The pcf-theory tag has no usage guidance.

**6**

votes

**1**answer

388 views

### A “good scale” that is not really a scale

I don't know much about singular cardinal combinatorics, so I apologize in advance if I write something that is wrong or looks funny. First let me recall some basic definitions.
Let $\lambda$ be a ...

**13**

votes

**0**answers

451 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

313 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 $|\mathbb{A}...

**6**

votes

**3**answers

412 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 ...

**7**

votes

**0**answers

200 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 } \...

**9**

votes

**1**answer

411 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

574 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) =^{+} \...

**4**

votes

**2**answers

234 views

### Existence of scales with special properties

Let $\kappa$ be a singular cardinal, and let $\langle \kappa_i \mid i<\mathrm{cf}(\kappa) \rangle$ be an increasing sequence of regular cardinals cofinal in $\kappa$. Recall that a scale on $\Pi_{i&...

**2**

votes

**0**answers

206 views

### a partial order not dense iff a measurable exists

For $\kappa>0$ a regular cardinal, let $Ht_\kappa$ denote the following partial quasi-order:
(i) elements(objects) of $Ht_\kappa$ are classes X of sets of size $\kappa$
with the property that,
($&...

**5**

votes

**1**answer

369 views

### Generalizations of pcf theory

Does anyone know of generalizations of pcf theory where we might consider products of the form:
$$\aleph_1 \times (\aleph_2 \times \aleph_2) \times (\aleph_3 \times \aleph_3 \times \aleph_3) \dots$$
...

**4**

votes

**1**answer

181 views

### Bounds on $\max \mathrm{pcf}(A)$ if $\Pi A$ is big

For concreteness, let $A = \{\aleph_n : n < \omega\}$. We know $\max \mathrm{pcf}(A) \in [\aleph_{\omega+1},\Pi A]$. My question is, if $\Pi A$ is big (say, $\aleph_{\omega_1+1}$), then which ...