 
Summary: Notes on 1
1conservativity, submodels, and
the collection schema
Jeremy Avigad
January 8, 2002
Abstract
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 secondorder arithmetic and T2 is 1
1 conservative over
T1, it is not necessarily the case that every countable model of T1 is an
submodel of a countable model of T2; this answers a question posed
in [3]. On the other hand, for n 1, every countable model of In
(resp. Bn+1 ) is an submodel of a countable model of WKL0 + In
(resp. WKL0 + Bn+1 ).
1 1
1conservativity and submodels
If T is a theory in the language of secondorder arithmetic, a Henkin model M of
T can be viewed as a structure M, SM , . . . , where firstorder variables are taken
to range over M, and secondorder variables are taken to range over some subset
