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
156 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
16 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 ...
3
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2answers
91 views

Non-well-founded models viewing well-founded models as non-well-founded.

I'm currently thinking about how different models of set theory view each other. In particular I'm looking at how well-foundedness behaves between different models. So we have the Axiom of ...
1
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1answer
51 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 ...
6
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1answer
37 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 ...
5
votes
1answer
38 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
37 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 ...
4
votes
1answer
116 views

Basic question about encoding ZFC into PA

1) Are ZFC and PA arithmetic mutually interpretable if we extend PA to PA+A , where A is the set formulas of PA that result from the translation of the axioms of ZFC (or any large cardinal axioms ...
2
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1answer
232 views

Axiom schema of specification - Existence of intersection and set difference

I want to prove existence of intersection $x\cap y=\{z\in x| z\in y\}$ and set difference $x\setminus y=\{z\in x| \neg z\in y\}$using an axiom schema of specification. My first thought was to use ...
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5answers
1k views

Foundation for analysis without axiom of choice?

Let's say I consider the Banach–Tarski paradox unacceptable, meaning that I would rather do all my mathematics without using the axiom of choice. As my foundation, I would presumably have to use ZF, ...
4
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0answers
43 views

General question: what happens if we replace the regularity stipulation in GCH with other conditions?

I went to bed last night pondering the following. We can formulate both a weak and a strong generalized continuum hypothesis. GCH0. If $\kappa$ is an infinite cardinal number, then $2^\kappa$ is ...
5
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0answers
36 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 ...
3
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6answers
603 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 ...
6
votes
1answer
69 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 $\mathbb{P}=T$ ...
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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 ...
3
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2answers
504 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 ...
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1answer
64 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
1answer
39 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
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 ...
25
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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
101 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
173 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
54 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
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 ...
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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 ...
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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
63 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
92 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
82 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
45 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
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 $\left < A_n: n \in \omega \right>$ ...
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1answer
33 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 ...
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vote
2answers
117 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
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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
votes
1answer
178 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 ...
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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
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 ...
1
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1answer
67 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
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0answers
47 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
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1answer
25 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
79 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
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 ...
5
votes
3answers
111 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)?