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

We give four examples of theories in which Kreisel 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 Conjecture.