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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Proving König's lemma (technical problems)

the aim of my exercise is to give a proof of the König's lemma. So, let $\kappa, \lambda$, be cardinals such that $cf(\kappa)\leq \lambda$. My professor's suggested us to prove that there exists a ...
7
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1answer
48 views

Maximal ordinal embeddable into poset of functions under eventual Domination

Consider the set of functions from $\mathbb{N}\to\mathbb{N}$. We can impose a partial order on this set by saying that $f>g$ if $f(n)>g(n)$ for all sufficiently large $n$. By a diagonalization ...
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1answer
41 views

Tarski's fixed point theorem

My professor stated the following "For every normal function $f \colon ON \to ON$ and for every ordinal $\beta$ there exist an ordinal $\gamma \ge \beta$ such that $f(\gamma)=\gamma$". (normal ...
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3answers
87 views

ZFC Axioms to be extended?

Sorry if this is going to be a really loaded question. I was told several times that for virtually all theorems/corollaries/propositions of mathematics (except those cases not compatible with ZFC ...
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1answer
52 views
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1answer
61 views

Simplify this expression of ordinal numbers: $(ω+1) ω²$ [on hold]

Let $ω$ be the ordinal of the natural numbers. Simplify $(ω+1) ω²$
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1answer
34 views

How can we show that $\omega_1$ is a regular cardinal?

A cardinal $\kappa$ is regular if and only if there is no $\lambda<\kappa$ for which there is a function $f:\lambda\rightarrow\kappa$ with range cofinal in $\kappa$. How can we see in ZFC that ...
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2answers
221 views

Non-constructive axiom of infinity

Looking at the ZFC axioms (in the Wikipedia version), the axiom of infinity stands out because it contains an extremely specific construction. This seems rather unelegant to me, therefore I've thought ...
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1answer
49 views

Examples for intensional set theories.

The normal set theory of todays mathematics (ZFC) is extensional, i.e. it has the axiom of extensionality $$\forall A \, \forall B \, ( \forall X \, (X \in A \iff X \in B) \Rightarrow A = B)$$ Are ...
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1answer
255 views

How can I prove that there is no set containing itself without using axiom of foundation?

I've already found some similar questions in here (and other sites), but in most of the case, the use of axiom of foundation is required to complete the proof. Is there any way to prove $\not\exists ...
2
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1answer
30 views

Preserve Cardinals and Adding No Bounded Subsets

In Chapter 15 at the bottom of page 228 of $\textit{Set Theory}$ by Jech, he writes that if $\kappa$ is a cardinal in $V$ and if $\kappa$ has no new bounded subsets in $V[G]$, then $\kappa$ remains a ...
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1answer
67 views

The cardinality of the set of all linear order types over $\omega$ is $2^{\aleph_0}+\aleph_1$ in ZF+AD?

In ZFC, cardinality of set of linear orders over $\omega$ is $2^{\aleph_0}$. By the argument given by here, we can prove (without the choice) the number of linear orders over $\omega$ is at least ...
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1answer
45 views

Large cardinals and $V$

I am confused by something: $\mu$ is a large cardinal if $\lambda<\mu\Rightarrow 2^{\lambda}<\mu$ and any union of less than $\mu$ sets of size less than $\mu$ is less than $\mu$. On Wikipedia ...
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0answers
41 views

Quotient Forcing in Iterations

I am trying to understand a proof of a lemma used to prove a preservation theorem for $^\omega \omega$-bounding for countable support iterations. In that quotient forcing is used to get a certain ...
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1answer
56 views

How many ways can we totally order a set $X$ (but not necessarily the whole set $X$, perhaps just a subset) up to isomorphism?

Here's Attempt #2 at asking this question. Suppose $X$ is a set (not necessarily finite), and let $T$ denote the collection of all totally ordered sets $(A,\leq)$ such that $A \subseteq X.$ Now let ...
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62 views
+50

What happens if we replace “regularity” in GCH with other conditions?

We can formulate both a weak and a strong generalized continuum hypothesis. GCH0. If $\kappa$ is an infinite cardinal number, then $2^\kappa$ is regular. GCH1. If $\kappa$ is an infinite cardinal ...
7
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0answers
45 views

Sets that are not $\infty$-Borel

I have seen a few techinques for proving that certain sets of real numbers are $\infty$-Borel (definition) but it just occurred to me that I don't know of any way to prove that a set of real numbers ...
1
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1answer
24 views

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 ...
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6answers
617 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 ...
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1answer
32 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
40 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
57 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? ...
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2answers
66 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
64 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
96 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 ...
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1answer
47 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 ...
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1answer
86 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 ...
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1answer
34 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 ...
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2answers
61 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
67 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
47 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
26 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 $\left < A_n: n \in \omega \right>$ ...
2
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1answer
66 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. ...
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2answers
120 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 ...
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0answers
52 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
85 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 ...
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3answers
125 views

Bijection between closed uncountable sets and R? [on hold]

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)?
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1answer
26 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 ...
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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
47 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.
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2answers
113 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
60 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
75 views

Existence of a regular 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
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1answer
36 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
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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 ...
4
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3answers
101 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). ...
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1answer
69 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 ...