Last time we got an idea of what scales are and why they’re useful. The next questions we then might ask is where do we find them, and how do we create new ones from existing ones? We’ll cover the ‘classical’ answers to these questions here, meaning the ones concerned with the projective hiearchy.
The first question, where we find the scales, is given by the following theorem, due to Novikov (Нóвиков), Kondô (近藤), Addison and Moschovakis (Μοσχοβάκης). We’ll give a full proof here to give an idea of how these scales are constructed.
Theorem (Novikov-Kondô-Addison-Moschovakis). has the scale property.
Proof. It suffices to prove that has the scale property, as the closure properties of allows us to define -scales from -scales, simply by including the relevant real parameter.
So let . Then for some recursive function it holds that iff , where is the set of all such that
is a wellordering (it can be shown that this set is a universal -set). For define and define, for and ,
and let be the norm corresponding to . We’ll show that is a -scale on .
Claim. is a scale.
Proof of claim. Let be a sequence of elements of and assume that for some and for some . We first show that ; i.e. that . We achieve this by showing that the mapping is an order-preserving map from the domain of to . Here is such that for sufficiently large , which is possible as .
Assume that . By continuity of we get that for sufficiently large , so that by definition and then . This shows that . Next, we have to show that for all . If we stare at the following equation for a while, we see that it’s sufficient to show that
where again for sufficiently large . By monotonicity of we get that , since implies , so there can be at most many -predecessors of by injectivity of . As holds for every we can conclude that
making a scale. QED
It remains to show that is a -scale; but this follows from the fact that there exist relations and such that for ,
iff iff .
Indeed, holds iff is a wellorder and given any map it holds that is injective iff it’s bijective. In other words, we’re simply saying that (which is ) and that , which we described in a fashion as well. The other formula, is defined as there exists an injective . Since we assumed that this automatically gives us that as well. QED
Okay, so we can find scaled pointclasses. Now, the question is how we move from one scaled pointclass to another. Say a pointclass is adequate if it contains all recursive sets and is closed under disjunction, conjunction, bounded number quantification of both kinds (i.e. over and ) and substitution of recursive functions. The usual arithmetical ( and ), analytical ( and ), Borel ( and ) and projective ( and ) hierarchies are all adequate. We then got the following theorem.
Theorem (Moschovakis). If is adequate and admits a -scale, then admits an -scale.
I won’t give the proof, but the scale in question is given by
where is the -scale given by assumption, and a coding of tuples of ordinals to ordinals. This then yields the corollary stating that for adequate scaled with , has the scale property. In particular we then get that has the scale property.
To cover the rest of the projective hierarchy we need to assume some determinacy. We arrive at the following second peridicity theorem.
Theorem (Moschovakis). Assume is adequate and holds. Then whenever admits a -scale, admits a -scale.
Here the construction of the scale is a bit more elaborate and will be omitted here — see “Notes on the theory of scales” in the first Cabal volume by Kechris (Κεχρής) and Moschovakis. Again we get that for adequate scaled such that holds, is scaled as well. This yields the following picture under projective determinacy, with the scaled pointclasses encircled:
This finishes the classical scale theory. The next step is to analyse the pointclasses of the form and , which is due to Steel. Note that and , so all the classical theorems cover the case where and Steel’s analysis focuses on the case. But that’s not until next time.