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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3
votes
0answers
121 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
201 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
219 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
201 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
233 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
137 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
159 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
159 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
386 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 ...
6
votes
2answers
367 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
217 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
308 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
111 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
264 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
556 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
247 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
194 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
262 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
289 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 ...
8
votes
3answers
879 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
131 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 ...
3
votes
1answer
309 views

Forcing exercise

This exercise is from Kunen, Set Theory. Let $M$ be a ctble transitive model of ZFC,$\kappa > \omega$, $\kappa$ regular, $P$ be a notion of forcing that is $\kappa$-closed (i.e. whenever $\gamma ...
2
votes
2answers
242 views

Why can't a model “say” of itself that it is countable?

Why can't a (standard?) model of ZFC "say of itself" that it is countable? That is, why is there no bijection $f$ ∈ 𝔐 between 𝔐 and $\omega^𝔐$? (I've read that it fails regularity, or even ...
3
votes
1answer
456 views

Cardinality of the set of ultrafilters on an infinite Boolean algebra

Let $\mathfrak B$ be a Boolean algebra with an infinite power $\kappa$. My question is how many ultrafilters does it have? $\kappa$ or $2^\kappa$? Or even smaller?
8
votes
6answers
2k views

Why accept the axiom of infinity?

According to my readings, Russell showed that a principle Frege used to reduce Peano arithmetic to logic lead to a contradiction. So, Russell tried to reduce mathematics to logic a different way but ...
0
votes
1answer
189 views

problem of a cardinality of a union

Let $\lambda$ a cardinal and $\delta<\lambda^+$. I want to proof there exists a increasing chain $$\{A^i_\delta : i< cf(\lambda)\}\subseteq[\delta\times\delta]^{<\lambda}$$ converging to ...
2
votes
1answer
213 views

Can we get uncountable ordinal numbers through constructive method?

As we know, $2^{\aleph_0}$ is a limit ordinal number, however, it is greater than $\omega$, $\omega+\omega$, $\omega \cdot \omega$, $\omega^\omega$, $\omega\uparrow\uparrow\omega$, and even $\omega ...
2
votes
2answers
1k views

Do $\omega^\omega=2^{\aleph_0}=\aleph_1$?

As we know, $2^{\aleph_0}$ is a cardinal number, so it is a limit ordinal number. However, it must not be $2^\omega$, since ...
6
votes
3answers
243 views

Axiom of Choice (for example in the Snake Lemma)

If we have to make a choice, but in the end it doesn't matter what choice we made, did we really make a choice to begin with? More explicitly, somewhere in the standard diagram-chasing proof of the ...
2
votes
4answers
329 views

Arithmetic on real numbers

So far, I have studied elementary set theory and I have some questions. I know how to add or multiply natural numbers and ordinals, but how do I subtract or divide or root or log? Is there any ...
3
votes
1answer
475 views

Question about Halmos' Naive Set Theory

Halmos proves shortly before the cited paragraph that finite subsets are not equivalent to themselves. He then says the following: The number of elements in a finite set E is, by definition, the ...
10
votes
4answers
811 views

Advantage of ZF over other set theories such as New Foundation

What would be the advantage of adopting ZF over other set theories such as New Foundation? I am very curious, since it seems that there is no reason just to stick with ZF. Edit: What about set ...
2
votes
2answers
198 views

Cofinality of infinite cardinals

I am looking for an answer to the question: "Show there exists an infinite cardinal $\kappa$ with $2^{cf(\kappa)}$ < $\kappa$ " Where $cf(\kappa)$ is defined as the least $\alpha$ such that there ...
0
votes
1answer
89 views

Factor in ultraproduct

The general method for getting ultraproducts uses an index set I, a structure $M_i$ for each element i of I (all of the same signature), and an ultrafilter U on I. The usual choice is for I to be ...
3
votes
2answers
430 views

Importance of cover (topology)

What would be importance of cover in topological space? And how is the concept of cover being used in other areas, more specifically set theory?
1
vote
2answers
448 views

Unnecessary property in definition of equivalence relation [duplicate]

Possible Duplicates: Symmetric, Transitive and reflexive Why isn't reflexivity redundant in the definition of equivalence relation? Dependence of Axioms of Equivalence Relation? Let ...
2
votes
1answer
429 views

Mostowski collapse lemma

Can anyone explain what Mostowski collapse lemma is? The Mostowski collapse lemma states that for any such $R$ there exists a unique transitive class (possibly proper) whose structure under the ...
5
votes
2answers
241 views

Does the assertion that every two cardinalities are comparable imply the axiom of choice?

If, for any two sets $A$ and $B$, Either $|A|<|B|, |B|<|A|$ or $|A|=|B|$ holds, does the axiom of choice holds? Why?
3
votes
2answers
297 views

How to show Tukey's Lemma proves Zorn's lemma?

I heard that Zorn's lemma is equivalent to Tukey's lemma. Now I've proved that Zorn's lemma implies that Tukey's lemma but I cannot prove that Tukey's lemma implies Zorn's lemma. How to show this?
2
votes
1answer
247 views

Countability in first-order logic is relative to what exactly?

Skolem's Paradox tells us that countability in first-order logic is relative. Relative to what? Below is what I've gathered. Countability it relative to: 1. what a model takes to be $\mathbb N$ 2. ...
9
votes
2answers
850 views

Using Replacement to prove transitive closure is a set without recursion

In the course on set theory I'm doing, I'm told that one of the main motivations behind the axiom of replacement is that the Axiom of Infinity asserts the existence of an infinite set, namely $\omega ...
3
votes
3answers
380 views

Importance of free ultrafilter

What would be the usage of free ultrafilter? And how is it important? BTW, I know the concept of free ultrafilter, so I only need explanation on usage and importance.
2
votes
1answer
221 views

Regressive injective function in a set of infinite cardinals

This is also from Kunen, Set Theory, ch. II: Let $A$ be a set of infinite cardinals such that for each $\lambda$ regular $A\cap\lambda$ is not stationary in $\lambda$. Show that there is an ...
6
votes
2answers
311 views

Is this a good way to explicate Skolem's Paradox?

Skolems Paradox shows an ostensible conflict between Cantor's Thoerem (CT) and the downward Löwenheim–Skolem Theorem (ST). CT: for any set $A$, the powerset of $A$, $P(A)$, has a strictly greater ...
7
votes
2answers
201 views

Topological restrictions on cardinality

From what I know, a Polish (completely metrizable separable) space has a cardinality at most of $\mathbb R$. Completeness assumption can be omitted here, because a completion of a metrizable separable ...
2
votes
2answers
441 views

If ZFC has a model, must it be at least a countable model?

(1) must ZFC have an infinite model? (2) if so, why? (3) is it because of the replacement schema? (4) if so, is it because we have a finite language and so we can only satisfy or describe ...
0
votes
2answers
128 views

Real line, field of real numbers and $\omega_1$ topological space difference

Real line is separable. Then, why is $\omega_1$ topological space not separable? IF this is true, doesn't this settle continuum hypothesis? Also, does the field of real numbers have anything to do ...