In classical descriptive set theory a need arose to analyse the analytical and projective hierarchies more abstractly, isolating the key *structural* properties of the various levels. I’ll describe these here and how these concepts can be generalised to the so-called *scale property*. This is going to be the first post in a series on scales, where we here will focus on *what* scales are and *why* they’re useful. The next ones will be concerned with *where* we find scales and *how* we construct new scales from existing ones.

Before we start let’s fix some terminology. Throughout, I’ll fix some space which is any space of the form for some . We call the subsets of such **pointsets**, and we also call sets of pointsets **pointclasses**, which we denote by . Okay, that being settled, let’s get started.

Let’s start off chronologically with some of the structural properties of pointclasses leading up to the scale notion. One of these properties is the **separation property**, which holds for a pointclass if any disjoint pair can be *separated* by some , meaning it contains one and is disjoint from the other. Taking , the analytic sets, this is a theorem due to Lusin (Лýзин): every disjoint pair of analytic sets can be separated by a Borel set.

Another such property is the **reduction property**, stating that any pair can be *reduced* by a pair , meaning that , and . A proto-typical example here is the coanalytic (i.e. ) sets. A connection between these two properties is the following.

Proposition.If a pointclass has the reduction property then has the separation property.

Proof.Let be disjoint. Then simply reduce to and then works.QED

Our last ‘basic’ property is the **uniformisation property**, stating that given any binary relation we can find a *uniformisation* , , such that holds iff holds. I.e. it’s a ‘choice principle for ‘.

A natural thing to do is try to find out what these properties have in common, to find out what *essential* properties are needed to carry out known classical proofs, enabling generalisation. The first such generalisation is the *prewellordering property*. Recall that a **prewellordering** is an ordering which is reflexive, transitive, total and wellfounded.

Then a pointclass is said to have the **prewellordering property** if every has a prewellordering such that there exist binary relations and satisfying that for every , iff iff . The prewellordering property is capturing some of the above properties.

Proposition.Let be a pointclass that contains all the clopen sets and is closed under continuous preimages as well as finite intersections and unions. If has the prewellordering property then

- has the reduction property;
- has the separation property;
- has the
number uniformisation property, i.e. that every in has a uniformisation in .

Proof.(1) Let and define the disjoint union as iff or . By the properties on we see that as well. Let be a prewellordering of , and then define and . Then and they reduce .(2) follows from the previous proposition. For (3), let , . Define as the set of with chosen to be the least integer such that has minimal -rank. Then uniformises . To see that , note that iff .

QED

So we get quite close to connecting the three properties now. To take this final step we consider *scales*. Note first that to every prewellordering on some set we can consider its corresponding rank function , and to every function we can define a prewellordering on as iff . We call these functions **norms**. The prewellordering property precisely postulates the existence of a “-norm” on every , so a way to generalise this is to postulate the existence of *many* norms on — this is exactly what a scale is.

More precisely, a **scale** on is a sequence of norms such that whenever we have a sequence of elements of satisfying that, for some ,

- (convergence) ;
- (norm-convergence) for all ;

then and for every . Now, a pointclass is then said to have the **scale property**, or to be **scaled**, if every has a -scale, which is simply that each prewellordering ,associated to every norm in the scale, satisfies the criteria given in the prewellordering property. This gives us enough strength to yield the full uniformisation property.

Proposition.Let be a scaled pointclass that contains all the Borel sets and is closed under Borel preimages as well as countable intersections, finite unions and universal quantifiers. Then

- has the prewellordering property;
- has the reduction property;
- has the separation property;
- has the uniformisation property.

Proof.(1) is clear by definition and (2)-(3) was shown in the above proposition, so we just have to show (4). Let be a binary relation in and let be -scale on . We first build a new scale out of such that (a) if then and (b) If and for all then for some (a scale satisfying these two conditions is called avery good-scale).Let be a continuous bijection and let . By our closure properties on it suffices to build a very good -scale on , which then implies that is a very good -scale on . In other words, we may assume without loss of generality that . We then define as

,

where we view as the Baire space as usual, and being the (-dimensional) Gödel pairing . It isn’t too hard to see that satisfies (a)-(b), so we need to show that it’s a -scale. Let . then note that holds iff

- ; or
- ; or
- ; or…
and the analogous property for holds as well, showing that is a -norm for all , making a very good -scale.

The great thing about these very good scales is that given such a scale on a set we can use it to pick a

canonicalelement of . Indeed, since the definition ensures that each prewellordering refines the previous one , we can define , so that for some — this is our canonical element.Going back to our binary relation , let be a very good -scale on , as produced above. Define new scales on the cross-sections as given by . These new scales are then also very good -scales on , for each . But then, by the procedure in the above paragraph, we can choose canonical elements for every , so that we can define a uniformisation as iff . Further, , since

iff .

QED

Sooo.. the conclusion for now is then that the scaled pointclasses really generalise all the previously mentioned properties. But can we even find scaled pointclasses anywhere? This leads us to Moschovakis’ *periodicity theorems* and Steel’s analysis of scales in . More on that next time!