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So I am trying to learn a little bit about iteration trees, and have decided to read a note by Steel, found here. On page 2, just after exercise 3, he uses the Hull "operator" in a way which I can't make sense of. To give some context:

Let $M\models \mathsf{ZFC}^-$, $\mathsf{ZFC}^-$ is $\mathsf{ZFC}$-$\mathsf{Powerset}$, $j:M\rightarrow N$ be elementary with $\kappa = \operatorname{crit}(j)$ and $\kappa \in \operatorname{wfp}(N)$(the well-founded part of $N$) and let $U_j$ be the usual ultrafilter derived from $j$ and let $k:\operatorname{Ult}(M, U_j)\rightarrow N$ be the factor map.

Now it is claimed that $\operatorname{Ult}(M, U_j)$ is isomorphic to $\operatorname{Hull}^N(\operatorname{ran}(j) \cup \{\kappa\})$ via $k$. What I don't understand is how do we construct the hull for general $N$ and what does it mean? In the case that we have a definable well-ordering of $N$, we can define something like "canonical" Skolem terms, just as we do for $L$. But in general, I don't see how this can be done.

And also another problem is a canonicity problem. Say we have managed to define the hull for general $N$ using some Skolem functions. But then it is claimed that the isomorphism is witnessed via $k$, and since different Skolem functions may give different hulls, we need a canonical way of finding such Skolem terms/functions. Which complicates the matter further for me.

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You do not need much to recover the full ultrapower. In fact, the $\Sigma_1$-weak Skolem hull should suffice, where the latter is defined by using not all Skolem functions but only those for $\Sigma_1$-formulas, and not even that, but only those functions defined as follows: given a $\Sigma_1$ formula $\varphi(t,y_1,\dots,y_n)$, let $f_\varphi:{}^nN\to N$ be the map given by $f_\varphi(a_1,\dots,a_n)=\emptyset$, unless there is a unique $b\in N$ such that $$N\models``\mbox{$b$ is the unique $t$ such that }\varphi(t,\vec a)\mbox{"},$$ in which case $f_\varphi(a_1,\dots,a_n)=b$. (Note that, even if $N$ is a proper class, we only need a partial satisfaction predicate to define this hull, and our theory should be enough to verify that $N$ has such a predicate.)

The point is that there is a very simple way of recovering every element of the ultrapower from the range of the embedding and its critical point.

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  • $\begingroup$ Thanks for the answer! After a little bit of thinking, I think I now understand what's going on except for one small detail. I don't know what a partial satisfaction relation is. Can you explain or give a reference(if needed), please? $\endgroup$ – Shervin Sorouri Nov 4 '20 at 13:31
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    $\begingroup$ For each $n$, you can define in $N$ the satisfaction relation $\models_n$ for $\Sigma_n$ formulas even though the full $\models$ is not definable in $N$. To define the relevant hull, you just need $\models_n$ for a very small $n$ ($n=3$ should be enough; all you need is to be able to define and refer to the functions $f_\varphi$ as above). $\endgroup$ – Andrés E. Caicedo Nov 4 '20 at 13:45
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    $\begingroup$ "I can get some satisfaction..." (The Standing Rocks) $\endgroup$ – Asaf Karagila Nov 4 '20 at 18:38

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