Tag Archives: Definability of IMH#

Re: Paper and slides on indefiniteness of CH

Dear Peter and Sy,

I would like to add a short comment about the move to \textsf{IMH}^\#. This concerns to what extent it can be formulated without consulting the hyper-universe is an essential way (which is the case for \textsf{IMH} since \textsf{IMH} can be so formulated). This issue has been raised several times in this thread.

Here is the relevant theorem which I think sharpens the issues.

Theorem. Suppose \textsf{PD} holds, Let X be the set of all ctm M such that M satisfies \textsf{IMH}^\#. Then X is not \Sigma_2-definable over the hyperuniverse. (lightface).

Aside: X is always \Pi_2 definable modulo being #-generated and being #-generated is \Sigma_2-definable. So X is always \Sigma_2\wedge \Pi_2-definable. If one restricts to M of the form L_{\alpha}[t] for some real t, then X is \Pi_2-definable but still not \Sigma_2-definable.

So it would seem that internalization \textsf{IMH}^\# to M via some kind of vertical extension etc., might be problematic or might lead to a refined version of \textsf{IMH}^\# which like IMH has strong anti-large cardinal consequences.

I am not sure what if anything to make of this, but I thought I should point it out.

Regards,
Hugh