6
$\begingroup$

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 with $\exists \kappa (o(\kappa) = \kappa^{++})$.

What else is known about the consistency strength of the failure of SSH?

$\endgroup$
8
$\begingroup$

Theorem. The following are equiconsistent:

1) SSH fails at some singular cardinal $\kappa>2^{\aleph_0}$,

2) there is a measurable cardinal $\kappa$ with $o(\kappa)=\kappa^{++}.$

The theorem follows from the following observations:

a) (Gitik) Suppose there is no inner model with a measurable cardinal $\kappa$ with $o(\kappa)=\kappa^{++}.$ Then for any singular cardinal $\kappa>2^{\aleph_0}, pp(\kappa)=\kappa^+.$

b) (Shelah) Let $\kappa$ be the least cardinal violating SCH. Then $\kappa$ has cofinality $\omega$ and $pp(\kappa) >\kappa^+.$

$\endgroup$
5
$\begingroup$

Shelah's strong hypothesis is that $\mathrm{pp}(\lambda)=\lambda^+$ for all singular cardinals $\lambda$. (For a quick review of the different pcf-theoretic assumptions in the neighborhood of SSH, see here.)

As pointed out by Mohammad, the failure of SSH (past the continuum) is equiconsistent with $o(\kappa)=\kappa^{++}$. This fits naturally with the thesis advanced in Cardinal arithmetic, section II.2, and is addressed explicitly in Gitik's original paper on the strength of SCH,

Moti Gitik. The strength of the failure of the singular cardinal hypothesis, Ann. Pure Appl. Logic, 51 (3), (1991), 215–240. MR1098782 (92f:03060).

Indeed, the main result of the paper is stated in terms of $\mathrm{pp}$ rather than $\mathsf{SCH}$:

Main theorem. Assume that $\lnot(\exists\alpha\,o(\alpha)=\alpha^{++})$. Let $\kappa>2^{\aleph_0}$ be a singular cardinal. Then $\mathrm{pp}(\kappa)=\kappa^+$.

Note that (as explained in the paper) what Gitik really means by $\lnot(\exists\alpha\,o(\alpha)=\alpha^{++})$ is that the core model $K$ exists, and that, in $K$, no $\alpha$ has Mitchell order $\alpha^{++}$.

See also the introduction to [Sh400] (Chapter IX in the book), where the history of this result is recalled:

Lately Gitik [Gi1] continuing works of Magidor [Mg] and Woodin, show[ed] $\mathsf{ZFC} + o(\kappa) =\kappa^{++}$ is an upper bound for consistency strength of "cardinal arithmetic non-trivial". Earlier Mitchell proved $$ ZFC + \kappa=\bigcup_{n<\omega}\kappa_n\ \&\ o(\kappa_{n+ l}) > \kappa_n $$ is a lower bound. When Mitchell visited Jerusalem he tried to convince me to get a complimentary consistency result (probably to show he is right) and I tried to convince him to show that the consistency strength of $\mathrm{pp}(\lambda) > \lambda^+\ \&\ (\lambda$ singular$)$ is $o(\kappa) =\kappa^{++}$ [...] (to show that I was right in [Sh-b, XIII,95,96)]); but it all comes to nothing. Also Gitik was not convinced and forgot it. But in Spring '89 he succeeded to get $o(\kappa) =\kappa^{++}$ as lower bound, for a closed but distinct problem; in his proof he was cornered to use $\mathrm{pp}(\lambda) > \lambda^+$ and $(\forall\mu<\lambda)\mu^{\aleph_0}<\lambda,\mathrm{cf}(\mu)=\aleph_0$.

The main recent news with respect to Shelah's pcf hypotheses is that Gitik announced around Summer 2011 that it is consistent to have a progressive set of cardinals $\mathfrak a$ with $|\mathrm{pcf}(\mathfrak a)|>|\mathfrak a|$. A preprint can be found at Gitik's page, as

Moti Gitik. Short extenders forcing. II, preprint, July 24, 2013.

That this is impossible is a consequence of SSH, and Gitik (perhaps surprisingly) shows that it can be violated from assumptions at the level of strong cardinals. The specific result is the consistency of the existence of a countable $\mathfrak a$ with $\mathrm{pcf}(\mathfrak a)$ of order type $\omega_1+1$. The assumption required for this is below one strong cardinal (but we need a sequence of cardinals with some -significant- degree of strongness). This also violates Shelah's Weak Hypothesis, as discussed here.

Beyond this, certain specific violations of this statement would require at least the strength of Projective Determinacy (and really, the expectation is that much more is needed, if at all possible). See

Moti Gitik, Ralf Schindler, and Saharon Shelah. Pcf theory and Woodin cardinals, in Logic Colloquium '02, Zoé Chatzidakis, Peter Koepke, and Wolfram Pohlers, eds., Lecture Notes in Logic, 27, Assoc. Symbol. Logic, La Jolla, CA, 2006, pp. 172-205.

$\endgroup$
4
$\begingroup$

Additional informations from

Pierre Matet. Large cardinals and covering numbers, Fundamenta Mathematicae 205 (2009), 45-75. doi:10.4064/fm205-1-3

on the first page:

And, as Moti Gitik pointed out to the author, one obtains a model of "$u (\omega_1 , \omega_{\omega}) > {\omega_{\omega}}^+$ (and hence $\neg$SSH) + SCH" by adding $\aleph_{\omega + 1}$ Cohen reals to a model of "for every infinite cardinal $\nu$, $2^{\nu}$ equals $\nu^{++}$ if $\nu = \omega_{\omega}$, and $\nu^+$ otherwise".

and

It is not known whether the failure of SSH is equiconsistent with that of SCH.

$\endgroup$
  • 1
    $\begingroup$ Yes, it is a curious phenomenon, that small cardinals (usually, $\mathfrak c$) end up being the ones difficult to treat sometimes. Other examples of this are the covering properties studied by Viale here (see lemma 6), or the lower bounds for $\lnot\square(\kappa)+\lnot\square_\kappa$ as studied here (see theorem 0.3). $\endgroup$ – Andrés E. Caicedo Nov 18 '13 at 19:46

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.