Questions tagged [pcf-theory]
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27
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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 $\langle\delta_n\rangle_{n<\omega}$, ...
15
votes
1
answer
419
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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<\...
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 ...
7
votes
0
answers
130
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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 $\langle \mu_n:n<\omega\rangle$ ...
11
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0
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275
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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 ...
11
votes
1
answer
304
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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 $[\mu]^\kappa$ ...
6
votes
1
answer
157
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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 $i_0$...
1
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0
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192
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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, ...
13
votes
1
answer
426
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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 $\...
17
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0
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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 ...
9
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2
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489
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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 $\...
9
votes
1
answer
539
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"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 ...
7
votes
0
answers
205
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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 ...
6
votes
1
answer
270
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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}, \...
8
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371
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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 ...
6
votes
1
answer
512
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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 ...
17
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757
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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
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435
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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}...
7
votes
3
answers
638
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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
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0
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255
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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:
$$
(\textrm{SWH}_n) \textrm{ There are no infinite } \nu ...
9
votes
1
answer
543
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Some variants of the Shelah's Weak Hypothesis
Are equivalent (in ZFC) the following two statements, for any infinite cardinal $\mu$?
(i) For every infinite cardinal $\kappa$, $|\{ \lambda \in \kappa : \lambda \textrm{ is a singular cardinal and} ...
10
votes
1
answer
677
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"cov vs pp" problem
This is the problem $(\beta)$ of the section 14.7 in the "Analytical Guide" of the Shelah's book "Cardinal Arithmetic":
$(\beta)$ Is $\operatorname{cov} ( \lambda , \lambda, \aleph_{1} , 2) =^{+} \...
4
votes
2
answers
302
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Existence of scales with special properties
Let $\kappa$ be a singular cardinal, and let $\langle \kappa_i \mid i<\mathrm{cf}(\kappa) \rangle$ be an increasing sequence of regular cardinals cofinal in $\kappa$. Recall that a scale on $\Pi_{i&...
2
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231
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a partial order not dense iff a measurable exists
For $\kappa>0$ a regular cardinal, let $Ht_\kappa$ denote the following partial quasi-order:
(i) elements(objects) of $Ht_\kappa$ are classes X of sets of size $\kappa$
with the property that,
($&...
5
votes
1
answer
550
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Generalizations of pcf theory
Does anyone know of generalizations of pcf theory where we might consider products of the form:
$$\aleph_1 \times (\aleph_2 \times \aleph_2) \times (\aleph_3 \times \aleph_3 \times \aleph_3) \dots$$
...
6
votes
1
answer
221
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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 ...
3
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2
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530
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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 ...