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
414 views

Is there a way to define the “size” of an infinite set that takes into account “intuitive” differences between sets?

The usual way to define the "size" of an infinite set is through cardinality, so that e.g. the sets $\{1, 2, 3, 4, \ldots\}$ and $\{0, 1, 2, 3, 4, \ldots\}$ have the same cardinality. However, is this ...
-6
votes
1answer
256 views

Order of products and order of multipliers [closed]

I asked this question (and have received an answer) at MathOverflow. Now a little more difficult question: Let $f$ and $g$ are binary relations (on some set $\mho$). The function $f\times^{C} g$ is ...
5
votes
1answer
303 views

Construction of special $\omega_1$-Aronszajn tree

Problem from Kunen II.40: The definition is the following: An $\omega_1$-Aronszajn tree $T$ called special iff $T$ is the union of $\omega$ antichains. Need to prove that $T$ is special iff there is ...
0
votes
1answer
73 views

From an equality to a comparison

Let $X$ be a set. Let $0$ be an element of $X$. For a function $P$ defined on tuples of $n$ elements of the set $X$ we know (for every tuples $f$ and $g$ each having $n$ elements) $$\forall i \in n : ...
2
votes
1answer
271 views

Are lexicographic and multiset the only extensions of sets that preserve well-foundedness?

It is well known that for a given set $S$ with well-founded order relation $R$, the lexicographic order that extends $R$ on tuples of $S$ is also well-founded. Also, the multiset order on the ...
4
votes
1answer
134 views

On $T_2$, first countable, countably compact space

As we know, For every $T_2$, first countable, compact space, its cardinality is not more than $2^\omega$. (See chapter 3 of Engelking's book.) However, I want to know whether the result is ...
2
votes
2answers
204 views

Is there more than one infinitessimal among the hyperreal numbers

Take $\mathbb{H}=\mathbb{R}^\mathbb{N}/\mathcal{U}$, where $\mathcal{U}$ is some ultrafilter. Questions: Are there more than one independent infinitessimal in this field. This means $\epsilon_1 > ...
5
votes
2answers
549 views

In set theory, what does the symbol $\mathfrak d$ mean?

What's meaning of this symbol in set theory as following, which seems like $b$? I know the symbol such as $\omega$, $\omega_1$, and so on, however, what does it denote in the lemma? Thanks ...
1
vote
2answers
120 views

On Countable chain condition

If the topological space $X$ has CCC (= countable chain condition ) with given a countable closed discreted subspace $Y$ of $X$, could we seperate the points in $Y$ by countable disjoint open sets in ...
2
votes
2answers
111 views

A counter-example of a set containing a club which is not a club

I wanted to prove that if $C$ is a club (in a $\kappa$ cardinal) and $C\subseteq C'$ then $C'$ is also a club (that is if $C$ is of "mesure" $1$ then $C'$ is too). It is easy to see that $C'$ is ...
4
votes
1answer
305 views

A question concerning on the axiom of choice and Cauchy functional equation

The Cauchy functional equation: $$f(x+y)=f(x)+f(y)$$ has solutions called 'additive functions'. If no conditions are imposed to $f$, there are infinitely many functions that satisfy the equation, ...
2
votes
2answers
198 views

Equivalence of two versions of diamond principle

Taken out of ch II, Kunen. Need to show the following two versions of $\Diamond$ are equivalent: $\Diamond_\kappa$: There are $A_\alpha \subset \alpha$ for $\alpha < \kappa$ such that for each $A ...
0
votes
0answers
249 views

Zorn's lemma (Halmos)

(I noticed that there was a very recent question which dealt with similar material here; I didn't want to query in the comments, but merged answers might be appropriate.) My expectation with this ...
3
votes
2answers
93 views

Question about cardinals without GCH

Without assuming the Generalized Continuum Hypothesis, how to show that there exists a uncountable cardinal $\kappa$ such that, for every $\lambda < \kappa$, one have $2^\lambda < \kappa$. With ...
0
votes
1answer
144 views

Simple Question about Borel Hierarchy

I'm a bit confused about the basics of the borel hierarchy. My question is this: if i have a closed set P and I make the set $\forall^\omega P$, is that $\Pi^0_3$? Similarly, if I have an open set P ...
5
votes
1answer
335 views

Question of an isomorphism of $\epsilon_ 0$ and a subset of the rationals.

I don't know if this question is appropriated for this site. Anyway, I'm searching for an isomorphism of order $f:K \longrightarrow \epsilon_o $, such that $(K, \leq)$ is a subset(proper or not) of ...
6
votes
1answer
175 views

Does the specification of a general sequence require the Axiom of Choice?

Many results in elementary analysis require some form of the Axiom of Choice (often weaker forms, such as countable or dependent). My question is a bit more specific, regarding sequences. For ...
7
votes
1answer
335 views

Number of well-ordering relations on a well-orderable infinite set $A$?

Given a well-orderable infinite set $A$, can we always say that the set $$\left\{R\in A\times A:\langle A,R\rangle\, \text{is a well-ordering}\right\}$$ has cardinality $2^{|A|}$? How much Choice is ...
2
votes
1answer
312 views

Explanation of Proof of Zorn's lemma in Halmos's Book- II- Definition of towers and $\mathscr{X}$

This question is continued from: Explanation of Proof of Zorn's lemma in Halmos's Book From the Book: Now we can forget about the given partial order in $X$. In what follows we ...
3
votes
1answer
237 views

Is the arithmetic most mathematicans use a modelled within first or a second order logic?

I often read that arithmetic in first order logic has problems and you really want to do it in second order logic. However, aren't the Zermelo–Fraenkel axioms written down in the language of first ...
3
votes
1answer
128 views

Absoluteness and categories

From the wikipedia article on the Skolem paradox: A central goal of early research into set theory was to find a first order axiomatisation for set theory which was categorical, meaning that the ...
2
votes
2answers
168 views

Decimal expression of reals

Let $x>0$ be real. Then $A_1=\{n\in \mathbb{N}\mid x<n\}$ is nonempty since $\mathbb{R}$ is dedekind complete. Since $\mathbb{N}$ is well ordered, $A_1$ has a least element $k$. Thus $k-1$ is ...
10
votes
1answer
183 views

Saturated Boolean algebras in terms of model theory and in terms of partitions

Let $\kappa$ be an infinite cardinal. A Boolean algebra $\mathbb{B}$ is said to be $\kappa$-saturated if there is no partition (i.e., collection of elements of $\mathbb{B}$ whose pairwise meet is $0$ ...
-4
votes
3answers
282 views

How to be sure a contradiction is not possible?

I have written a formal proof of the theorem: $$\forall U \exists r(\forall a(a\in r \leftrightarrow (a\in U \wedge a\notin a)) \wedge r\notin U \wedge r\notin r)$$ See: ...
6
votes
3answers
1k views

Is there a difference between allowing only countable unions/intersections, and allowing arbitrary (possibly uncountable) unions/intersections?

As in the title, I am asking if there is a difference between allowing set-theoretic operations over arbitrarily many sets, and restricting to only countably many sets. For example, the standard ...
2
votes
1answer
328 views

On compact space with order topology

We are familar with that for the first uncountable cardinality $\omega_1$, the topological space $[0,\omega_1]$ is compact. I find the proof for the $\omega_1$, is also for every regular cardinality. ...
3
votes
0answers
117 views

$\mathbb{N}$ with uncountably many infinite subsets [duplicate]

Possible Duplicate: Countable set having uncountably many infinite subsets Can the set $\mathbb{N}$, the set of natural numbers, contain uncountably many infinite subsets ...
0
votes
2answers
192 views

On the space of ultrafilters on $N$

I meet the space $X$ of ultrafilters on $N$ with the topology generated by sets of the form $\{p\}\cup A$ where $A\in p \in X$. I can't understand the definition of the topology. Is the points in $N$ ...
4
votes
1answer
195 views

Width and height of partial ordered sets

The width $w$ of a partial ordered set(poset) is defined as the cardinality of the maximum antichain. By Dilworth Theorem, we know it is equivalent to the minimum number of chains in any partition. ...
4
votes
0answers
145 views

Set theoretic arguments to prove the existence of a certain null set

Let me recall the well-known Carleson's theorem. Theorem (Carleson). Let $f$ be any periodic $L^2[0, 2\pi]$ function. Let $\hat{f}(n)$ be its Fourier coefficients. Then we have $$\lim_{N \to ...
2
votes
0answers
196 views

Sum and product of an ultrafilter

I know the following simple fact is true, but I can't find a good proof: Over the naturals, the only ultrafilter $\mathcal U$ such that $\mathcal U \oplus \mathcal U = \mathcal U \odot \mathcal U$ is ...
2
votes
2answers
225 views

Measure on Baire space

Inspired by the first parenthetical sentence of Joel's answer to this question, I have the following question: is there any useful notion of measurability in the Baire space $\omega^\omega$? Some ...
4
votes
1answer
131 views

Total order in the power set of the real line

Is it possible to define constructively a total order in the power set of the real line ?
-3
votes
1answer
157 views

Find an elegant proof of a set-theoretic equiality about relations

I am now attempting to prove the following theorem. I am in half-underway of the proof and it seems I can do it by myself. But the proof I am now constructing is not elegant. Could anyone provide a ...
5
votes
3answers
158 views

Always a value with uncountably many preimages? (for a continuous real map on the plane)

Let $f$ be a continuous map ${\mathbb R}^2 \to {\mathbb R}$. For $y\in {\mathbb R}$, denote by $P_y$ the preimage set $\lbrace (x_1,x_2) \in {\mathbb R}^2 | f(x_1,x_2)=y \rbrace$. Is it true that ...
3
votes
0answers
118 views

Resolving set builder equations

Much of mathematics is about solving and resolving equations, most prominently algebraic equations. But is there a general theory of resolving set builder equations? To give an example, the equation ...
2
votes
1answer
372 views

On the generalized Sierpinski space

In Sierpiński topology the open sets are linearly ordered by set inclusion, i.e. If $S=\{0,1\}$, then the Sierpiński topology on $S$ is the collection $\{ϕ,\{1\},\{0,1\} \}$ such that ...
5
votes
2answers
337 views

How to resolve Skolem's Paradox by realizing what can be said of a set is relative to what is in the domain of some model?

I apologize in advanced if I'm hopelessly confused... Skolem's Paradox, I suppose, can be put like this: $M$ is a countable model of ZFC and $M$ implies the existence of uncountable sets. I suppose ...
2
votes
1answer
215 views

The axiom of choice and connected groupoids

Recall the two definitions of equivalence of categories: Definition 1. An equivalence of categories is a quadruplet $(F, G, \alpha, \beta)$, where $F : \mathbb{C} \to \mathbb{D}$ and $G : \mathbb{D} ...
3
votes
3answers
304 views

An element not in a set

Having a set I need to take an arbitrary element which is not in this set. I know that existence of such elements for every set can be proved in ZF. My question: Are there any established ...
2
votes
0answers
108 views

Berkovich analytification of Robinson fields

Let $\rho$ be an infinitesimal and let $^\rho \mathbb{R}$ be a (non-archimedean) Robinson valued field. Is there anything known about the topological structure of $\mathbb{A}^{1,an}_{^\rho ...
2
votes
2answers
255 views

What did Cantor take to be the relationship between the countable ordinals and the power set of the naturals?

I've been told that Cantor sees a relationship between the countable ordinals (Cantor's second number class) and the powerset of the natural numbers. I've read the "Grundlagen" a few times, but can't ...
1
vote
0answers
519 views

Dedekind cut multiplicative inverse

I posted this problem yesterday and Brian gave me really nice answer using Bernoulli inequality, but I think this can be proved with the concept of Archimedean property of $ \mathbb{Q}$ and greatest ...
5
votes
1answer
246 views

Why is this forcing notion closed?

I'm studying a forcing argument which produces a generic extension in which GCH holds, but I am, somewhat embarrassingly, stuck on a minor detail. I hope someone can point out the thing I'm missing. ...
3
votes
1answer
190 views

Explanation of how models can differ on $\omega$?

Assuming set theory (here, ZF) is consistent, there is a model $V$ of ZF, the universe of all sets. So, there is a $\omega^V\in V$. A set $A\in V$ is countable iff a bijection $f\in V$ exists ...
5
votes
2answers
242 views

Consistency strength: If Con($T+A$) implies Con($T+B$), can we infer anything about $A$ and $B$?

To be more specific, let $T$ be a first order theory and let $A$ and $B$ be two different first-order sentences, both in the same language as $T$ but independent of $T$. Additionally, suppose we have ...
1
vote
4answers
281 views

What's the definition of limit of sets(esp. ordinals) in set theory?

By definition of exponent operator on ordinals, we have $$0^\omega=\lim_{\xi\to\omega}0^\xi$$ However, Note that $0^\xi$ is not increasing, so if we still let ...
2
votes
1answer
66 views

question about cofinality and function

In a paper, I want to prove a result that seems to me general. Let $g:\delta\longrightarrow cf(\lambda)$ where $\delta$ is an ordinal less than $\lambda^+$ and $\lambda$ a cardinal. Suppose that ...
7
votes
3answers
826 views

Banach-Tarski theorem without axiom of choice

Is it possible to prove the infamous Banach-Tarski theorem without using the Axiom of Choice? I have never seen a proof which refutes this claim.
6
votes
2answers
130 views

What's the name of this operator?

Let $f,g$ be functions in $C^A$ and $C^B$ respectively. Let $\boxtimes:C^A \times C^B \to (C\times C)^{A \times B}$ s.t. $f\boxtimes g(a,b)=(f(a),g(b))$ It seems not the tensor product, nor ...