This article presents a proof that Buss’s $S_2^2$ can prove the consistency of a fragment of Cook and Urquhart’s PV from which induction has been removed but substitution has been retained. This result improves Beckmann’s result, which proves the consistency of such a system without substitution in bounded arithmetic $S_2^1$.

Our proof relies on the notion of “computation” of the terms of PV. In our work, we first prove that, in the system under consideration, if an equation is proved and either its left- or right-hand side is computed, then there is a corresponding computation for its right- or left-hand side, respectively. By carefully computing the bound of the size of the computation, the proof of this theorem inside a bounded arithmetic is obtained, from which the consistency of the system is readily proven.

This result apparently implies the separation of bounded arithmetic because Buss and Ignjatović stated that it is not possible to prove the consistency of a fragment of PV without induction but with substitution in Buss’s $S_2^1$. However, their proof actually shows that it is not possible to prove the consistency of the system, which is obtained by the addition of propositional logic and other axioms to a system such as ours. On the other hand, the system that we have considered is strictly equational, which is a property on which our proof relies.

Kripke recently suggested viewing the intuitionistic continuum as an expansion in time of a definite classical continuum. We prove the classical consistency of a three-sorted intuitionistic formal system IC, simultaneously extending Kleene’s intuitionistic analysis I and a negative copy C° of the classically correct part of I, with an “end of time” axiom ET asserting that no choice sequence can be guaranteed not to be pointwise equal to a definite (classical or lawlike) sequence. “Not every sequence is pointwise equal to a definite sequence” is independent of IC. The proofs are by Crealizability interpretations based on classical ω-models ${\cal M}$ = $\left( {\omega ,{\cal C}} \right)$ of C°.