Nonstandard Models and Constructivity

Abstract

Publisher Summary This chapter reviews an instance of constructivization. Constructive mathematics is the legacy of Brouwer's philosophy, and the relation between constructive and nonconstructive mathematics is the legacy of Hilbert's programme. The best general-purpose conservation result relating classical and intuitionistic systems of mathematics is the conservation of a classical system over a corresponding intuitionistic one with respect to Π 2 sentences, that is, the closure of the intuitionistic system under Markov's Rule (MR). The chapter presents a nonconstructive proof , and to get a constructive proof out of it, it is to be seen that it can be formalized in a classical theory conservative over an acceptable intuitionistic theory with respect to a nice class of sentences. Peano arithmetic (PA) serves as a sufficiently strong classical theory, and conservation with respect to either the negative sentences (the result of Godel and Gentzen) or sentences (closure under Narkov's Rule) yields the provability of the result in Heyting arithmetic (HA). The key to the above constructivization is the arithmetization of model theory, specifically the (classical) construction of models with arithmetic truth definitions via the arithmetization of the Completeness Theorem.