MODELS OF PT^–WITH INTERNAL INDUCTION FOR TOTAL FORMULAE

Abstract

We show that a typed compositional theory of positive truth with internal induction for total formulae (denoted by PTtot) is not semantically conservative over Peano arithmetic. In addition, we observe that the class of models of PA expandable to models of PTtot contains every recursively saturated model of arithmetic. Our results point to a gap in the philosophical project of describing the use of the truth predicate in model-theoretic contexts.

Models of Positive Truth

“…This is the unique way of using · in this paper. 4. to enhance readability we suppress the formulae representing the syntactic operations.…”

“…And UTB to be the extensions of UTB − with all instantiations of induction scheme with L T formulae. In [6], [7] and [8] (this last philosophical motivation was summarized also in [4]) authors motivated the need for a weak theory of truth which would be able to prove in a single sentence the fact that every arithmetical formula satisfy the induction scheme. Such a fact can be naturally expressed by an L T sentence…”

“…The theory PT − + INT ↾ tot was claimed to be model-theoretically conservative in [5] (and then used in [6], [7] and [8] as such). However, as shown in [4], the proof of its conservativity contained an essential gap and no prime model of PA 9 admits an expansion to the model of PT − + INT ↾ tot . Moreover, it was shown that every recursively saturated model of PA can be expanded to a model of this theory.…”

“…More generally, a countable model is short recursively saturated if and only if it has a recursively saturated elementary end extension, see [18], Theorem 2.8. The proof of our theorem will closely parallel the proof of Theorem 4 from [4]. In particular, we will again use a propositional construction invented by Smith.…”

“…

This paper is a follow-up to [4]. We give a strenghtening of the main result on the semantical non-conservativity of the theory of PT − with internal induction for total formulae (PT − + INT(tot)).

…”