Definability and witnessing in second order theories
Jan Krajı́ček · Cambridge University Press eBooks · 1995
This chapter is devoted primarily to proving several definability and witnessing theorems for the second order system and analogous to those in Chapters 6 and 7. Our tool is the RSUV isomorphism (Theorem 5.5.13), or rather the definition of (Definition 5.5.3), together with the model-theoretic construction of Lemma 5.5.4. The first section discusses and defines the second order computations. In the second section are proved some definability and witnessing theorems for the second order systems and further conservation results for first order theories (Corollaries 8.2.5-8.2.7). The proofs are sketched and the details of the RSUV isomorphism arguments are left to the reader. Second order computations Let A (a, β t (b) ) be a second order bounded formula and ( K, X ) a model of. By Definition 5.5.3 we may think of K as of K = Log( M ) for some M ⊨, with X being the subsets of K coded in M . Pick some a, b ∈ K of length n and some β t (b) . Then if and only if (see Theorem 5.5.13 for the notation) where u codes β t (b) The length of u is thus.