# Tagged Questions

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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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? ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### (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 ...
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### 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 $y \leq x$?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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.
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### 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 ...
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 ...
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### 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 ...
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### $\Delta_0$-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 $\check y$ ...
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### $\kappa$-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 ...
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### 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?
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### 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 ...
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### 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 ...
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### 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 ...
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?
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### 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.
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### 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, ...
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### Cardinal exponentation

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