Questions tagged [cardinal-arithmetic]

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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 ...
Mohammad Golshani'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
9 votes
0 answers
236 views

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 ...
Alec Rhea's user avatar
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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?
Dominic van der Zypen's user avatar
7 votes
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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
6 votes
0 answers
161 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 $\langle\delta_n\rangle_{n<\omega}$, ...
Seba Thei's user avatar
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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 ...
Hannes Jakob's user avatar
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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}|...
Paolo Leonetti's user avatar
4 votes
0 answers
126 views

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 ...
Seba Thei's user avatar
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4 votes
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172 views

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-...
Burak's user avatar
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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 ...
James Hanson's user avatar
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4 votes
1 answer
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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 ...
Veky's user avatar
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3 votes
0 answers
178 views

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 $\...
new account's user avatar
2 votes
0 answers
276 views

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 ...
Anixx's user avatar
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2 votes
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235 views

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-...
Gabriel Medina's user avatar
-4 votes
0 answers
155 views
+50

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 ...
E8 Heterotic's user avatar