Notes on Pi^1_1 Conservativity, Omega-Submodels, and the Collection Schema

These are some minor notes and observations related to a paper by Cholak, Jockusch, and Slaman [3]. In particular, if T1 and T2 are theories in the language of second-order arithmetic and T2 is Π 1 1 conservative over T1, it is not necessarily the case that every countable model of T1 is an ω-submodel of a countable model of T2; this answers a question posed in [3]. On the other hand, for n ≥ 1, every countable ω-model of IΣn (resp. BΣn+1 ) is an ω-submodel of a countable model of WKL0 + IΣn (resp. WKL0 + BΣn+1 ). 1 Π1-conservativity and ω-submodels If T is a theory in the language of second-order arithmetic, a Henkin modelM of T can be viewed as a structure 〈M,SM , . . .〉, where first-order variables are taken to range overM , and second-order variables are taken to range over some subset SM of the power set of M . If M = ω and M has the standard interpretations of +, ×, etc., then M is said to be an ω-model. If M1 = 〈M1, SM1 , . . .〉 and M2 = 〈M2, SM2 , . . .〉 are models, then M1 is said to be an ω-submodel of M2 if M1 =M2 and SM1 ⊆ SM2 (note that M1 and M2 need not be ω!). The theories RCA0 , WKL0 , ACA0 are fragments of second-order arithmetic in which induction is restricted to Σ1 formulae with parameters, and in which comprehension is replaced by recursive comprehension, a weak version of König’s lemma, or arithmetic comprehension, respectively. From here on the general reference for subystems of second-order arithmetic is Simpson [10]. It is not hard to see that if T1 and T2 are theories in the language of secondorder arithmetic and every countable ω-model of T1 is an ω-submodel of a countable model of T2, then T2 is Π1-conservative over T1: if ψ is Π1 and T1 does not prove ψ, let M1 be a countable model of T1 + ¬ψ; find a model M2 of T2 such that M1 is an ω-submodel of M2; then M2 is a model of T2 + ¬ψ. ∗Carnegie Mellon Technical Report CMU-PHIL-125. Section 3 modified slightly, January 8, 2002.