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
184 views

The collection of continuous functions between any 2 compact Hausdorff spaces forms a set

I would like to show precisely what I have stated in the title (assuming that it is correct; I have reason to suspect it is, thanks to a tricky past exam paper I'm trying to surmount); namely, that ...
27
votes
4answers
1k views

Is Banach-Alaoglu equivalent to AC?

The Banach-Alaoglu theorem is well-known. It states that the closed unit ball in the dual space of a normed space is $\text{wk}^*$-compact. The proof relies heavily on Tychonoff's theorem. As I have ...
0
votes
1answer
397 views

the sup of a set of ordinals

Let $A$ a set of ordinals. We know that $\sup A:=\bigcup A$ is an ordinal. Frequently, in proofs, one use that it is a limit ordinal. I would want to know when it is. To show that it is limit : let ...
2
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1answer
191 views

Equivalences to “D-finite = finite”

By a D-finite set, we mean a set admitting no injection from the natural numbers (or equivalently, a set not in bijection with any proper subset). I have encountered a proof that the following are ...
5
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1answer
537 views

Do sets whose power sets have the same cardinality, have the same cardinality? [duplicate]

Possible Duplicate: power set cardinal equality Let $X$ and $Y$ be sets, and suppose that $|\mathscr{P}(X)| = |\mathscr{P}(Y)|$ (where $\mathscr{P}$ denotes the power set). Does it follow ...
2
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3answers
514 views

Axiom of subsets and Russell's paradox

Russell, with his paradox, proves that the set $\{x:x\notin x\}$ of all sets that are not members of themselves doesn't exist. So, he demonstrates that the set $\{x:p(x)\}$ doesn't exist necessary (it ...
8
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1answer
276 views

Definable order types without infinity axiom.

Denote by $ZF^\times$ the theory of $ZF$ without the axiom of infinity. We know that $V_\omega$, the set of all hereditarily finite sets in a model of $ZF$, is a model of $ZF^\times$. We further know ...
14
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1answer
262 views

Are sets constructed using only ZF measurable using ZFC?

Suppose $S$ is a subset of $\mathbb{R}$ which can be defined without using the axiom of choice, i.e. which can be proved to exist using only the axioms of ZF. Does it follow that $S$ is measurable? ...
2
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1answer
214 views

proof of diamond

I have many questions about page 2 of this paper http://www.cs.elte.hu/~kope/ss3.pdf. First, on the top, I try to prove that, if $cf(\delta)\neq\kappa$, then we can choose the $\alpha, \beta$ in an ...
2
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1answer
234 views

consistency of large cardinal axiom

It is known that if ZFC is consistent, ZFC+"no such large cardinal exists" is consistent. Then, is ZFC+"such large cardinal exists" known to be consistent? This would imply that proving large cardinal ...
11
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1answer
1k views

Axiom of choice and calculus

I thought many results in calculus need axiom choice. For example, I thought one needs AC to prove that a bounded sequence in the real line has a convergent subsequence. Recently I was taught that one ...
5
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1answer
1k views

Proving the pairing axiom from the rest of ZF

In ZF, the pairing axiom states that for every $x,y$ there exists the set $\{x, y\}$. Wikipedia also tells us we can dispense this axiom: This axiom is part of Z, but is redundant in ZF because it ...
2
votes
3answers
2k views

Using setminus notation with set elements

The "correct" way to write a set without a specific element is as follows: $S \setminus \{s\}$ But in some contexts this is cumbersome to write/type or read, and it detracts from the flow of the ...
21
votes
2answers
2k views

Cardinality of a Hamel basis

What is the cardinality of a Hamel basis of $\ell_1(\mathbb{R})$? Is it deducible in ZFC that it is seemingly continuum? Does it follow from this that each Banach space of density $\leqslant ...
1
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1answer
239 views

Formal Definition/counter part in mathematics for “Objects” of Object Oriented Models [closed]

I'm a newbie in both formal mathematics and theoretical computer science, so please bear with me if you find my question is not properly framed. Object Oriented Modeling seems very useful in defining ...
1
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1answer
185 views

Can PA+Con(ZFC), prove everything ZFC can?

Can PA+Con(ZFC) prove every theorem of ZFC?
31
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9answers
3k views

Where to begin with foundations of mathematics

I would like to know more about the foundations of mathematics, but I can't really figure out where it all starts. If I look in a book on axiomatic set theory, then it seems to be assumed that one ...
4
votes
1answer
268 views

What is the smallest possible value of $\omega_1$ in $\mathrm{ZF}$?

It is consistent with $\mathrm{ZF}$ that a countable union of countable sets may be uncountable. As far as I understand it, this is because in absence of $\mathrm{AC}$ we cannot necessarily choose a ...
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2answers
76 views

A decomposition of an ordinal

Again http://www.cs.elte.hu/~kope/ss3.pdf . After the Remark 2, I have some problem to prove that there exists a increasing decomposition of $\delta<\lambda^+$ ($\delta$ ordinal ? or cardinal ?) ...
3
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2answers
154 views

How to interpret $1 \to 0$ in ${\bf Set}^\mathrm{op}$, and ${\bf Set}^\mathrm{op}$ itself?

How to interpret the morphism $1 \to 0$ in ${\bf Set}^\mathrm{op}$, dual to $\bf Set$, with the standard meanings of the initial and terminal objects? Since the objects have the same interpretation ...
5
votes
3answers
359 views

Is there a branch of mathematics that requires the existence of sets that contain themselves?

I notice that Russell's paradox, Burali-Forti's paradox, and even Cantor's paradox, all depend on our tolerance of sets that contain themselves (at one level of depth or another). Thus, I was thinking ...
2
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2answers
524 views

Paradox of General Comprehension in Set Theory, other than Russell's Paradox

As is well known, the General Set Comprehension Principle (any class is a set) leads to the Russell Paradox (the class $x \notin x$ cannot be a set). As a result, set theories must restrict the ...
3
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1answer
334 views

Given an injection $\mathbb{N}\to\mathcal{P}(X)$, how can we construct a surjection $X\to\mathbb{N}$?

I goofed on my earlier post, here Given a surjection $f:\mathbb{N}\to\mathcal{P}(X)$, how can one construct an injection $X\to\mathbb{N}$? I am trying to show that there is an injection ...
1
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1answer
174 views

Given a surjection $f:\mathbb{N}\to\mathcal{P}(X)$, how can one construct an injection $X\to\mathbb{N}$?

I'm proving that $|X|\leq^*\mathbb{N}$ iff $\mathbb{N}\leq\mathcal{P}(X)$. One direction is trivial, but I'm not sure how to proceed for the other direction. Any suggestions? Update My apologies. ...
33
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7answers
9k views

Difference between a class and a set

I know what a set is. I have no idea what a class is. As best as I can make out, every set is also a class, but a class can be "larger" than any set. (A so-called "proper class".) This obviously ...
7
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1answer
391 views

Epsilon induction

I recently came upon this technique called epsilon induction, and was searching for some proof using the same. But I saw no such proof. Does someone know of any proof using this technique?
1
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1answer
82 views

Maximal set of pairwise disjoint elements of a dense subset.

Let $B$ be a complete Boolean algebra, and suppose $D \subseteq B$ is a dense subset. How is it possible to construct a maximal set of pairwise disjoint elements of $D$? Is it true that $\sum S = \sum ...
-4
votes
1answer
669 views

A counter-example for a set-theoretic problem?

I have proved the below conjecture for the special cases $n\in\{0,1,2\}$. The cases $n\ge 3$ (finite and infinite) are unknown. If the following conjecture is true, I don't expect that you will be ...
0
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1answer
42 views

(final) episinks preserved under pullbacks in SET

How do I prove the following: We work in SET. If $(f_{i}:Y_{i} \rightarrow Y)_{i \in I}$ is a (final) episink, $f: X \rightarrow Y$, and $h_{i}: X_{i} \rightarrow Y_{i}$ are functions such that for ...
3
votes
2answers
167 views

Existence of a particular well-ordering of [0,1]

How do you show, assuming the Axiom of Choice and the Continuum Hypothesis, that there exists a well-ordering on $[0,1]$ such that for all $x$, there are only countably many $y$ such that $y \leq x$?
4
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3answers
116 views

Can the truth value of an independent property be changed at will by enlarging the model?

Let $\phi$ be a property that's independent of $ZFC$, so that there are strcutures ${\mathfrak A}=(A,{\in}_A)$ (where $A$ is a set or class and ${\in}_A$ is a binary relation on $A$) that are models ...
2
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2answers
1k views

set-theoretic function definition; recursion theorem

I am an undergraduate student, currently studying axiomatic set theory (I am reading Halmos' Naive Set Theory as an overview, and consulting other sources recommended to me to supplement the sparser ...
7
votes
1answer
206 views

Shelah's proof of diamond

This paper is http://www.cs.elte.hu/~kope/ss3.pdf . In Remark 1 : I want to prove that the set $$D=\{\delta<\lambda^+:\pi\restriction \delta \text{ is a bijection onto }\lambda\times\delta\}$$ is ...
2
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2answers
68 views

Elementary Question

Let be $f:X\to X$ a bijection, an $A\subset X$ a invariant subset of $X$, i.e $f(A)\subset A.$ How can see that $$f(A)=A$$ I'm trying to show that $$f(A^{c})\subset A^c$$ but I can not.
0
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1answer
89 views

enumeration of subsets

Good evening. Let $\lambda>\omega$ a cardinal. We know that there is a bijection $\pi$ between $\lambda^+$ and $\lambda\times\lambda^+$. I don't understand in Remark 1 of the paper Shelah's proof ...
5
votes
2answers
274 views

Analytic versus Analytical Sets

Browsing MathOverflow I came across a question about analytical sets. Through the discussion following a comment made by our very own Asaf, I learned that bold face $\mathbf{\Sigma^1_1}$ and light ...
5
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1answer
227 views

How do we justify functions on the Ordinals

In ZFC there seem to be two ways to define a function, as an ordered triple of domain, codomain and the set of ordered pairs that make its graph, or a relation over the two sets. Whichever we use ...
1
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0answers
67 views

$\Delta_0$-formulas in the Boolean-valued model $V^B$

I want to show that $\Vert \check x = \check y \Vert = 1$ implies $x = y$, where $\check x$ is the canonical name for $x$ in $V^B$. I'd like try induction over the ranks of $\check x$ and $\check y$ ...
4
votes
1answer
251 views

$\kappa$-Suslin (Aronszajn, Kurepa) subtree of the complete binary tree, $2^{\lt \kappa}$

This is from chapter 2 of Kunen, Set theory: an introduction to independence proofs. Given $\kappa$ a regular cardinal, and the existence of a $\kappa$-Suslin (Aronszajn, Kurepa) tree, show that ...
8
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2answers
2k views

Difference between ZFC & NBG

Can someone tell me what's the advantages and disadvantages of using NBG rather than ZFC and what's the advantages and disadvantages of using ZFC rather than NBG?
2
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1answer
130 views

Question about Cuts in Boolean Algebras

Let $A$ be a Boolean algebra, and let $A^+$ denote the set of non-zero elements of $A$. A cut $U \subseteq A^+$ is a set such that if $q\in U$, then $p\le q \implies p \in U$, for all $p \in A^+$. A ...
3
votes
1answer
282 views

Provability and truth

The following is quoted from Set Theory and the Continuum Problem by Raymond M. Smullyan and Melvin Fitting. So far, no attempts have been the slightest bit successful in determining whether the ...
5
votes
1answer
980 views

Union of ordinals

From Jech (pg. 20): If $X$ is a nonempty set of Ordinals, than $\bigcup X$ is an ordinal. This should be easy enough to prove but I don't see how; also, I guess this is not the case if $X$ is a ...
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2answers
466 views

Example of total order with some properties that is not well ordered

Is there an example of a total order with properties there is a least element and every element has a (unique) successor not is not also a well ordering?
5
votes
1answer
208 views

Cardinal power $\kappa^\kappa$. When is it equal to $2^\kappa$?

Under what assumptions on an infinite cardinal $\kappa$ we have $$\kappa^\kappa= 2^\kappa?$$ Please delete this question. I know the answer.
2
votes
1answer
241 views

Forcing questions

I have been looking at a proof of a technical forcing lemma, and I have a couple of questions. Here is the setup: $(N, \epsilon) \prec (\mathbf{H}( \chi ), P, \epsilon ) $ is a countable submodel, ...
2
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1answer
154 views

Cardinal exponentation

Let $\lambda, \kappa$ be infinite cardinals. Does it follow from the inequality $\kappa\lt2^\lambda$ that $2^\kappa\lt2^{2^\lambda}$?
2
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1answer
159 views

Is the notation for the definition of a set stricly formalized?

In the 800 pages set theory book by Jech, he uncommented starts using $$Y=\{ u\in X : \phi(u)\}$$ as equivalent to $$Y=\{ u:u\in X \wedge\phi(u)\}$$ on the first few pages. The fact that, in the ...
1
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1answer
274 views

Do the proofes in set theory rely on the semantics of the formulas used in the axioms?

Motivation: The Axiom of separation $$\forall w_1,\ldots,w_n \, \forall A \, \exists B \, \forall x \, ( x \in B \Leftrightarrow [ x \in A \wedge \phi(x, w_1, \ldots, w_n, A) ] )$$ is used to ...
3
votes
1answer
212 views

Measurable cardinal existence condition

We know that a cardinal $k > \omega$ is measurable if there is a measure function $\mu :2^k\mapsto \{0, 1\}$ that satisfies the following 3 conditions: 1.$<k$ -additive: for every set I of ...