Re: Paper and slides on indefiniteness of CH

Dear Sy,

On Oct 15, 2014, at 3:34 AM, Sy David Friedman wrote:

Getting as far as the \textsf{SIMH}^\# is genuine progress. I provided a direction for further progress in my Maximality Protocol. Be patient, Hugh! Working out the mathematical features of Maximality will take time, and the programme has only just begun.

If things are this tentative then the conjecture “CH is false based on \textsf{SIMH}^\# or some variant thereof” seems a bit curious at best.

Look we have competitor principle: \textsf{IMH}^\#(\text{card-arith}) which we know is consistent. There is the possibility that \textsf{IMH}^\#(\text{card-arith}) implies the GCH. If \textsf{SIMH}^\# is inconsistent then this possibility certainly looks more likely.

As Pen has implied, it is good to have different programmes in set theory, whether they be motivated by sophisticated issues emanating from large cardinal theory and descriptive set theory, like your Ultimate-L programme, or by an “intrinsic heuristic” like the Maximality of V. Your programme is also extremely hard, but I would not fault it for that reason and I hope that it works out as hoped.

It is not whether the questions are hard which is the issue, it is whether at this stage the principles can even be discussed.

The mathematical implications of the Ultimate L Conjecture are clear and there are many. It is just the conjecture which is hard.

The mathematical implications of \textsf{SIMH}^\# are not clear at all beyond failures of the GCH which are trivial. So it is somewhat difficult to have a mathematical discussion about it.

Please re-read the Maximality Protocol: Height Maximality, Cardinal Maximality, Width Maximality, in that order. I gave precise suggestions for Height and Cardinal Maximality; Width Maximality is obviously trickier but at least I made a tentative proposal with the \textsf{SIMH}^\#. The problem with Strong-\textsf{SIMH}^\# was given toward the end of my Max story (Max isn’t happy with \omega_1 being captured by a single real).

So I assume you are referring to this:

The set-theorists tell him that maybe his mistake is to start talking about preserving cardinals before maximising the notion of cardinal itself. In other words, maybe he should require that \aleph_1 is not equal to the \aleph_1 of L[x] for any real x and more generally that for no cardinal \kappa is \kappa^+ equal to the kappa^+ of L[A] when A is a subset of \kappa. In fact maybe he should go even further and require this with L[A] replaced by the much bigger model HOD_A of sets hereditarily-ordinal definable with the parameter A! [Sy's Maximality Protocol, Part 2]

Interesting. If for each uncountable cardinal \kappa and for each $latex A \subset \kappa$, (\kappa^+)^{\text{HOD}_A} is strictly less than \kappa^+ then PD holds.

About the \textsf{SIMH}^\# issue. Sy, you wrote on Sept 27 in your message to Pen:

The \textsf{SIMH}^\# is a “unification” of the \textsf{SIMH} and the \textsf{IMH}^\#. The SIMH is not too hard to explain, but the \textsf{IMH}^\# is much tougher. (I don’t imagine that you found my e-mail to Bob very enlightening!). Let me do the \textsf{SIMH} now, and if you haven’t heard enough I’ll give the \textsf{IMH}^\# a go in my next e-mail.

SIMH

The acronym denotes the Strong Inner Model Hypothesis. For the sake of clarity, however, I’ll give you a simplified version that doesn’t quite imply the original IMH; please forgive that.

A cardinal is “absolute” if it is not only definable but is definable by the same formula in all cardinal-preserving extensions (“thickenings”) of V. For example, \aleph_1 is absolute because it is obviously “the least uncountable cardinal” in all cardinal-preserving extensions. The same applies to \aleph_2, aleph_3, \cdots, \aleph_\omega, \dots for a long way. But notice that the cardinality of the continuum could fail to be absolute, as the size of the continuum could grow in a cardinal-prserving extension (this is what Cohen did when he used forcing to make CH false; Bob Solovay got the ultimate result).

Now recall that the IMH says that if a first-order sentence without parameters holds in an outer model (“thickening”) of V then it holds in an inner model (“thinning”) of V. The SIMH says that if a first-order sentence with absolute parameters holds in a cardinal-preserving outer model of V then it holds in an inner model of V (of course with the same parameters). The SIMH implies that CH is false: By Cohen’s result there is a cardinal-prserving outer model of V in which the continuum has size at least \aleph_2 of V and therefore using the SIMH we conclude that there is an inner model of V in which the continuum has size at least \aleph_2 of V; it follows that also in V, the continuum has size at least \aleph_2, i.e. CH is false. In fact by the same argument, the SIMH implies that the continuum is very, very large, bigger than aleph_alpha for any ordinal alpha which is countable in Gödel’s universe L of constructible sets!

The SIMH# is the same as the SIMH except we require that V is #-generated (maximal in height) and instead of considering all cardinal-preserving outer models of V we only consider outer models of V which are #-generated (maximal in height). It is a “unification” of height maximality with a strong form of width maximality.

The attraction of the \textsf{SIMH}^\# is that it is a natural criterion that mirrors both height and width maximality and solves the continuum problem (negatively).

This to me seems to clearly indicate that at that time \textsf{SIMH}^\# was Strong-\textsf{SIMH}^\#. Sy, you wrote that “\textsf{SIMH}^\# is the same as \textsf{SIMH} except we require that V is #-generated…” I assumed the restriction to cardinal preserving #-generated outer models was just to simplify the discussion since otherwise \textsf{SIMH}^\# would not obviously imply \textsf{IMH}^\#.

So fine, my assumption was not correct or you have changed. Nothing wrong with changing things, it just complicates the discussion.

In any case, perhaps it would be more efficient to postpone our discussion of HP until HP has passed the embryonic stage and things are bit more settled.

Regards.
Hugh

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