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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6
votes
1answer
68 views

$n$th-power of ccc posets

We know that it is relatively consistent with $\textbf{ZFC}$ that there is a ccc poset $\mathbb{P}$ such that its cartesian square $\mathbb{P} \times \mathbb{P}$ is not ccc. Indeed, if $\mathbb{P}=T$ ...
0
votes
1answer
16 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 ...
3
votes
2answers
501 views

What are the consequences if Axiom of Infinity is negated?

What mathematics can be built with standard ZFC with Axiom of Infinity replaced with its negation? Can the analysis be built? Is there special name for "ZFC without Infinity" set theory? I also ...
1
vote
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. ...
3
votes
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 ...
3
votes
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} & ...
2
votes
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 ...
25
votes
1answer
600 views

Showing a filter on the Power set of $\mathbb{Z}$ is a one point Filter

Let $\mathcal{P}_0(X)$ the Power set of $X$ without the empty set and let $\dot{x}:=\{A\subseteq X: x \in A\}$ the one point filter generated by $x$. Furtermore let $$ \mathcal{A} := \{ f \in ...
3
votes
1answer
100 views

Size Of Proper Classes

There is a well-known hierarchy of infinite cardinalities for sets. I've heard it said that proper classes are from a certain point of view "too large" to be sets. Are some proper classes larger ...
2
votes
2answers
171 views

Zorn's lemma implies the well-ordering principle

I am little confused about the proof given here http://euclid.colorado.edu/~monkd/m6730/gradsets05.pdf On the second page, when defining $P$, the author says that $B\subset A$ and $(B,<)$ is a ...
4
votes
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
votes
2answers
104 views

A confusion on Axiom of infinity

I'm currently working "the elements of advanced mathematics" by steven g. krantz, currently on Chapter 5. I came to "Axiom of Infinity" which roughly states: $$\exists A \; s.t. \; \phi \in A \; and ...
1
vote
1answer
100 views

The comprehension axioms follows from the replacement schema.

I hope to show that the comprehension axioms follows from the replacement schema. This is a solution that professor wrote. $P(u,u)$: every set $u$, exists an unique $u$ such that $\psi(u)$. Then ...
1
vote
1answer
72 views

What is the cardinality of $X$?

Let $X=\{x\in D^\mathfrak{c}: 0<|\{\xi<\mathfrak{c}:x(\xi)=1\}|\le\omega_1\}$, where $D=\{0,1\}$. What is the cardinality of $X$? I think it is $\mathfrak c$, however I'm not sure. Also I don't ...
2
votes
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
votes
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
votes
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 ...
6
votes
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
vote
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 ...
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). ...
2
votes
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
vote
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 ...
2
votes
2answers
84 views

Truth and Definability Lemmas

I'm slightly confused about truth and definability lemmas (sometimes called forcing theorem A and forcing theorem B) of forcing. I've been using Kunen's new text and from his remarks in the matter I ...
2
votes
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 ...
1
vote
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 ...
1
vote
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
votes
1answer
177 views

Better representaion of natural numbers as sets?

Natural numbers can be represented as $0=\emptyset$ $1=\{\emptyset\}$ $2=\{\{\emptyset\}\}$ $...$ or as $0=\emptyset$ $1=\{0\}=0\cup\{0\}$ $2=\{0,1\}=1\cup\{1\}$ $...$ What are the names of ...
1
vote
2answers
56 views

Union axiom in ZF+ur(elements) - example

I am reading this: http://de.wikipedia.org/wiki/Zermelo-Fraenkel-Mengenlehre#ZF_mit_Urelementen in particular "Vereinigungsaxiom"!! and I thinking: "if $K:=\{e,g,f,\{a,b\},\{c,d\},\{z,g\}\}$, with ...
3
votes
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 ...
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 ...
3
votes
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 ...
0
votes
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 ...
2
votes
1answer
126 views

Help with definition: partition mod ultrafilter.

I do not see how the partition is well-defined. By definition $A\neq\varnothing\mbox{ mod }D\iff A\notin I_{D}$. Since D is a maximal filter $A\notin I_{D}\iff A\in D$ . So ...
3
votes
1answer
56 views

Why do these stationary subsets union to the entire set?

In proving the following theorem, I do not see why $S$ is the union of the pairwise disjoint stationary sets $S'_\eta$. It seems that for this to hold, you need every $\alpha_\xi$ to be equal to some ...
8
votes
1answer
313 views

Is the sentence “$(A,\in)\models ZFC$” absolute?

I know that we can assume that formulas are objects in $V_\omega$, and that notions such as formula and satisfiability for a standard model (when the universe is a set) are definable and absolute. ...
4
votes
0answers
78 views

Solovay Theorem

Solovay's Theorem states If $\kappa$ is regular uncountable, then any stationary set in $\kappa$ can be partitioned into $\kappa$-many pairwise disjoint stationary sets. For regular uncountable ...
2
votes
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
votes
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)?
1
vote
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 ...
0
votes
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 ...
4
votes
1answer
72 views

Other criteria for a set to be an ordinal.

I'm reading up on ordinals, and I've come across various definitions for what it means to be an ordinal. The main book that I'm following is Kunen's Set Theory book. He defines an ordinal as a set ...
3
votes
2answers
144 views

Prove the principle of mathematical induction in $\sf ZFC $

How does one prove the principle of mathematical induction using the standard axioms of $\sf ZFC $?
1
vote
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.
6
votes
1answer
177 views

Problem 24 from Chapter 1 of Kunen's Set Theory: An Introduction to Independence Proofs

Just want to make sure I'm tracking Kunen here, and hopefully the proof I have is correct. Comments / Suggestions welcome. Thanks! Problem 24. Let T be any consistent set of axioms extending ZF. ...
2
votes
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 $\mathsf{ZF}$ ...
-1
votes
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 $G\notin M$. ...
0
votes
1answer
80 views

What is the Axiom of choice [duplicate]

I was learning Set Theory for fun and I came across something called the axiom of Choice, What is the axiom of choice?
16
votes
1answer
982 views

Why is the continuum hypothesis believed to be false by the majority of modern set theorists?

A quote from Enderton: One might well question whether there is any meaningful sense in which one can say that the continuum hypothesis is either true or false for the "real" sets. Among those ...
16
votes
5answers
683 views

Alternative set theories

This is a (soft!) question for students of set theory and their teachers. OK: ZFC is the canonical set theory we all know and love. But what other, alternative set theories, should a serious student ...
8
votes
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 ...