# Skolem hulls in arbitrary models of some fragments of ZFC

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.

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.)
• 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). Nov 4, 2020 at 13:45