Dear Sy,
On Oct 15, 2014, at 3:34 AM, Sy David Friedman wrote:
Getting as far as the 
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 or some variant thereof” seems a bit curious at best.
Look we have competitor principle: 
which we know is consistent. There is the possibility that implies the GCH. If 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 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 . The problem with Strong- was given toward the end of my Max story (Max isn’t happy with 
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 
is not equal to the of ![L[x]](//s0.wp.com/latex.php?latex=L%5Bx%5D&bg=ffffff&fg=000&s=0 "L[x]")
for any real 
and more generally that for no cardinal 
is 
equal to the 
of ![L[A]](//s0.wp.com/latex.php?latex=L%5BA%5D&bg=ffffff&fg=000&s=0 "L[A]")
when 
is a subset of . In fact maybe he should go even further and require this with replaced by the much bigger model 
of sets hereditarily-ordinal definable with the parameter ! [Sy's Maximality Protocol, Part 2]
Interesting. If for each uncountable cardinal and for each $latex A \subset \kappa$, 
is strictly less than then PD holds.
About the issue. Sy, you wrote on Sept 27 in your message to Pen:
The is a “unification” of the 
and the 
. The SIMH is not too hard to explain, but the is much tougher. (I don’t imagine that you found my e-mail to Bob very enlightening!). Let me do the now, and if you haven’t heard enough I’ll give the 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, is absolute because it is obviously “the least uncountable cardinal” in all cardinal-preserving extensions. The same applies to 
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 
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 of V; it follows that also in V, the continuum has size at least , 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 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 was Strong-. Sy, you wrote that “ is the same as 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 would not obviously imply .
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
takes more work, but can be done. Of course the difficulty is that we are not dealing with actual objects but with the consistency of theories. I’ll write more about this when I get a chance.
in 
or even in 
where 
witnesses that 
which we know is consistent. There is the possibility that 
being captured by a single real).
is not equal to the ![L[x]](http://s0.wp.com/latex.php?latex=L%5Bx%5D&bg=ffffff&fg=000&s=0 "L[x]")
for any real 
and more generally that for no cardinal 
is 
equal to the 
of ![L[A]](http://s0.wp.com/latex.php?latex=L%5BA%5D&bg=ffffff&fg=000&s=0 "L[A]")
when 
is a subset of 
of sets hereditarily-ordinal definable with the parameter 
is strictly less than 
and the 
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).
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 
seems absolutely critical to me which is why I kept pressing you on this.![\omega_1 = \omega_1^{L[x]}](http://s0.wp.com/latex.php?latex=%5Comega_1+%3D+%5Comega_1%5E%7BL%5Bx%5D%7D&bg=ffffff&fg=000&s=0 "\omega_1 = \omega_1^{L[x]}")
. The latter is a very deep theorem which is an immediate corollary of the sophisticated machinery of coding by class forcing that you have developed.
latex \textsf{SIMH}^\# instead of to Strong-
.
holds of 
in a #-generated outer model of V which preserves 
.