$\omega$-small and properly small premice.

Question

Let $\mathcal M$ be a premouse. $\mathcal M$ is said to be $\omega$-small if and only if whenever we have that $\kappa=crit(E)$ for some extender $E$ on the sequence of $\mathcal M$, then $\mathcal J^{\mathcal M}_{\kappa} \not \models$ There are $\omega$-many Woodins. $\mathcal M$ is said to be properly small if and only if $\mathcal M$ is $\omega$-small and $\mathcal M \models$ "there is a largest cardinal and $ZF^-$ (i.e without the power set axiom) and there are no Woodins".

Now if $\mathcal M$ is $\omega$-small then so is any of the $\mathcal J^{\mathcal M}{\beta}$ for any $\beta$ but if $\mathcal M$ is properly small then it could be that there is some $\beta$ for which we have $\mathcal J^{\mathcal M}_{\beta} \models$ there are $\omega$ Woodins.

I am confused about how that can be, since in the definition of properly small we also have that the premouse has to be $\omega$-small. Can someone show me an example? Thx

Answer 1

If $\mathcal M$ is $\omega$-small, then many $\mathcal J^{\mathcal M}_\beta$ may think that (a large fragment of $\mathsf{ZF}$ holds and) there are plenty of Woodin cardinals. What matters is that for any such $\beta$ there is a larger $\tau$ where we see that this is no longer the case.

This may happen for a variety of reasons: Maybe the relevant cardinals got collapsed along the way. Or maybe they remain cardinals but new subsets $A$ got added, for which we do not have witnesses to $A$-strongness. Or maybe the previous witnesses for various sets $A$ do not actually give rise to $A$-strong embeddings of $\mathcal J^{\mathcal M}_\tau$.

The only serious requirement is that $\tau$ must be reached before we arrive at an active stage (one where we add an extender to the sequence).

Properly small mice have the additional requirement that we in fact certify that no cardinals are Woodin by the time we reach the height of the mouse.

Perhaps a simple-minded analogy may help: (Assume $V=L$ if you want.) If $\kappa$ is inaccessible, and $X\prec L_\kappa$ is countable, then the collapse of $X$ is an $L_\beta$ with $\beta$ countable such that $L_\beta$ is a model of $\mathsf{ZFC}$. However, there is an $\alpha$ with $\beta<\alpha<\omega_1$ such that $L_\alpha$ sees that $\beta$ is countable and, of course, by the time we reach $\omega_1$ we have certified that no smaller ordinal (past $\omega$) is a cardinal.