A note on bootstrapping intuitionistic bounded arithmetic

This paper, firstly, discusses the relationship between Buss’s definition and Cook and Urquhart’s definition of BASIC axioms and of IS 2 . The two definitions of BASIC axioms are not equivalent; however, each intuitionistically implies the law of the excluded middle for quantifier-free formulas. There is an elementary proof that the definitions of IS 2 are equivalent which is not based on realizability or functional interpretations. Secondly, it is shown that any negated positive consequence of S 2 is also a theorem of IS 2 . Some possible additional axioms for IS 1 2 are investigated.