Questions tagged [pcf-theory]

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Shelah’s Representation Theorem: existence of scales

Let $\lambda$ be a singular cardinal of countable cofinality. Shelah’s Representation Theorem states that there is an increasing sequence of regular cardinals $\langle\delta_n\rangle_{n<\omega}$, ...
Seba Thei's user avatar
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15 votes
1 answer
419 views

Can $\kappa^\lambda$ be large if $2^\lambda$ is small and $\lambda<\mathrm{cof}(\kappa)$?

We work in ZFC throughout. The following question was posed to me by a friend: Can there exist cardinals $\kappa,\lambda$ such that $\lambda<\mathrm{cof}(\kappa)$ and $2^\lambda<\kappa<\...
Wojowu's user avatar
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2 votes
1 answer
209 views

Continuum function maximum

Easton's theorem can give a very weak nontrivial constraint on continuum function, but it does not hold for singular cardinals. So: What are the non-trivial constraints on continuum function in ...
Ember Edison's user avatar
7 votes
0 answers
130 views

Better scales and Failures of SCH

Assume $\mu$ is a singular cardinal of countable cofinality. Recall that a scale for $\mu$ consists of an increasing sequence $\vec{\mu}$ of regular cardinals $\langle \mu_n:n<\omega\rangle$ ...
Todd Eisworth's user avatar
11 votes
0 answers
275 views

Can we bound $2^{\aleph_\omega}$ without pcf theory?

One of the famous applications of pcf theory is that if $\aleph_\omega$ is a strong limit cardinal then $2^{\aleph_\omega}<\aleph_{\omega_4}$. I'm curious whether any weaker result with the same ...
Noah Schweber's user avatar
11 votes
1 answer
304 views

Can this result in cardinal arithmetic be established without using pcf theory?

Suppose $\kappa\leq\mu$ are infinite cardinals. Let us agree to call a family $\mathcal{P}\subseteq[\mu]^{<\mu}$ a countably generating family for $[\mu]^\kappa$ if every member of $[\mu]^\kappa$ ...
Todd Eisworth's user avatar
6 votes
1 answer
157 views

Regular limit points of possible cofinalities

Let $A$ be a non-empty set of regular cardinals such that $\vert A\vert <\text{min}\ A$, and $\{\nu_i\mid i<i_0\}\subseteq \text{pcf}\ A$ be a strict increasing sequence having limit length $i_0$...
Sho Banno's user avatar
1 vote
0 answers
192 views

A categorial PCF theory?

I'm not an expert in PCF theory, so please forgive me if this question makes no sense. I'm looking for a categorial version of PCF theory. Specifically, if we replace $Set$ with another category, ...
Omer Rosler's user avatar
13 votes
1 answer
426 views

What are some good lower bounds on the consistency of the failure of the PCF conjecture?

Shelah's celebrated theorem states that $\aleph_\omega$ is a strong limit cardinal, then $2^{\aleph_\omega}<\aleph_{\omega_4}$. But the conjecture is that $\omega_4$ can be provably replaced by $\...
Asaf Karagila's user avatar
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17 votes
0 answers
541 views

Gitik's work on Shelah's weak hypothesis

It seems that Moti Gitik has recently refuted some variants of Shelah's weak hypothesis. For this see the title and abstract of his talk at the Set Theory, Model Theory and Applications conference. I ...
Mohammad Golshani's user avatar
9 votes
2 answers
489 views

PCF theory and good points in scales

If $\kappa$ is a singular cardinal, a scale for $\kappa$ consists of an increasing sequence $\langle \kappa_i : i < \mathrm{cf}(\kappa) \rangle$ converging to $\kappa$ and a sequence of functions $\...
Monroe Eskew's user avatar
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9 votes
1 answer
539 views

"Towers" on singular cardinals with countable cofinality

Let $\lambda$ be a singular cardinal of countable cofinality. Is there necessarily a sequence $\{A_\alpha\mid\alpha<\lambda^+\}$ of countable subsets of $\lambda$, such that $\alpha<\beta$ if ...
Asaf Karagila's user avatar
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7 votes
0 answers
205 views

Possible cofinalities of cuts of ultraproducts

Suppose $\kappa$ is a regular cardinal, $\bar{\mu}=(\mu_i: i<\kappa)$ is an increasing sequence of regular cardinals ($>\kappa$) and set $pcut(\bar \mu)=\{ (\lambda_1, \lambda_2):$ for some ...
Mohammad Golshani's user avatar
6 votes
1 answer
270 views

Reference for Chang's Conjecture at $\aleph_{\omega}$

The following theorem is well known: Theorem: $(\aleph_{\omega + 1}, \aleph_{\omega}) \not\twoheadrightarrow (\aleph_{n + 1}, \aleph_n)$ for every $n \geq 3$. Under CH, $(\aleph_{\omega + 1}, \...
Yair Hayut's user avatar
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8 votes
0 answers
371 views

PCF conjecture and fixed points of the $\aleph$-function

Recently Moti Gitik refuted Shelah's PCF conjecture, by producing a countable set $a$ of regular cardinals with $|\operatorname{pcf}(a)| \geq \aleph_1.$ See his papers Short extenders forcings I and ...
Mohammad Golshani's user avatar
6 votes
1 answer
512 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 ...
Trevor Wilson's user avatar
17 votes
0 answers
757 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 ...
Mohammad Golshani's user avatar
5 votes
2 answers
435 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}...
Mohammad Golshani's user avatar
7 votes
3 answers
638 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 ...
Alberto Levi's user avatar
7 votes
0 answers
255 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 } \nu ...
Alberto Levi's user avatar
9 votes
1 answer
543 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} ...
Alberto Levi's user avatar
10 votes
1 answer
677 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) =^{+} \...
Alberto Levi's user avatar
4 votes
2 answers
302 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&...
Chris Lambie-Hanson's user avatar
2 votes
0 answers
231 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, ($&...
o a's user avatar
  • 468
5 votes
1 answer
550 views

Generalizations of pcf theory

Does anyone know of generalizations of pcf theory where we might consider products of the form: $$\aleph_1 \times (\aleph_2 \times \aleph_2) \times (\aleph_3 \times \aleph_3 \times \aleph_3) \dots$$ ...
Amit Kumar Gupta's user avatar
6 votes
1 answer
221 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 ...
Amit Kumar Gupta's user avatar
3 votes
2 answers
530 views

Some Pcf Theory

Let $pcf(a)$ denote the set of regular cardinals such that $J_{\leq \lambda} - J_{<\lambda} \neq \emptyset$ and let $maxpcf(a)$ denote the maximum of $pcf(a)$. The $J_{\leq \lambda}$ are the usual ...
Rachid Atmai's user avatar
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