Scales 101 – part IV: leaving a gap

So far we’ve characterised the scaled pointclasses among the projective hierarchy as well as establishing Steel’s result that \Sigma_1^{J_\alpha(\mathbb R)} is scaled for all \alpha>0 such that \text{Det}(J_\alpha(\mathbb R). 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 [\alpha,\beta] is a gap if J_\alpha(\mathbb R)\prec_1^{\mathbb R} J_\beta(\mathbb R) and this stops being the case when we extend the interval. Here \prec_1^{\mathbb R} means that the left-hand side is a \Sigma_1-elementary substructure with parameters allowed from \mathbb R\cup\{V_{\omega+1}\}. This is also sometimes called a \Sigma_1-gap, but since we’re not going to need the \Sigma_n-gaps for anything we stick to the ‘gap’ terminology.

For convenience we also say that [(\bf\delta^2_1)^{L(\mathbb R)},\Theta^{L(\mathbb R)}] is a gap, where we note that (\bf\delta^2_1)^{L(\mathbb R)} is least such that J_{(\bf\delta^2_1)^{L(\mathbb R)}}(\mathbb R)\prec_1^{\mathbb R} L(\mathbb R). Our first lemma states that gaps can be used to characterise ordinals.

Lemma. The gaps partition the ordinals below \Theta^{L(\mathbb R)}.

Proof. Let \gamma<\Theta^{L(\mathbb R)} and let \alpha\leq\gamma be least such that J_\alpha(\mathbb R)\prec_1^{\mathbb R}J_\gamma(\mathbb R) and \beta\geq\alpha the supremum of all \delta<\Theta^{L(\mathbb R)} such that J_\gamma(\mathbb R)\prec_1^{\mathbb R}J_\delta(\mathbb R). Then [\alpha,\beta] is clearly a gap with \gamma\in[\alpha,\beta].

Now say that two gaps [\alpha,\beta] and [\alpha',\beta'] overlapped at some \gamma. This means that J_\alpha(\mathbb R)\prec_1^{\mathbb R}J_\gamma(\mathbb R) and J_\gamma(\mathbb R)\prec_1^{\mathbb R}J_{\beta'}(\mathbb R) but J_\alpha(\mathbb R)\not\prec_1^{\mathbb R}J_{\beta'}(\mathbb R). But this is impossible as we’re requiring that all parameters lie in \mathbb R\cup\{V_{\omega+1}\}QED

We can thus split an arbitrary \alpha<\Theta^{L(\mathbb R)} into three cases:

  1. \alpha begins a gap;
  2. \alpha ends a gap;
  3. \alpha 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 [\alpha,\beta] be a gap and assume \text{Det}(J_{\alpha+1}(\mathbb R)). Then there’s a \Pi_1^{J_\alpha(\mathbb R)} subset of \mathbb R\times\mathbb R with no \bf\Sigma_1^{J_\beta(\mathbb R)} uniformisation. In particular, if \gamma\in(\alpha,\beta) and \text{Det}(J_\gamma(\mathbb R)) then none of the classes \bf\Sigma_n^{J_\gamma(\mathbb R)} or \bf\Pi_1^{J_\gamma(\mathbb R)} are scaled, for any n<\omega.

We thus only have to analyse case 1 and 2. Let’s start off with the first case where \alpha begins a gap. Again we may assume \alpha>0, since we already covered the \alpha=0 case. The case where n=1 turns out to just be a corollary of the lightface result that we covered last time.

Corollary. If \alpha>0 begins a gap and \text{Det}(J_\alpha(\mathbb R)) holds then \bf\Sigma_1^{J_\alpha(\mathbb R)} has the scale property.

Proof. Note that since \alpha begins a gap there’s a partial \bf\Sigma_1^{J_\alpha(\mathbb R)} surjection \mathbb R\to J_\alpha(\mathbb R). Indeed, by forming the hull \mathcal H:=\text{cHull}_1^{J_\alpha(\mathbb R)}(\mathbb R\cup\{V_{\omega+1}\}) we see that \mathcal H=J_\alpha(\mathbb R) by condensation and minimality, as \alpha begins a gap, so the associated \bf\Sigma_1^{J_\alpha(\mathbb R)} Skolem function is then as desired. Steel’s lightface result then gives the desired scale. QED

When we move to n>1 then the existence of scales turn out to depend on the admissibility of \alpha, where we say that \alpha is admissible if J_\alpha(\mathbb R)\models\textsf{KP}, or equivalently that there’s no \Sigma_1^{J_\alpha(\mathbb R)} cofinal map \gamma\to\omega\alpha for some \gamma<\omega\alpha. Technically speaking the terminology admissible is reserved for the L-hierarchy, so we could’ve called it something like \mathbb R-admissible — but as we won’t need to consider the L-variants of admissibility, there won’t be any confusion. Now, the key lemma is the following.

Lemma. Assume that \alpha begins a gap and is inadmissible (not admissible). Then for all n\geq 1,

\bf\Sigma_{n+1}^{J_\alpha(\mathbb R)}=\exists^{\mathbb R}\bf\Pi_n^{J_\alpha(\mathbb R)} and \bf\Pi_{n+1}^{J_\alpha(\mathbb R)}=\forall^{\mathbb R}\bf\Sigma_n^{J_\alpha(\mathbb R)}.

I won’t give the proof here (a full proof can be found in my note), but the idea is that the conjunction of \alpha beginning a gap and being inadmissible implies that we get a cofinal map \mathbb R\to\omega\alpha. The above two results then give us that \bf\Sigma_{2n+1}^{J_\alpha(\mathbb R)} and \bf\Pi_{2n+2}^{J_\alpha(\mathbb R)} have the scale property, whenever \alpha>0 begins a gap, is inadmissible, \text{Det}(J_{\alpha+1}(\mathbb R)) holds and n<\omega. To see that inadmissibility is necessary, Martin comes to the rescue again.

Theorem (Martin). Assume \alpha begins a gap, is admissible and \text{Det}(J_{\alpha+1}(\mathbb R)) holds. Then there’s a \Pi_1^{J_\alpha(\mathbb R)} subset of \mathbb R\times\mathbb R with no uniformisation in J_{\alpha+1}(\mathbb R). In particular, none of the classes \bf\Sigma_n^{J_\alpha(\mathbb R)} or \bf\Pi_n^{J_\alpha(\mathbb R)} are scaled for n>1.

This finishes the case where \alpha begins a gap. We’ll need a definition for the gap endings. We say that a gap [\alpha,\beta] is strong if for every b\in J_\beta(\mathbb R) there’s a \gamma<\beta and an a\in J_\gamma(\mathbb R) such that for every \varphi(v)\in\Sigma_1\cup\Pi_1 it holds that

J_\gamma(\mathbb R)\models\varphi[a] iff J_\beta(\mathbb R)\models\varphi[b].

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 [\alpha,\beta] be a strong gap and assume \text{Det}(J_{\alpha+1}(\mathbb R)). Then there’s a \Pi_1^{J_\alpha(\mathbb R)} relation which has no uniformisation in J_{\beta+1}(\mathbb R). In particular, none of the classes \bf\Sigma_n^{J_\beta(\mathbb R)} or \bf\Pi_n^{J_\beta(\mathbb R)} are scaled for n<\omega.

So we can focus solely on weak gaps then. Furthermore, if \rho_k(J_\beta(\mathbb R))\neq\mathbb R (i.e. we’re not projecting to 0) for every k\leq n then we don’t get any new subsets of reals of interest, meaning that

\bf\Sigma_n^{J_\beta(\mathbb R)}=\bf\Sigma_n^{J_\alpha(\mathbb R)} and \bf\Pi_n^{J_\beta(\mathbb R)}=\bf\Pi_n^{J_\alpha(\mathbb R)},

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 [\alpha,\beta] be a weak gap and assume \text{Det}(J_{\alpha+1}(\mathbb R)). If n<\omega is least such that \rho_n(J_\beta(\mathbb R))=\mathbb R then

\bf\Sigma_{n+2k}^{J_\beta(\mathbb R)} and \bf\Pi_{n+2k+1}^{J_\beta(\mathbb R)}

are scaled, for every k<\omega.

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 \bf\Sigma_n^{J_\alpha(\mathbb R)} is scaled, we can now use the following handy flowchart (pdf available here):