Theories very close to PA where Kreisel's Conjecture is false

Abstract

AbstractWe give four examples of theories in which Kreisel's Conjecture is false: (1) the theory PA(-) obtained by adding a function symbol minus, ‘—’, to the language of PA, and the axiom ∀x∀y∀z (x − y = z) ≡ (x = y + z ∨ (x < y ∧ z = 0)); (2) the theory L of integers; (3) the theory PA(q) obtained by adding a function symbol q (of arity ≥ 1) to PA, assuming nothing about q; (4) the theory PA(N) containing a unary predicate N(x) meaning ‘x is a natural number’. In Section 6 we suggest a counterexample to the so called Sharpened Kreisel's Conjecture.