that every countable ...
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2answers
106 views

Does the law of the excluded middle imply the existence of “intangibles”?

First off, I'm not sure if "intangible" is standard terminology, Wikipedia defines an intangible object to be: "objects that are proved to exist, but which cannot be explicitly constructed". So if ...
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3answers
332 views

Are the natural numbers implicit in the construction of first-order logic? If so, why is this acceptable?

I have recently been reading about first-order logic and set theory. I have seen standard set theory axioms (say ZFC) formally constructed in first-order logic, where first-order logic is used as an ...
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vote
1answer
41 views

Axiom Of Choice to create a sequence of right inverses

I want to construct a sort of sequence of right inverses. My question is whether the construction uses the Axiom Of Choice correctly. Suppose I have a sequence of surjective functions $$ ...
3
votes
1answer
73 views

Is there a standard set of axioms of set theory in which set complements exist?

If $S$ is a set, let $S^c = \{ x | x \notin S \}$ be the complement of $S$. In ZFC, if $S$ is any set, then $S^c$ doesn't exist because $S \cup S^c$ would be the universal set, which does not exist ...
4
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1answer
115 views

Model of homotopy type theory in ZFC

There is a model of ZFC in homotopy type theory Does exist a model of homotopy type theory in ZFC? Is there a proof of "equal logical expressivity" of these theories? p.s. I use word "model" in ...
2
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0answers
32 views

“Nested independence” of $\mathsf{ZFC}$? [duplicate]

I have a question about undecidable statements in $\mathsf{ZFC}$. I know there are true statements like: $$X\text{ is independent of }\mathsf{ZFC}.$$ But is it also possible that such a statement ...
8
votes
2answers
112 views

Is there a forcing extension $M[G]$ of $M$ that adds a new $\omega$-sized subset to $\omega_2$ without adding any new subsets of $\omega$?

I should add that the forcing extension must preserve the cardinals $\aleph_1$ and $\aleph_2$. Note that such a forcing extension cannot add any new $\omega$-sized subsets to $\omega_1$, and also ...
0
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1answer
33 views

Why are concretizable categories locally small?

I have seen it mentioned in a few places that concrete (or concretizable) categories are locally small, but never seen any proof. Is it particularly trivial? If not, does anybody have some reference ...
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votes
2answers
162 views

What's an example of a vector space that doesn't have a basis if we don't accept Choice?

I've read that the fact that all vector spaces have a basis is dependent on the axiom of choice, I'd like to see an example of a vector space that doesn't have a basis if we don't accept AoC. I'm ...
3
votes
2answers
70 views

What is the axiom of quasideterminacy?

This is something mentioned in the "See also" of the wikipedia page for the axiom of determinacy, but when you click on it it takes you to the page for "Determinacy" and the section for ...
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0answers
38 views

Is every topos equivalent to a full subtopos of U-small objects in another topos?

Let $\mathcal{E}$ be a topos, and $\mathcal{U}$ an $\mathcal{E}$-universe (as discussed on this nLab page). Let $\mathcal{E}_\mathcal{U}$ be the full subcategory of $\mathcal{U}$-small objects in ...
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votes
1answer
168 views

Can all Power Sets be Limit Cardinals?

Is it possible to create a model of ZFC, so that the cardinality of each power set is a limit cardinal (as opposed to GCH where they are always successor cardinals)? Take for example the following ...
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vote
1answer
57 views

Weird partitions of the real line

Partition $\mathbb{R}$ into subsets $S_r$ indexed by $r\in\mathbb{R}$ with the following property: For all $r₁, r_2\in\mathbb{R}$: $$\forall x\in S_{r_1}\forall y\in S_{r_2}:\; x+y\in S_{r_1+r_2}$$ ...
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0answers
38 views

Set-theoretical problems of regarding Grothendieck topologies as presheaves

I apologise for the vagueness of this question - it's just something I was idly wondering about. Let $(\mathcal{C},J)$ be a site with $\mathcal{C}$ small. The axioms for the Grothendieck topology $J$ ...
5
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1answer
76 views

Is it consistent that every set is the countable union of sets with smaller cardinality, or is it just alephs?

(note: in what follows by "consistent" I mean "consistent relative to large cardinals") My question regards the exact statement of result which Gitik has proven in his paper "All Uncountable ...
3
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0answers
78 views

Examples of toposes in which the Axiom of Determinacy holds.

I just stumbled upon the Axiom of Determinacy which is an axiom in set theory - inconsistent with the Axiom of Choice, consistent with the Axiom of Dependent Choice, that states that for every subset ...
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1answer
173 views

Cardinality of power sets decides all of cardinal arithmetic?

Assuming ZFC, is it possible to have two models which agree on the cardinality of all the power sets, but disagree on the cardinality of some other cardinal exponentiation (meaning that they agree on ...
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1answer
92 views

Description of the universe of sets

Here the mathematician Caicedo recommended an article in the book Mathematical Logic where Shoenfield describes that the universe of sets is devided into stages. The first stage contains nothing. The ...