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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1answer
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Splitting Frechet filter into two proper filters

Let $\Omega$ be a Frechet filter (=cofinite filter) on an infinite set. Do there exist proper filters $\mathcal{A}$ and $\mathcal{B}$ such that $\mathcal{A}\cap\mathcal{B} = \Omega$ and ...
3
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5answers
555 views

Question about the Continuum Hypothesis

The Continuum Hypothesis hypothesises There is no set whose cardinality is strictly between that of the integers and the real numbers. Clearly this is either true or false - there either exists ...
2
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1answer
31 views

Question concerning a statement about separability

So here is my question, Let $X$ be topological space. If every disjoint familiy of open sets is at most countable, then $X$ is separable. I am pretty sure that this is true but I still wanted to ask ...
3
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1answer
37 views

Finite-case symmetry leads to infinite-case asymmetry

Formulas for sines or cosines of sums superficially appear to have a certain symmetry, specifically it looks as if sine and cosine play something like symmetrical roles: $$ \begin{align} & ...
4
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1answer
50 views

Equivalence of Axiom of Regularity

So Axiom of regularity states: every non-empty set A contains an element that is disjoint from A I'm wondering if this is equivalent as any set is not a member of itself? If so, how do we prove it? ...
2
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2answers
64 views

In axiomatic set theory every set is a “collection” of “empty sets”?

Based on the answers of this question: How elements are defined in axiomatic set theory and this part of this book: (page 9) I will examine this reasoning in depth: Let's take a random example: ...
3
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1answer
62 views

Size of topological space depending on the size of local basis. (With elementary submodels)

Recall that the character of a topological space $\chi(X)$ is the minimum cardinal $\kappa$ such that every point in $X$ has a local basis of size $\kappa$. I need to prove that if $X$ is compact ...
8
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1answer
90 views

Can all theorems of $\sf ZFC$ about the natural numbers be proven in $\sf ZF$?

I know a proof of Hindman's theorem that uses ultrafilters on the natural numbers, and ultimately, the axiom of choice. But the theorem itself is essentially a combinatorial property of the natural ...
1
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1answer
43 views

Question about rank into rank cardinals

I do not understand why λ is smaller than $\kappa_0$ if λ is the supremum of a growing sequence that starts at $\kappa_0$ (see definition below). From cantor's attic ...
6
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1answer
79 views

Building normal filters around a stationary set

Recently I've been looking at connections between Laver functions on large cardinals and diamonds. While $\diamondsuit$-like principles tend to readily generalize to Laver function-like concepts, I've ...
1
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1answer
32 views

Defining forcing relation in base transitive model $M$

In page 177 of Set Theory for the Working Mathmatician, on chapter forcing it says: Theorem 9.2.7 For every formula $\varphi(x_1,..., x_n)$ of set theory there exists another formula ...
1
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2answers
59 views

An equivalence of AC

I have to prove the following: In $ZF^-$ the axiom of choice implies: For every set X there exist $Y \subseteq \bigcup X$ such that: Y has at most one element in common with each of X Y is maximal ...
2
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1answer
66 views

Viewing forcing as a result about countable transitive models

I think of forcing in the following context: Truth and Definability Lemmas. So forcing is a schema in the meta theory. Now in his Set Theory book (the first edition), Kunen claims that setting up the ...
3
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1answer
44 views

almost disjoint functions from $\aleph_{\omega+1}$ to $\aleph_\omega$

Is it consistent that any collection of almost-disjoint functions $\aleph_{\omega+1}$ to $\aleph_\omega$ has size at at most $\aleph_{\omega+1}$? "Almost-disjoint functions" are also called ...
2
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1answer
25 views

Equivalent definitions of P-Points (Ultrafilters)

It is stated in lots of places (including this one) that these two are equivalent definitons for an ultrafilter $u$ to be a p-point. If for every sequence $\left < A_n: n \in \omega \right>$ ...
1
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1answer
61 views

Can set-theoretic forcing exist without law of excluded middle?

Of course the law of excluded middle is accepted by almost every mathematician except a few constructivists, but then I was wondering if set-theoretic forcing can exist without law of excluded middle. ...
1
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2answers
114 views

Existence of an axiom question in relation to $\mathsf{Infinity}$

Original Post This may be a stupid question, but does there axist an axiom $\phi$ that is independent of $\mathsf{ZFC}$, and not equivalent to the axiom of $\mathsf{Infinity}$, such that ...
3
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0answers
45 views

Is it known if “Homotopy type theory” (HTT) can consistently model objects beyond V?

I read the free book on HTT but could not find an answer to this question. If that were the case HTT would be stronger than ZFC+LC (for any choice of the Large Cardinal axiom). That would be ...
2
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1answer
82 views

Well-ordering the reals: finding a certain model of $\mathsf{ZFC}$

How would one go about constructing a model $\mathfrak{M}$ of $\mathsf{ZFC}$ such that under $\mathfrak{M}$, no formula defining a well-ordering of $\mathbb{R}$ exists? I am certain such models are ...
5
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3answers
105 views

Bijection between closed uncountable sets and R?

This is probably a really stupid question but I'm hoping it's true: can we find a bijection between every closed uncountable set and R (real number line)?
0
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1answer
24 views

subset of dedekind infinite set is infinite [closed]

I got a question: Is any subset of dedekind infinite set is infinite? or if I remove a singleton set from dedekind infinite set, is the set left infinite? Can anyone give me an example of injective ...
1
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0answers
44 views

The Categories $\mathrm{Set}\times\mathrm{Set}$ and $\mathrm{Set}+\mathrm{Set}$

I'm thinking about the following problem. In $\mathrm{Cat}$ I can form the product $\mathrm{Set}\times\mathrm{Set}$. Elements are tuples, say $(A,X)$. I think that inner products and coproducts are ...
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0answers
43 views

The lexicographic order [duplicate]

If it is given ordinals $\alpha$ and $\beta$, the lexicographic order on $\alpha \times \beta$,$\leq_{lex}$ is given by: $(\gamma_0,\delta_0)<_lex(\gamma_1,\delta_1)$ if and only if either ...
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0answers
30 views

Elementary embeddings, elementary substructures,category of sets

I would like to characterize elementary embeddings AND elementary substructures in the category of sets and functions, Set. Not only characterize, but also justify this characterization.
2
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2answers
102 views

$\mathsf{ZF}$ is not finitely axiomatizable

As we know a first order theory $T$ is finitely axiomatizable if there is a finite set $F\subseteq T$ of axioms such that $F\vdash \sigma$ for every $\sigma \in T$. How we can prove if $\mathsf{ZF}$ ...
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1answer
54 views

P-generic filter [closed]

Let $M$ be a countable transitive model of $ZF$ and let $P\in M$ be a partial order then how can we see If $P$ is non atomic partial order and $G$ is a P-generic filter over $M$, then $G\notin M$. ...
5
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1answer
66 views

Existence of a real uncountable $\aleph_{\alpha}$ without $\mathsf{AC}$

Set theory (Jech) $\text{p.}\;27:$ It is an open problem whether one can prove without the axiom of choice that there exists a regular uncountable $\aleph_{\alpha}\;($the informed guess is that ...
2
votes
1answer
33 views

Can we define ordinals such that the following sentences are independent of ZFC?

Can we explicitly define two ordinals $\alpha$ and $\beta$ in the language of $\{\in\}$ such that the following hold? ZFC proves that $\alpha$ and $\beta$ exist. ZFC proves that $\beth_\beta \neq ...
4
votes
2answers
59 views

Different definitions of P-Points (ultrafilters)

Recently I have been exposed to the concept of P-Points. I have three definitions: A non-principal ultrafilter $u$ is a P-Point if: For every sequence $\left < A_n \right >_{n\in \omega}$ of ...
3
votes
3answers
97 views

How are models constructed?

As I understand, a model $\langle S,\sim\rangle$ of $\mathsf{ZFC}$ is a set $S$ coupled with some binary operation $\sim$, in which the axioms of $\mathsf{ZFC}$ hold (please correct me if I am wrong). ...
1
vote
1answer
66 views

How to extract the sets “produced” by a axiomatic set theory, into some newly introduced collection?

Say I know well how to reason in a set theory, which for the sake of this question I'll call $\bf{ST}$, say one of those, it can by $\bf ZFC$. It's principally written down via first order logic and ...
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0answers
38 views

An other question about filters and ultrafilters

Let $a$ be an ultrafilter on a set $\mho$. Let $\mathcal{L}$ be an $a$-indexed family of filters (that is $\mathcal{L}$ is a function from $a$ to the set of filters on $\mho$). Conjecture: For every ...
0
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1answer
33 views

A problem about filters and ultrafilters

Let $a$ be an ultrafilter on a set $\mho$. Let $\mathcal{L}$ be an $a$-indexed family of filters (that is $\mathcal{L}$ is a function from $a$ to the set of filters on $\mho$). Are the following two ...
3
votes
1answer
57 views

Book/Books leading up to the the axiom of choice?

I am familiar with the axioms of ZF set theory and some basic uses of them to completely formally construct more complex objects such as natural numbers etc. However I have pretty much no background ...
1
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1answer
56 views

Is the size of any set constructible from the set of natural numbers?

I like to prove that for any set there exists a set of simple graphs that have the trivial automorphism group and such that there are no homomorphisms between them. I have an idea how to prove this, ...
1
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1answer
85 views

How do we externalize an element of a model of ZFC?

Let $\mathbf{Z} =(V^\mathbf{Z},\in^\mathbf{Z})$ denote a model of ZFC (not necessarily well-founded; but, its a set). Question 0. If $\mathbf{G} \in V^\mathbf{Z}$ has the property that "$\mathbf{Z} ...
2
votes
1answer
82 views

Kunen “Set Theory” 2011 versus 1980 edition - worth buying again?

What are the differences between the original edition (1980) of Kunen's famous book and the new edition (2011)? Is the updated version worth buying? (I hope this kind of question is allowed here. I ...
1
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1answer
40 views

Are there situations outside of set theory where it would be useful if $\mathrm{ICF}$ were true?

Write $\mathrm{ICF}$ for the "injective continuum function hypothesis" i.e. the sentence of ZFC expressing that $$2^X \cong 2^Y \rightarrow X \cong Y$$ for all sets $X$ and $Y$, where $\cong$ ...
6
votes
1answer
111 views

Unions and the axiom of choice.

Is the following equivalent to the axiom of choice? Let $A = \{a_i: i \in I\}$ be a collection of pairwise-disjoint non-empty sets indexed by $I$. Similarly, let $B = \{b_i : i \in I \}$. Further ...
0
votes
2answers
114 views

Is Continuum Hypothesis false? [closed]

The continuum hypothesis states that there is no set whose cardinality is strictly between that of the integers and the real numbers. However, it seems that it is possible to construct sets that ...
0
votes
2answers
27 views

Boolean-valued model and the use of generic ultrafilter in ZFC

So I asked the question about generic filter; but I was also reading http://math.mit.edu/~tchow/forcing.pdf which is a forcing (in ZFC) guide for dummies. Then I was struck with the part where it ...
3
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2answers
57 views

How generic ultrafilter is used in forcing

So I just learned what ultrafilter is and generic filter is. As all maths are for beginners, it just looks like pure concepts, and I don't see how they are going to be applied in forcing. I am looking ...
3
votes
1answer
59 views

Maximal pruned subtree - an absolute notion?

Fix a tree $ p $ over $ \omega $. Let $ [p] $ denote the set of all branches of $ p $. Given a set of reals $ F \subseteq \omega^\omega $, let $ T(F) := \{ x \mathord{\upharpoonright} n : x \in F ...
1
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1answer
25 views

What does it mean for ultrafilter to be $\kappa$-complete?

What does it mean when ultrafilter is said to be $\kappa$-complete? I cannot find suitable Internet resource, so I am asking here.
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2answers
45 views

Proving there is a sequence convergent to a limit point of a set without axiom of countable choice?

Often, we use a construction like this: Given a subset $ A $ of a metric space and its limit point $ a $, we know that for every $ \epsilon > 0 $ there is another point $ x $ different from $ a $ ...
8
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2answers
81 views

Is there always a bijection between a universe of set theory and its ordinal numbers?

Assume ZFC (and AC in particular) as the background theory. If $(M,\in^M)$ is a model of ZFC (not necessarily transitive or standard), must there exist a bijection between $M$ and $$\{x \in M \mid ...
4
votes
1answer
63 views

What exactly is AD+ axiom and does this axiom contradict axiom of choice?

I know what axiom of determinacy is, but I am having a hard time finding out information regarding AD+. Wikipedia page seems a lot confusing and Jech's set theory book does not seem to have AD+, ...
2
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1answer
71 views

What is the weakest notion of “set” that we need, so that we can say the Yoneda lemma implies something about sets?

We set up a set theory axiomatically by fixing certain statements about "$\in$". There are many different set theories. The Yoneda lemma (the theorem about functors to Set) is a early but central ...
2
votes
1answer
47 views

Are the objects associated with large cardinals still sets?

When set theorists speak of large cardinals, are they still referring to the cardinality of some collection? If so, is this object a (hypothetical) set, a proper class, or something else?
5
votes
1answer
52 views

Ineffable Cardinals and Critical Point of Elementary Embeddings

A cardinal $\kappa$ is a ineffable if and only if for all sequences $\langle A_\alpha : \alpha < \kappa\rangle$ such that $A_\alpha \subseteq \alpha$ for all $\alpha < \kappa$, then there exists ...