# Why do we require soundness for fine structural ultrapowers

set-theory

So for this question you may assume the background fine structure is that of "Fine structure and iteration trees".

I actually have two questions which are related to properties of the degree of some node in an iteration tree.

First is that when constructing the $$n$$th fine structural ultrapower of some ppm, we require it to be $$n$$-sound. But as far as I see it, if the critical point of the extender is below the $$n$$th projectum, then the extender measures all the $$r\Sigma_n$$ subsets of its critical point, so we can actually go ahead with the construction. So why do we require $$n$$-soundness?

Secondly, do all fine structural ultrapowers provide new information? i.e. if $$\mathcal{M}$$ is some ppm and $$E$$ is an extender over some $$\kappa < \rho_{n+1}(\mathcal{M})$$, is it possible that $$\operatorname{Ult}_{n+1}(\mathcal{M}; E) = \operatorname{Ult}_{n}(\mathcal{M}; E)$$? Does soundness play a role here?

First question: I haven't personally considered situations where taking an ultrapower at degree higher than the soundness degree seemed to be relevant. Of course that certainly doesn't mean that it's not, and I think something like that is done in Jensen's $$\Sigma^*$$ fine structure. However, there is an example in one of the standard texts showing that at the level of infinitely many measurable cardinals, there is a $$2$$-sound mouse $$M$$, with $$M$$ having a measurable cardinal $$\mu<\rho_2^M$$, such that given any real $$x$$, there is an linear iteration $$M$$, with critical points above $$\rho_1^M$$, such that $$x$$ is boldface-$$\Sigma_2$$ definable over the eventual iterate (but is not over $$M$$, since $$\omega<\mu<\rho_2^M$$). So, as they say there, random information can be coded into $$\Sigma_2$$ over $$0$$-sound structures, so one should take the $$1$$-core before considering $$\Sigma_2$$ (or $$r\Sigma_2$$). As I say, though, I think that one forms larger ultrapowers via $$\Sigma^*$$ fine structure, but I don't know the details.

Second question: Assuming $$M$$ is $$(n+1)$$-sound and $$\mathrm{crit}(E)<\rho_{n+1}^M$$, it can certainly be that $$\mathrm{Ult}_{n}(M,E)=\mathrm{Ult}_{n+1}(M,E)$$, even with the natural factor map $$\sigma:\mathrm{Ult}_n(M,E)\to\mathrm{Ult}_{n+1}(M,E)$$ being the identity. Equivalently, it can be that all pairs $$[a,f]$$ used in forming the degree $$n+1$$ ultrapower are represented (mod measure 1) with pairs $$[a,f]$$ used in the degree $$n$$ ultrapower. For a simple example, this happens (for all extenders over $$M$$ and $$n<\omega$$ simultaneously) if $$M$$ models $$\mathrm{ZF}^-$$. But there are also more interesting examples (for more details on the following things see e.g. $$\S$$2 of A premouse inheriting strong cardinals from $$V$$ and $$\S$$3 of Fine structure from normal iterability): Suppose $$\kappa=\mathrm{crit}(E)<\rho_{n+1}^M$$ and there is no boldface-$$\mathrm{r}\Sigma_{n+1}^M$$-cofinal function from $$\kappa$$ to $$\rho_n^M$$. Then $$U_n=\mathrm{Ult}_n(M,E)=\mathrm{Ult}_{n+1}(M,E)=U_{n+1}$$ and the natural factor map $$\sigma$$ is the identity. To see this, use the following observations: (i) $$\rho_n^{U_{n+1}}=\sup\sigma\rho_n^{U_n}$$, since both are the sups of the ultrapower maps applied to $$\rho_n^M$$ (for $$U_n$$, just because it is a degree $$n$$ ultrapower; for $$U_{n+1}$$, by the cofinality assumption) and as the maps commute. (ii) $$\sigma$$ is an $$n$$-embedding (use (i) and commutativity for the $$\mathrm{r}\Sigma_{n+1}$$-elementarity). (iii) $$U_n,U_{n+1}$$ are both $$(n+1)$$-sound and $$\rho_{n+1}^{U_{n+1}}=\rho_{n+1}^{U_n}\leq\mathrm{crit}(\sigma)$$ and $$\sigma$$ preserves $$\vec{p}_{n+1}$$. This leads to (iv) $$U_{n+1}\subseteq\mathrm{rg}(\sigma)$$, which is enough.