We 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 𝒵 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.
"Theories very close to PA where Kreisel's Conjecture is false." J. Symbolic Logic 72 (1) 123 - 137, March 2007. https://doi.org/10.2178/jsl/1174668388