This paper constitutes a short report on the proof theory which developed in the course of an investigation of Essenin-Volpin's foundational studies (see his paper in Infinitistic methods.) The Infinitistic methods paper is primarily concerned with proving the consistency of ZF. His arguments employ some very unusual ideas, especially that of the possibility of different length natural number series. The modal setting which he uses for these ideas and which is really all important is not reflected in the present paper. Only a simple representation of this idea is considered, namely, the use of *N, the nonstandard natural numbers, together with N, the standard natural numbers, to play the roles of a “long” and a “short” number series.

It will be useful to make a few general remarks before beginning on the technical details. These remarks are especially directed towards those people who have read Essenin-Volpin's work, the above-mentioned paper in particular and the one appearing in the proceedings of the 1968 Conference on Intuitionism and Proof Theory at Buffalo, New York.

I want to draw a comparison between the ultra-intuitionistic position of Essenin-Volpin on the one hand with the intuitionistic and realistic positions on the foundations of mathematics on the other hand. Intuitionism makes a transition from the traditional mathematical position (realism) that the rules of mathematical reasoning must be constructed so as to preserve truth (as known by an omniscient being about “the real world” of abstract mathematical objects) to the position that the rules of mathematical reasoning must be in accord with the process of knowing (performed by the “ideal mathematician” or “creative subject”) about the world of mental constructions. Ultra-intuitionism deepens and extends this transition to the position that the fundamental rules of mathematical reasoning must be in accord with processes of knowing accessible to human beings.