Dear Sy,

On Oct 26, 2014, at 7:39 PM, Sy David Friedman wrote:

Dear Peter,

But probably there’s a proof of no Reinhardt cardinals in ZF, even without Ultimate-L:

Conjecture: In ZF, the Stable Core is rigid.

Note that V is generic over the Stable Core.

I took a brief look at your paper on the stable core and did not immediately see anything that genuinely seemed to argue for the conjecture you make above. (Maybe I just did not look at the correct paper).

Are you just really conjecturing that there is no (nontrivial) , or more generally that if V is a generic extension of an inner model N (by a class forcing which is amenable to N) then here is no nontrivial ? Or is there extra information about the Stable Core which motivates the conjecture?

I would think that based on HP etc., you would actually conjecture that there *is* a nontrivial . This would have the added advantage explaining why follows from maximality considerations etc. (This declaration you have made at several points in this thread and which I must confess I have never really understand the reasons for.)

This seems like a perfect opportunity for you to use your conception of HP and boldly make a conjecture. (i.e. that the existence of is consistent because by the HP protocols, the class free version, stated as (1) below, must be true in the preferred ct.’s and these are #-generated).

The axiom (that there is such a ) surely transcends the hierarchy we have now. So this HP insight if well grounded would be a remarkable success.

You could then very naturally go further and modify your unreachability property to:

if is a proper inner model of then there is a nontrivial .

If fact you could combine everything and go with the following perfect pairing:

1) For all sufficiently large , there is a nontrivial elementary embedding

.

2) If is a proper inner model of (definable from parameters) then for some set .

In the Ultimate-L approach one faces a similar choice but there one is compelled to reject

such non-rigidity conjectures (since they must be false in that approach).

But why do you? You did after all write to Peter on Oct 19:

Well, since this thread is no stranger to huge extrapolations beyond current knowledge, I’ll throw out the following scenario: By the mid-22nd cenrury we’ll have canonical inner models for all large cardinals right up to a Reinhardt cardinal. What will simply happen is that when the LCs start approaching Reinhardt the associated canonical inner model won’t satisfy AC. The natural chain of theories leading up the interpretability hierarchy will only include theories that have AC: they will assert the existence of a canonical inner model of some large cardinal. These theories are better than theories which assert LC existence, which give little information.

Regards,

Hugh