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-9
votes
0answers
61 views

Using the intuition from the arithmetic of cardinal numbers to ascribe values to functions on the extended real line? [closed]

Please note, this question is about ascribing exact values to indefinite forms, not about fining corresponding limits, so please do not answer "there is no such limit, so it is indeterminate". First ...
14
votes
1answer
562 views

Is it possible to define higher cardinal arithmetics

In number theory there are several operators like ‎addition, ‎multiplication and ‎exponentiation defined from ‎$‎‎‎\omega‎‎\times‎‎\omega‎$ ‎to ‎‎$‎‎‎\omega‎$. Each ‎of ‎them ‎is defined as an ...
6
votes
2answers
371 views

Exponentiation and Dedekind-finite cardinals

It is known that the sum and the product of two Dedekind-finite cardinals are also Dedekind-finite cardinals. What about cardinal exponentiation ? Question: Let A and B be two Dedekind-finite ...
11
votes
0answers
350 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 ...
5
votes
2answers
266 views

For which cardinal numbers $\kappa$ is it consistent with ZFC that $\kappa^{\mathrm{cf}(\kappa)} < \kappa^\kappa$?

ZFC proves that $\kappa^{\mathrm{cf}(\kappa)} \leq \kappa^\kappa$ for all infinite cardinal numbers $\kappa$. Further, it is consistent with ZFC that we always have equality (e.g. assume GCH). ...
5
votes
2answers
257 views

When are all greater cardinals sharply greater?

Makkai and Paré introduced the following binary relation on regular cardinals: given $\kappa$ and $\lambda$, $\kappa \vartriangleleft \lambda$ (read, $\kappa$ is sharply less than $\lambda$) when ...
5
votes
2answers
279 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 ...
6
votes
3answers
347 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 ...
8
votes
0answers
187 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 } ...
22
votes
1answer
1k views

Possible troubles in Shelah's book “Cardinal Arithmetic”

I found some possible troubles in Observation 5.3(7) in the Chapter II of the Shelah's book "Cardinal Arithmetic" (page 86). For convenience, I quote the result and the proof in the book here ...
9
votes
1answer
386 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} ...
10
votes
1answer
533 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) =^{+} ...
3
votes
3answers
143 views

Subquotients in ZF

In ZF we have the two relations $A \leq B$ and $A \leq^\ast B$ which relate the size of sets: the first says there is an injection from $A$ to $B$, the second that there is a surjection from $B$ to ...
5
votes
4answers
333 views

do behavior of gimel or GCH determine all infinte products of cardinals?

Let $Card$ be the class of infinite cardinals and $p\colon Card^2\to Card$ be given by $(\kappa,\lambda)\mapsto\kappa^\lambda$. Assuming GCH it is known that $p(\kappa,\lambda)$ is either $\kappa$ (if ...
1
vote
1answer
370 views

Cardinal Arithmetic, foundations and constructive math

This is not my area but a question occurred to me that I can not find the answer to. There is a very strong axiom of constructibility which ironically gives us highly non-constructive math (GCH is one ...
1
vote
1answer
366 views

About the notions of Grothendieck Universe and Tarski Universe

I assume ZFC. Let $U$ a set with the following (1), (2), (3): 1) $\omega\in U$ 2) $x\in U\ \Rightarrow x\subset U$ 3) $x\in U\ \Rightarrow \mathcal{P}(x)\in U$ (where $\mathcal{P}(x):=${$y| ...
2
votes
1answer
307 views

A question about cardinal arithmetic

Let [J]: Jech "Set theory" (Millenium edition) Let $\kappa$ a limit ordinal. From [J], T.3.11, p. 33 we have that $\kappa<\kappa^{cf(\kappa)}$. I improved that proof, and obtain : $\kappa ...
7
votes
1answer
424 views

Does this “jumping-ahead” ordinal function exist?

While working on a project in operator algebras with a collaborator (and fellow MO user), we are able to successfully complete a transfinite induction assuming that the following has an affirmative ...
7
votes
1answer
374 views

Is there always an uncountable $\kappa$ such that $\kappa^{\aleph_0}=2^\kappa$?

The cardinal equation $\kappa^{\aleph_0}=2^\kappa$ is satisfied by $\kappa=\aleph_0$. It is also satisfied by any $\kappa$ for which $MA(\kappa)$ holds. Under $GCH$, the equation is satisfied by ...