The theory of countable analytical sets

Abstract

ABSTRACT. The purpose of this paper is the study of the structure of countable sets in the various levels of the analytical hierarchy of sets of reals. It is first shown that, assuming projective determinacy, there is for each odd n a largest countable II set of reals, <?" (this is also true for n even, replacing 11 n" by £" and has been established earlier by Solovay for n = 2 and by Moschovakis and the author for all even n > 2). The internal structure of the sets €" is then investigated in detail, the point of departure being the fact that 1 each <2n is a set of An-degrees, wellordered under their usual partial ordering. Finally, a number of applications of the preceding theory is presented, covering a variety of topics such as specification of bases, uj-models of analysis, higherlevel analogs of the constructible universe, inductive definability, etc.It is a classical theorem of effective descriptive set theory that a 2 J thin (i.e. containing no perfect set) subset of the continuum is countable and in fact contains only A{ reals. As a consequence, among the countable 2} sets of reals there is no largest one. Solovay [41] showed that every thin 2] set contains only constructible reals and therefore (in sharp contrast with the previous case), assuming a measurable cardinal exists, there is a largest countable 1,\ set of reals, namely the set of constructible ones. In [19] Moschovakis and the author extended Solovay's theorem to all even levels of the analytical hierarchy. It was proved there that, assuming projective determinacy (PD), there is a largest countable X\n set f°r all n> 1, which we denote by C2"-The first main result we prove in this paper (see §1) is the existence of a largest countable Il2" + 1 set (which we denote by C2w + i) for all n>0, assuming PD again. For w = 0 our proof shows, in ZF + DC only, the existence of a largest thin II} set of reals C,, a fact which was also independently discovered by D. Guaspari [12] and G. E. Sacks [38]. (To complete the picture, we remark here that it was shown in [17], using PD, that no largest countable 2L+i or n2n sets exist.) Once the existence of the largest countable sets C" is established the rest of § 1 is devoted to the study of their internal structure. We show here that C" is actually a set of A¿-degrees which is iwell ordered under the usual ordering of An-degrees.

Smooth sets for a Borel equivalence relation

“…There is another structural property of the co-analytic thin sets that has been studied, namely the existence of the largest nj thin set, i.e., there is a nj thin subset C\ of X such that if A is a nj thin subset of X then A ç Ci. A theorem of Kechris (Theorem 1A-2 [7]) gives a sufficient condition for the existence of such largest thin sets with respect to a given hereditary family of subsets of X (in our case, the family of closed smooth sets). The two conditions are: The family has to be nj on the codes of Z¡ set and it has to be nj-additive (see [7] for the definition).…”

“…A theorem of Kechris (Theorem 1A-2 [7]) gives a sufficient condition for the existence of such largest thin sets with respect to a given hereditary family of subsets of X (in our case, the family of closed smooth sets). The two conditions are: The family has to be nj on the codes of Z¡ set and it has to be nj-additive (see [7] for the definition). Since sparse sets have measure zero with respect to the collection of non-atomic, ergodic measures then they are nj-additive ( [7]) and from Theorem 2.7 (ii) we get that the other condition is also satisfied.…”

Higher Randomness and Genericity

“…Π 1 1 -randomness, lowness and cupping. As mentioned above, there is a greatest null Π 1 1 set (Stern and independently Kechris [Ste75,Kec75], and later rediscovered in [HN07]). In fact, this greatest set can be described succinctly.…”

“…We mainly deal with what is called Π 1 1 -randomness and Σ 1 1 -genericity. The notion of Π 1 1 -randomness goes back to Sacks [Sac90] and Kechris [Kec75], and it started to be studied formally by Hjorth and Nies [HN07]. It is a notion of interest because of some remarkable properties shared with no other randomness notion.…”

Minimal Covers and Hyperdegrees

“…It can be characterized as the maximum 11} class of reals having no perfect subclass. More folklore on C can be found in Kechris [17]. Proof.…”