So far we’ve characterised the scaled pointclasses among the projective hierarchy as well as establishing Steel’s result that is scaled for all such that . We now move on to boldface territory, finishing off this series on scales.
We’re going to need the notion of a gap, where we say that an interval is a gap if and this stops being the case when we extend the interval. Here means that the left-hand side is a -elementary substructure with parameters allowed from . This is also sometimes called a -gap, but since we’re not going to need the -gaps for anything we stick to the ‘gap’ terminology.
For convenience we also say that is a gap, where we note that is least such that . Our first lemma states that gaps can be used to characterise ordinals.
Lemma. The gaps partition the ordinals below .
Proof. Let and let be least such that and the supremum of all such that . Then is clearly a gap with .
Now say that two gaps and overlapped at some . This means that and but . But this is impossible as we’re requiring that all parameters lie in . QED
We can thus split an arbitrary into three cases:
- begins a gap;
- ends a gap;
- lies properly within a gap.
When it comes to finding scaled pointclasses, the following result shows that we can disregard the third possibility.
Theorem (Martin). Let be a gap and assume . Then there’s a subset of with no uniformisation. In particular, if and then none of the classes or are scaled, for any .
We thus only have to analyse case 1 and 2. Let’s start off with the first case where begins a gap. Again we may assume , since we already covered the case. The case where turns out to just be a corollary of the lightface result that we covered last time.
Corollary. If begins a gap and holds then has the scale property.
Proof. Note that since begins a gap there’s a partial surjection . Indeed, by forming the hull we see that by condensation and minimality, as begins a gap, so the associated Skolem function is then as desired. Steel’s lightface result then gives the desired scale. QED
When we move to then the existence of scales turn out to depend on the admissibility of , where we say that is admissible if , or equivalently that there’s no cofinal map for some . Technically speaking the terminology admissible is reserved for the -hierarchy, so we could’ve called it something like -admissible — but as we won’t need to consider the -variants of admissibility, there won’t be any confusion. Now, the key lemma is the following.
Lemma. Assume that begins a gap and is inadmissible (not admissible). Then for all ,
I won’t give the proof here (a full proof can be found in my note), but the idea is that the conjunction of beginning a gap and being inadmissible implies that we get a cofinal map . The above two results then give us that and have the scale property, whenever begins a gap, is inadmissible, holds and . To see that inadmissibility is necessary, Martin comes to the rescue again.
Theorem (Martin). Assume begins a gap, is admissible and holds. Then there’s a subset of with no uniformisation in . In particular, none of the classes or are scaled for .
This finishes the case where begins a gap. We’ll need a definition for the gap endings. We say that a gap is strong if for every there’s a and an such that for every it holds that
It turns out that for us to have any chance of finding scales at the end of gaps, we have to require our gaps to be weak. Martin returns with a counterexample.
Theorem (Martin). Let be a strong gap and assume . Then there’s a relation which has no uniformisation in . In particular, none of the classes or are scaled for .
So we can focus solely on weak gaps then. Furthermore, if (i.e. we’re not projecting to 0) for every then we don’t get any new subsets of reals of interest, meaning that
just reducing it to the begins-a-gap case. Taking this into account, we do have the following theorem, whose proof is a technical tour de force based on the same ideas as the lightface proof that I’ve written up here.
Theorem (Steel). Let be a weak gap and assume . If is least such that then
are scaled, for every .
Aaaand that finishes the complete characterisation, which is used to organise the cases in the core model induction. So if you’re ever wondering if is scaled, we can now use the following handy flowchart (pdf available here):