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6
votes
2answers
350 views

Exponentiation and Dedekind-finite cardinals

It is known that the sum and the product of two Dedekind-finite cardinals are also Dedekind-finite cardinals. What about cardinal exponentiation ? Question: Let A and B be two Dedekind-finite ...
11
votes
0answers
337 views

Ideas behind Gitik's solution of PCF conjecture

Recently Moti Gitik has refuted Shelah's PCF conjecture (see Short extenders forcings II ) by proving the following theorem: Theorem. Assuming the consistency of infinitely many strong cardinals, one ...
5
votes
2answers
261 views

For which cardinal numbers $\kappa$ is it consistent with ZFC that $\kappa^{\mathrm{cf}(\kappa)} < \kappa^\kappa$?

ZFC proves that $\kappa^{\mathrm{cf}(\kappa)} \leq \kappa^\kappa$ for all infinite cardinal numbers $\kappa$. Further, it is consistent with ZFC that we always have equality (e.g. assume GCH). ...
5
votes
2answers
255 views

When are all greater cardinals sharply greater?

Makkai and Paré introduced the following binary relation on regular cardinals: given $\kappa$ and $\lambda$, $\kappa \vartriangleleft \lambda$ (read, $\kappa$ is sharply less than $\lambda$) when ...
5
votes
2answers
276 views

Prevalent singular cardinals hypothesis

The following notion is introduced by Assaf Rinot: Definition. A singular cardinal $\kappa$ is a prevalent singular cardinal iff there exists a family $\mathbb{A}\subset P(\kappa)$ with ...
6
votes
3answers
345 views

Consistency strength of the failure of Shelah's Strong Hypothesis (SSH)

Some known facts about SSH (Shelah's Strong Hypothesis): i) "$0^\sharp$ does not exist" implies SSH. ii) SSH implies SCH (Singular Cardinal Hypothesis). iii) The failure of SCH is equiconsistent ...
8
votes
0answers
186 views

Other variants of the Shelah's Weak Hypothesis

The paper Menachem Kojman. Splitting families of sets in ZFC. arXiv:1209.1307 presents these variants of the Shelah's Weak Hypothesis: $$ (\textrm{SWH}_n) \textrm{ There are no infinite } ...
22
votes
1answer
1k views

Possible troubles in Shelah's book “Cardinal Arithmetic”

I found some possible troubles in Observation 5.3(7) in the Chapter II of the Shelah's book "Cardinal Arithmetic" (page 86). For convenience, I quote the result and the proof in the book here ...
9
votes
1answer
384 views

Some variants of the Shelah's Weak Hypothesis

Are equivalent (in ZFC) the following two statements, for any infinite cardinal $\mu$? (i) For every infinite cardinal $\kappa$, $|\{ \lambda \in \kappa : \lambda \textrm{ is a singular cardinal and} ...
10
votes
1answer
526 views

“cov vs pp” problem

This is the problem $(\beta)$ of the section 14.7 in the "Analytical Guide" of the Shelah's book "Cardinal Arithmetic": $(\beta)$ Is $\operatorname{cov} ( \lambda , \lambda, \aleph_{1} , 2) =^{+} ...
3
votes
3answers
143 views

Subquotients in ZF

In ZF we have the two relations $A \leq B$ and $A \leq^\ast B$ which relate the size of sets: the first says there is an injection from $A$ to $B$, the second that there is a surjection from $B$ to ...
5
votes
4answers
330 views

do behavior of gimel or GCH determine all infinte products of cardinals?

Let $Card$ be the class of infinite cardinals and $p\colon Card^2\to Card$ be given by $(\kappa,\lambda)\mapsto\kappa^\lambda$. Assuming GCH it is known that $p(\kappa,\lambda)$ is either $\kappa$ (if ...
1
vote
1answer
364 views

Cardinal Arithmetic, foundations and constructive math

This is not my area but a question occurred to me that I can not find the answer to. There is a very strong axiom of constructibility which ironically gives us highly non-constructive math (GCH is one ...
1
vote
1answer
361 views

About the notions of Grothendieck Universe and Tarski Universe

I assume ZFC. Let $U$ a set with the following (1), (2), (3): 1) $\omega\in U$ 2) $x\in U\ \Rightarrow x\subset U$ 3) $x\in U\ \Rightarrow \mathcal{P}(x)\in U$ (where $\mathcal{P}(x):=${$y| ...
2
votes
1answer
302 views

A question about cardinal arithmetic

Let [J]: Jech "Set theory" (Millenium edition) Let $\kappa$ a limit ordinal. From [J], T.3.11, p. 33 we have that $\kappa<\kappa^{cf(\kappa)}$. I improved that proof, and obtain : $\kappa ...
7
votes
1answer
420 views

Does this “jumping-ahead” ordinal function exist?

While working on a project in operator algebras with a collaborator (and fellow MO user), we are able to successfully complete a transfinite induction assuming that the following has an affirmative ...
7
votes
1answer
370 views

Is there always an uncountable $\kappa$ such that $\kappa^{\aleph_0}=2^\kappa$?

The cardinal equation $\kappa^{\aleph_0}=2^\kappa$ is satisfied by $\kappa=\aleph_0$. It is also satisfied by any $\kappa$ for which $MA(\kappa)$ holds. Under $GCH$, the equation is satisfied by ...