Tag Archives: Large cardinals from HP

Re: Paper and slides on indefiniteness of CH

Dear Sol,

On Thu, 7 Aug 2014, Solomon Feferman wrote:

Dear Sy,

I’m very pleased that my paper has led to such a rich exchange and that it has brought out the importance of clarifying one’s aims in the ongoing development of set theory. Insofar as it might affect my draft, I still have much to absorb in the exchange thus far, and there will clearly be some aspects of it that are beyond my current technical competence. In any case, I agree it would be good to bring the exchange to a conclusion with a summary of positions.

Thanks again for triggering the discussion with your interesting paper.

In the meantime, to help me understand better, here is a question about HP: if I understand you properly, if HP is successful, it will show the consistency of the existence of large large cardinals in inner models.

To be clear, the HP does not produce a single criterion for preferred universes, but a family of them, and each must be analysed for its consequences. But many such criteria will indeed produce inner models with at least measurable cardinals and I would conjecture that an inner model with a Woodin cardinal should also come out. However the programme achieves this only via the core model theory and not directly on its own. In particular I see no scenario for it to produce an inner model with a supercompact, as the core model theory seems unable to do that.

On the other hand all of the criteria seem to be compatible with the existence of arbitrarily large cardinals in inner models, even if they fail to produce such iner models.

However I don’t consider the creation of inner models with large cardinals, or even the confirmation of the consistency of large cardinals, to be a central goal of the programme. The programme will likely have more valuable consequences for understanding problems like CH whose undecidability does not hinge on large cardinal assumptions.

Then how would it be possible to establish the success of HP without assuming the consistency of large large cardinals in V?  If so, isn’t the program circular?  If not, it appears that one would be getting something from nothing.

The answer is given by core model theory: Without assuming the consisency of large cardinals one can use this theory to show that various set-theoretic properties yield inner models with large cardinals. A nice example is the failure of the singular cardinal hypothesis, which without any further assumptions produces inner models with many measurable cardinals.

All the best,
Sy

Re: Paper and slides on indefiniteness of CH

Dear Sy,

I’m very pleased that my paper has led to such a rich exchange and that it has brought out the importance of clarifying one’s aims in the ongoing development of set theory. Insofar as it might affect my draft, I still have much to absorb in the exchange thus far, and there will clearly be some aspects of it that are beyond my current technical competence. In any case,  I agree it would be good to bring the exchange to a conclusion with a summary of positions.

In the meantime, to help me understand better, here is a question about HP: if I understand you properly, if HP is successful, it will show the consistency of the existence of large large cardinals in inner models. Then how would it be possible to establish the success of HP without assuming the consistency of large large cardinals in V?  If so, isn’t the program circular?  If not, it appears that one would be getting something from nothing.

Best,
Sol