On formulas of one variable in intuitionistic propositional calculus

Abstract

McKinsey and Tarski [3] described Gödel's proof that the number of Brouwerian-algebraic functions is infinite. They gave an example of a sequence of infinitely many distinct Brouwerian-algebraic functions of one argument, which means that there are infinitely many non-equivalent formulas of one variable in the intuitionistic propositional calculus LJ of Gentzen [1]. However they did not completely characterize such formulas. In § 1 of this note, we define a sequence of basic formulas P_∞(X), P₀(X), P₁(X), … and prove the following theorems.