Definability of IMH# | Set Theory — The Search for New Axioms

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

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