Questions tagged [pcf-theory]
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30 questions
6 votes
0 answers
164 views
A Silver-type lemma
I'm looking for a reference/proof of the following Silver-type lemma. Lemma. Let $\mu$ be a singular cardinal of uncountable cofinality and let $\delta<\mu$ be a regular cardinal. Suppose that $$\{\...
- 665
6 votes
1 answer
254 views
Computing the cofinality above the first strongly compact cardinal
It is well known that Shelah's Strong Hypothesis (SSH) holds above the first strongly compact cardinal. On the other hand, SSH implies that $$(\star)\text{ cof}([\alpha]^{<\delta},\subseteq)\leq\...
- 665
7 votes
1 answer
664 views
Destroying scales
Suppose $\lambda$ is a cardinal with uncountable cofinality and $C\subseteq\lambda$ is a club of ordertype cf$(\lambda)$. Let $f=\langle f_\xi\mid\xi<\lambda^+\rangle$ be a scale consisting of ...
- 665
7 votes
0 answers
228 views
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 $\langle\delta_n\rangle_{n<\omega}$, ...
- 665
16 votes
1 answer
587 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<\...
- 31.4k
2 votes
1 answer
318 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 ...
- 1,435
7 votes
0 answers
185 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 $\langle \mu_n:n<\omega\rangle$ ...
- 7,487
11 votes
0 answers
462 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 ...
- 19.5k
11 votes
1 answer
405 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 $[\mu]^\kappa$ ...
- 7,487
6 votes
1 answer
203 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 $i_0$...
- 63
1 vote
0 answers
225 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, ...
- 773
14 votes
1 answer
541 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 $\...
- 41.8k
17 votes
0 answers
597 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 ...
- 33.5k
9 votes
2 answers
658 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 $\...
- 20.9k
9 votes
1 answer
826 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 ...
- 41.8k
7 votes
1 answer
310 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 ...
- 33.5k
6 votes
1 answer
308 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}, \...
- 5,152
8 votes
0 answers
415 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 ...
- 33.5k
6 votes
1 answer
611 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 ...
- 5,561
16 votes
0 answers
826 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 ...
- 33.5k
5 votes
2 answers
521 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}...
- 33.5k
7 votes
3 answers
745 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 ...
- 627
6 votes
0 answers
282 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: $$ (\textrm{SWH}_n) \textrm{ There are no infinite } \nu \...
- 627
9 votes
1 answer
600 views
Some variants of the Shelah's Weak Hypothesis
Are equivalent (in ZFC) the following two statements, for any infinite cardinal $\mu$? (i) For every infinite cardinal $\kappa$, $|\{ \lambda \in \kappa : \lambda \textrm{ is a singular cardinal and} ...
- 627
9 votes
1 answer
739 views
"cov vs pp" problem
This is the problem $(\beta)$ of the section 14.7 in the "Analytical Guide" of the Shelah's book "Cardinal Arithmetic": $(\beta)$ Is $\operatorname{cov} ( \lambda , \lambda, \aleph_{1} , 2) =^{+} \...
- 627
4 votes
2 answers
368 views
Existence of scales with special properties
Let $\kappa$ be a singular cardinal, and let $\langle \kappa_i \mid i<\mathrm{cf}(\kappa) \rangle$ be an increasing sequence of regular cardinals cofinal in $\kappa$. Recall that a scale on $\Pi_{i&...
- 1,663
2 votes
0 answers
252 views
a partial order not dense iff a measurable exists
For $\kappa>0$ a regular cardinal, let $Ht_\kappa$ denote the following partial quasi-order: (i) elements(objects) of $Ht_\kappa$ are classes X of sets of size $\kappa$ with the property that, ($&...
- 468
5 votes
1 answer
624 views
Generalizations of pcf theory
Does anyone know of generalizations of pcf theory where we might consider products of the form: $$\aleph_1 \times (\aleph_2 \times \aleph_2) \times (\aleph_3 \times \aleph_3 \times \aleph_3) \dots$$ ...
- 4,122
6 votes
1 answer
277 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 ...
- 4,122
3 votes
2 answers
618 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 ...
- 3,904