Questions tagged [cardinal-arithmetic]
The cardinal-arithmetic tag has no usage guidance.
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questions with no upvoted or accepted answers
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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 ...
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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 ...
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Categorifying Hyperoperations
Is there some categorical version of tetration or higher hyperoperations?
This is motivated by the fact that coproducts categorify addition of finite cardinals, and products/exponentials categorify ...
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Model for "$\kappa$ limit cardinal iff $2^\kappa$ limit cardinal"
Is there a model of ${\sf (ZFC)}$ such that in the model we have that $\kappa$ is a limit cardinal if and only if $2^\kappa$ is a limit cardinal?
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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:
$$
(\textrm{SWH}_n) \textrm{ There are no infinite } \nu ...
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Shelah’s Representation Theorem: existence of scales
Let $\lambda$ be a singular cardinal of countable cofinality. Shelah’s Representation Theorem states that there is an increasing sequence of regular cardinals $\langle\delta_n\rangle_{n<\omega}$, ...
5
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Does this proof by Shelah use any "hidden assumptions"?
Recall that the approachability ideal for $\kappa$, denoted $I[\kappa]$, consists of all sets $A\subseteq\kappa$ such that there is a sequence $\overline{a}=(a_{\alpha})_{\alpha\in\kappa}$ of bounded ...
5
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Existence of a large family of sets with big differences
Let $\lambda\leq\kappa$ be two cardinals and $\Gamma$ be a set with $|\Gamma|=\kappa$.
Question. Does there exist a family $\mathscr{A}\subseteq \{A\subseteq \Gamma: |A|=\lambda\}$ with $|\mathscr{A}|...
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Characterizations for SSH and SCH above an uncountable cardinal
SSH asserts that pp$(\lambda) =\lambda^+$ for every singular cardinal $\lambda$. There are two nice characterizations for SCH and SCH in terms of covering numbers (see for example "Large ...
4
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Reference request for an elementary identity in cardinal arithmetic
Let $\kappa$ be an infinite cardinal. Then we have
$$\beth_{\kappa+1} = \prod_{\alpha<\kappa} \beth_{\alpha} = \beth_{\kappa}^{\kappa}$$
The only non-trivial inequality is the first less-than-or-...
4
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Generic two-cardinal behavior of first-order sentences
This is a hopefully improved version of a question I asked before and then deleted because it was based on some fundamentally incorrect assumptions.
Some first-order theories are able to control the ...
4
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Does Tarski's squaring theorem imply Axiom of Choice in NFU?
I'm trying to see which results from mainstream set theory (ZF) about Axiom of Choice can be proved in New Foundations with Urelements (U is added simply because ...
3
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Basic cardinal arithmetic without choice
Do we know everything about addition and multiplication of cardinalities in choiceless set theory?
For example, let $M$ be a model of $\textsf{ZF}+\textsf{AD}+V=L(\mathbb{R})$, consider the sets $\...
2
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Can be this "handwaving" idea about "counting" reals somehow put on solid ground?
We know that the Cantor's cardinality of a countable set is $\aleph_0$ and the cardinality of continuum is $2^{\aleph_0}=\aleph_0^{\aleph_0}$. Unfortunately, this measure is based on the idea of ...
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Cardinal numbers and the Bolzano-Weierstrass theorem
Let $\kappa$ be a cardinal number, define $\textsf{M}(\kappa)$ and $\textsf{BW}(\kappa)$ as follows:
$\textsf{M}(\kappa)$ : For every sequence $(f_{n}:\kappa\to \mathbb{R})_{n\in\mathbb{N}}$ of real-...
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CH vs Not CH, What is the Consequence?
EDIT: Altered (3) to say that $\beth_1$ can't be $\aleph_{\omega}$ or $\aleph_0$ but removed statement that it must be less than $\aleph_{\omega}$.
Let us assume ZFC. We now consider 2 transfinite ...