Intuitionistic Analysis: The Search for Axiomatization and Understanding

An intuitionistic theory of lawlike, choice and lawless sequences

“…We assume that Θ do not occur as bound variables in the given deduction Γ E. For each axiom E of IDLS − containing free only Θ, Ψ which is "new" (in the sense that it is not an axiom of FIM extended to the language of IDLS − ) we give a realization-Θ/ D function ϕ = λΨ ϕ[Ψ] = λΨλtϕ(Θ, Ψ, t) partial recursive in Θ and finitely many functions from D. Assuming that a realization-Θ/ D function ϕ exists for each premise of a new rule of inference, and that no variable in Θ is varied in the use of the rule, we give a realization-Θ/ D function ϕ for the conclusion. 16 10D. ∀aA(a) ⊃ A(g) where g is a D-functor free for a in A(a).…”

“…Especially since the publication twenty-five years ago of Kleene's and R. E. Vesley's metamathematical investigation [8], much effort has been devoted to axiomatizing parts of intuitionistic mathematics beyond number theory. Troelstra's and D. van Dalen's two recent volumes [15], taken together with Vesley's address [16] to the 1979 Kleene Symposium, provide an excellent guide to the history and current state of this work. In particular, Chapters 4 and 12 of [15] give the background of Kreisel and Troelstra's work on lawlike and lawless sequences; Chapter 12 also describes other special classes of choice sequences which have recently been studied by Troelstra, van Dalen, G. F. van der Hoeven, and others.…”