The cardinal-arithmetic tag has no wiki summary.

**6**

votes

**2**answers

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

**0**answers

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

**2**answers

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

**2**answers

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

**2**answers

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

**3**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**3**answers

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

**4**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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 ...