5
votes
1answer
52 views
1
vote
1answer
34 views

How can we show that $\omega_1$ is a regular cardinal?

A cardinal $\kappa$ is regular if and only if there is no $\lambda<\kappa$ for which there is a function $f:\lambda\rightarrow\kappa$ with range cofinal in $\kappa$. How can we see in ZFC that ...
2
votes
1answer
30 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 ...
1
vote
1answer
56 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 ...
4
votes
0answers
62 views
+50

What happens if we replace “regularity” in GCH with other conditions?

We can formulate both a weak and a strong generalized continuum hypothesis. GCH0. If $\kappa$ is an infinite cardinal number, then $2^\kappa$ is regular. GCH1. If $\kappa$ is an infinite cardinal ...
5
votes
1answer
75 views

Existence of a regular uncountable $\aleph_{\alpha}$ without $\mathsf{AC}$

Set theory (Jech) $\text{p.}\;27:$ It is an open problem whether one can prove without the axiom of choice that there exists a regular uncountable $\aleph_{\alpha}\;($the informed guess is that ...
2
votes
1answer
36 views

Can we define ordinals such that the following sentences are independent of ZFC?

Can we explicitly define two ordinals $\alpha$ and $\beta$ in the language of $\{\in\}$ such that the following hold? ZFC proves that $\alpha$ and $\beta$ exist. ZFC proves that $\beth_\beta \neq ...
2
votes
2answers
108 views

A property of strong limit cardinal

Suppose $\lambda$ is a strong limit cardinal, i.e. $\forall \alpha<\lambda \ 2^\alpha<\lambda$, and the cofinality of $\lambda$: $cf(\lambda)=\omega$. How do we show that $2^\lambda \leq ...
4
votes
2answers
94 views

about the smallest $k$ that $V_k$ is a model of ZFC

Let $k$ to be the smallest ordinal that $V_k$ is a model of ZFC. I know that $k$ need not to be inaccessible cardinal,and $k$ has confinality $\omega$. Then how big is $k$? How to write down $k$ in ...
5
votes
4answers
107 views

Cardinality of Irrational Numbers

I know and I have proved more than once that the set of irrational numbers ($\mathbb{I}$) is uncountable, but now I'm given to solve this problem: Show that $|\mathbb{I}|=|\mathbb{R}|$, How can I ...
4
votes
2answers
110 views

The regularity of successor cardinal

I was looking at two different proofs of the fact that successor cardinals are regular. It struck me as odd that both proofs used AC. Looking at the concepts involved in defining cofinality I feel as ...
5
votes
3answers
215 views

Uncountable Cardinals without AC

I am doing an exercise, proving that without AC or Replacement that there are uncountable cardinals. As a point of reference I looked at the proof in Kunen's "The Foundations of Mathematics" that ...
1
vote
1answer
59 views

About alephs and beths

If $2^{\aleph_{0}} \ge \aleph_{\omega_1}$, show that $\beth_{\aleph_{\omega}} = 2^{\aleph_{0}}$ , and that $\beth_{\aleph_{\omega_1}} = 2^{\aleph_{1}}$ I don´t know how to start, can you give me a ...
3
votes
1answer
309 views

A question about splitting sets

I've been looking into combinatorics and small cardinals, in particular, the splitting number $\mathfrak{s}$. By definition, a set $X \subseteq \omega$ splits an infinite set $Y \subseteq \omega$ if ...
2
votes
1answer
44 views

Are there ordinals other than the set of natural numbers which satisfy this property?

Let $\alpha$ be an ordinal. We say that $\alpha$ is good iff for every $\beta\in \alpha$, there exists $\gamma\in \alpha$ such that $|\scr{P}(\beta)|\leq |\gamma|$. Question: Is the set of natural ...
2
votes
1answer
45 views

Cofinality assuming GCH

There is this statement that GCH holds iff any pair of regular cardinals $\kappa,\lambda$ such that $\kappa<\lambda$ satisfy that $\lambda^\kappa = \lambda$. Assume we do have two such cardinals. ...
2
votes
1answer
28 views

Identity on singular strong limit cardinals

Let $\lambda$ be a singular strong limit cardinal. Prove that $2^\lambda = \lambda^{\mbox{cf}\lambda}$. It has been a while since I had to prove anything relating to cardinals, and I am not sure ...
1
vote
1answer
88 views

Continuum Hypothesis $\iff ?$?

I have read that CH cannot be proved nor disproved within ZFC, and I was wondering: Which (If any) branches/fields of Mathematics are built upon CH being true? Are there any subjects built upon ...
2
votes
1answer
62 views

$\aleph$ function fixed points below a weakly inaccessible cardinal are a club set

I am throwing yet another one of my solutions out here for the internets to debug and for future set-theory students. Let $\aleph_\delta$ a weakly inaccessible cardinal. Prove that $A =\{\alpha ...
10
votes
2answers
2k views

What's “the catch” in this question?

I am solving old exam questions and I came across this question: Let $\langle A_n \mid n < \omega\rangle$ disjoint sets such that $\bigcup_{n < \omega}A_n = \mathbb{R}$. Prove that there ...
2
votes
0answers
71 views

$\mathfrak b \leq \mathfrak r$. Is this proof correct?

I am trying to prove that $\mathfrak b \leq \mathfrak r$ where $\mathfrak r$ is the minimal cardinality of a reaping family of sets. A family $\mathcal R \subseteq [\omega]^\omega$ of sets $\mathcal ...
4
votes
1answer
73 views

How to show that $\mathfrak s \leq \mathfrak d$

I am trying to understand why $\mathfrak s \leq \mathfrak d$. Can anyone state a proof of it? I have a proof , which I don't understand yet. My question regarding that proof is here below: At the ...
5
votes
1answer
71 views

A verification for a proof that $\omega_1 \leq \mathfrak s$

I am trying to prove that $\omega_1 \leq \mathfrak s$ where $\mathfrak s$ is the splitting number which is the smallest cardinality of any splitting family. This statement was left as an ...
4
votes
1answer
77 views

How to prove that $\omega_1 \leq \mathfrak p$

I am trying to understand the stated above Theorem. $\mathfrak p$ is the smallest cardinality of any family $\mathcal F \subseteq [\omega]^\omega$, which has the strong finite intersection property, ...
3
votes
2answers
61 views

The cardinal characteristic $\mathfrak d$

I am reading a chapter in a book of Andreas Blass which is called: "Combinatorial Cardinal Characteristics of the Continuum". In there, the cardinal characteristic $\mathfrak d$ is defined as folows: ...
5
votes
1answer
108 views

Injective or constant function on measurable cardinal

I'm working on the following question: Let $\kappa$ regular cardinal, and $f:\kappa \rightarrow \kappa$. Prove that $\{\alpha < \kappa\mid f|_\alpha : \alpha \rightarrow \alpha\}$ is a ...
3
votes
2answers
128 views

Ordinals that satisfy $\alpha = \aleph_\alpha$ with cofinality $\kappa$

I am trying to solve the following question: Prove that for every regular cardinal, $\kappa \gt \aleph_0$, there is a exists an $\alpha$ with cofinality $\kappa$ such that $\alpha = ...
2
votes
1answer
63 views

chain A s.t. $|X|<|A|\leq |P(X)|$ [duplicate]

Can we prove that there exists at least one chain $A$ in P(X), where X is a non-empty set (finite or infinite), s.t. $ |X|<|A|\leq |P(X)|$? If you can't solve it, ideas/possible directions are ...
6
votes
1answer
124 views

What would a world where $\mathsf{CH}$ is false look like?

My question is a little more specific than the title may lead to believe. In the article The set-theoretic multiverse (J.D. Hamkins), the author writes the following: [...] the continuum is ...
2
votes
1answer
133 views

Every club of $\kappa$ in $M[G]$ contains a club in $M$.

I'm trying to solve exercise (H1) of chapter VII on Kunen's Introduction to Independence Proofs and I would like some hint. I would prefer a hint instead of the full solution :) Assume in M that ...
3
votes
1answer
65 views

$(k^{<\lambda})^{<\lambda}=k^{<\lambda}$ if $\lambda$ is regular

I'm in need of some help... Why does $(k^{<\lambda})^{<\lambda}=k^{<\lambda}$ if $\lambda$ is regular? I can't see why... Any hints?
4
votes
1answer
99 views

Bijection between countable ordinals and reals

The set of all countable ordinals is $\omega_1$, which has a cardinality of $\aleph_1$. When accepting the continuum hypothesis, $2^{\aleph_0} = \aleph_1$, so a bijection between countable ordinals ...
1
vote
1answer
69 views

Russell's definition of finite cardinals

whether the thought had been previously adumbrated, perhaps confusedly, i know not, but the name of Bertrand Russell has become associated with the assertion that: the number $2$ is the set of all ...
8
votes
2answers
64 views

Proof of non-existence of non-principal $\kappa$-complete ultrafilter

Let $\lambda$ be a cardinal. I would like to prove that for all cardinals $\lambda < \kappa \leq 2^\lambda$, there can't be a $\kappa$-complete non-principal ultrafilter on $\kappa$. Here is my ...
2
votes
1answer
90 views

Godel's pairing function and proving c = c*c for aleph cardinals

I have a few questions about Godel's pairing function and proving that c = c * c for aleph cardinals. Mostly, though, I'm concerned that most of the proofs I've seen are erroneous, and this concerns ...
3
votes
2answers
52 views

Regularity of Limit Cardinals

My intuition is that co-finality is a non-decreasing function on the cardinals. If that's true, it seems to follow that all infinite cardinals are regular. In particular, $\aleph_0$ is clearly regular ...
4
votes
1answer
110 views

Jech's proof of Silver's Theorem on SCH

Jech's textbook proves Silver's Theorem–that if the Singular Cardinals Hypothesis holds for all singular cardinals of countable cofinality, then it holds for all singular cardinals–by breaking up the ...
1
vote
1answer
59 views

Cardinals, bijections, and a general inquiry

I'm trying to solve the following problem from Schimmerling's "A Course on Set Theory." (Problem) Prove that there exists a family $\mathcal G\subseteq\mathcal P(\omega)$ such that $|\mathcal ...
1
vote
1answer
50 views

Addition of Alephs

Prove: $\aleph_{\alpha} + \aleph_{\alpha} = \aleph_{\alpha} $ The textbook I am using has a long proof done by transfinite induction. I am looking for a direct proof. Can I do this: ...
0
votes
1answer
85 views

Cardinal of $X^2$

The axiom of choice is equivalent to the following statement: if $X$ is an infinite set then the cardinal of $X^2$ is the same as that of $X$. Is there an elementary proof of this statement?
0
votes
1answer
61 views

Largest Useful Sets [closed]

I'm just asking this out of curiosity, what are the largest sets that are actually meaningful (infinite sets)? I know that there is no highest cardinal number, but there must come a point where we ...
0
votes
1answer
63 views

How big is the set of hyper-naturals?

Consider the set $\mathbb N^*$, the set of hypernaturals. How big is this set? Is it the same size as $\mathbb R^*$?
5
votes
1answer
94 views

Aleph arithmetic question

We want to prove that: $$\aleph_2^{\aleph_0} = \aleph_2\aleph_1^{\aleph_0}$$ My idea was to approach this by doing a Schroder-Bernstein style argument and proving this by showing two inequalities, ...
-3
votes
1answer
270 views

Question about Cantors Diagonal Argument [closed]

Lets be honnest I don't understand cantors diagonal argument. http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument While I can understand that this proof proves that you cannot make the list of ...
1
vote
1answer
54 views

The set of all countably-infinite subsets of an infinite set

Let $A$ be an infinite set and $D(A)$ denote the set of all countably-infinite subsets of $A$ and let $P(A)$ denote ,as usual, the power set of $A$, then (i) does there exist a surjection of $D(A)$ ...
7
votes
2answers
191 views

Does $\mathcal P ( \mathbb R ) \otimes \mathcal P ( \mathbb R ) = \mathcal P ( \mathbb R \times \mathbb R )$?

Obviously the answer's yes if I replace $\mathbb R$ by a countable set. If $\mathbb R$ is replaced by a set $X$ having a greater cardinality, then the diagonal $\{ (x,x) , \ x\in X\}$ is not in the ...
2
votes
1answer
47 views

Infinite cardinal comparation

Let $\alpha$ and $\beta$ two infinite cardinal numbers. Can we have $\alpha = \beta^\alpha$? This problem comes from a situation where I am dealing with the cardinal of a set of functions.
7
votes
1answer
148 views

weak consequence of GCH

Can ZFC prove that there is a regular uncountable cardinal $\kappa$ such that $2^{<\kappa} < 2^\kappa$? Note, if the answer is no, it would require a strong global violation of SCH, so large ...
10
votes
3answers
296 views

Can sets of cardinality $\aleph_1$ have nonzero measure?

$\aleph_1$ is the cardinality of the countable ordinals. It is the least cardinal number greater than $\aleph_0$, and assuming the continuum hypothesis it's equal to $\mathfrak{c}$, the cardinality of ...
1
vote
1answer
64 views

Cartesian product of large sets

For a non-empty set $A$ let $A'$ denote the Cartesian product of $A$ with itself taken denumerably many times. Now given a set $S$ whose cardinality is strictly greater than the cardinality of ...