Thanks, that helps. But just to be clear, does imply the following statement?
If holds of in a #-generated outer model of which preserves then holds of in an inner model of .
I don’t see why it should.
The reason I ask is that for , the analogous statement (deleting #-generated) holds and is implied by .
Hugh, the HP is (primarily) a study of maximality criteria of the sort we have been discussing. As I have been trying to explain, it is essential to the programme to formulate, analyse, compare and synthesise different criteria, discovering their mathematical consequences. I referred to my formulation of the as “crude and uncut” as it may have to be modified later as we learn more. Changes in its formulation do not mean a defeat for the programme, but rather progress in our understanding of maximality.
So it makes no sense to assert that if a particular formulation of maximality coming out of the programme contradicts large cardinal existence then the programme is a failure and therefore irrelevant to the resolution of CH. Indeed the first HP criterion, the IMH, did contradict large cardinals, but it was later for compelling reasons synthesised with #-generation into the , which does not. It is not yet clear if the optimal maximality criterion will be compatible with large cardinal existence. It is certainly not the intention of the programme to take a stance on large cardinal existence “in advance” before seeing what maximality criteria are out there.
Hugh Woodin wrote to Sy Friedman:
I will try one more time. At some point HP must identify and validate a new axiom. Otherwise HP is not a program to find new “axioms”. It is simply part of the study of the structure of countable wellfounded models no matter what the motivation of HP is.
It seems that to date HP has not done this. Suppose though that HP does eventually isolate and declare as true some new axiom. I would like to see clarified how one envisions this happens and what the force of that declaration is. For example, is the declaration simply conditioned on a better axiom not subsequently being identified which refutes it? This seems to me what you indicate in your message to Pen.
Out of LC comes the declaration “PD is true”. The force of this declaration is extreme, within LC only the inconsistency of PD can reverse it.”
As I understand it, the original IMH put forward by Sy has the following drawbacks. Both of you were involved in getting upper and lower bounds on the strength of original IMH.
- The official presentation is in terms of countable transitive models, and not a sentence or scheme in set theory itself. Thus in a careful presentation of IMH, one defines what a countable IMH model is, and the IMH states that there is a countable IMH model.
- Sy proves that no countable IMH model satisfies that there exists an inaccessible cardinal.
- There are versions of IMH, not the ones emphasized, that are statements in class theory – incidentally, not set theory – involving class forcing. Then one can talk about a theory extending NBG + Global Choice. And this theory, as I understand it, proves that there are no inaccessible cardinals.
- Of course, the existence of an inaccessible cardinal represents (a consequence of) a much clearer and convincing kind of maximality than IMH with proper class forcing. Countable model IMH – the official version – doesn’t represent any kind of maximality principle whatsoever involving sets and classes. So to turn this around, inaccessible cardinals refute IMH (class forcing).
- As I understand it, there has been a reworking of IMH so that the new form(s) are known to be, or conjectured to be, consistent with some large cardinals. In order to address the first sentence above from Hugh Woodin, these new form(s) need to be formulated not as modified IMH (countable models) but as modified IMH (class forcing), in order for people to examine what their value is for “intrinsic” foundations of set theory. I understand that, technically, there may be little or no difference between the two formulations of modified IMH (at least in terms of results), but there is a major difference in terms of direct relevance to the “intrinsic”.
- I glanced briefly at some account of one of these variants of IMH – yes for countable models, but presumably it has a class forcing version — and it was at least one layer more involved than the original IMH (any form). My concern is that accommodating the IMH idea with large cardinals will look like an ad hoc glueing of separate ideas, but what I briefly glanced at was not friendly reading.
- Of course, you can simply start with ZFC + LCs, and look at only “outer models” satisfying ZFC + the same LCs. This would be the most obvious plan. Does anything interesting come out of this simple minded fix (countable models or class forcing forms)?
- What would you two recommend in terms of the clearest statement of “second generation” IMH that accommodates at least some large cardinals so that we can judge how natural this fix (or multiple fixes) of IMH is?
- I have no criticism per se of people treating countable transitive models of ZFC in a way roughly akin to, say, group theorists treating countable groups (although of course there are very substantial connections with other areas of mathematics, which we do not have for ctm’s) – as specialized mathematical activity. What is at issue is whether such investigations rise to the level of affecting our understanding of the foundations of set theory – or whether they hold any promise for doing so.
Any comments from any of you two should be illuminating.