Gentzen proved the consistency of elementary arithmetic (i.e. first-order Peano arithmetic) in 1936 before his most famous and influential proof in 1938. The consistency proof in 1936 contains some ambiguous parts and seems to be quite different from his consistency proof in 1938. The aim of the consistency proof in 1936 is "to give finitist sense" to provable formula. In this paper, we give an exact reconstruction of the consistency proof in 1936 and claim that "to give finitist sense" is a uniform idea behind Gentzen's three consistency proofs including the proof in 1938. First we explain Gentzen's basic ideas of the proof in 1936 in detail. In particular, the idea of finitist interpretation and the main structure of the proof are explained. Secondly, we define a reduction step via the modern method of proof theory called "finite notation for infinitary derivations" due to Mints-Buchholz. It is shown that the reduction essentially coincides with Gentzen's reduction in 1936. Especially we give a definition of "normalization tree" describing Gentzen's reduction step. Moreover, the well-foundedness of this tree is proved. The well-foundedness of the normalization tree implies the consistency of elementary arithmetic. Together with Buchholz's analysis of Gentzen's 1938 consistency proof, this shows that the proof in 1938 is just a special case of the proof in 1936. Thirdly, we clarify what the normalization tree is. According to Gentzen, the normalization tree makes us possible to see the "correctness" of a provable formula in elementary arithmetic. Then we propose a uniform reading of three consistency proofs as based on the same spirit. Finally we discuss some relationship between Gentzen's idea, the method of "finite notation for infinitary derivations", and Gödel's idea of his famous Dialectica interpretation. According to our analysis, Gentzen's idea and the method of "finite notation for infinitary derivations" can be explained in the same way as "carrying out finite proof as program". Moreover, we suggest that Gödel's interpretation (no-counterexample interpretation) should be obtained by describing the normalization tree as functionals.
We can regard Russell's theory of classes in The Principles of Mathematics as a dual theory of classes based upon the distinction between a class as one and a class as many. In this paper, I shall show the following (A)-(D) from the viewpoint of how Russell dealt in the Principles with the paradox that bears his own name. (A) The dual theory of classes is one that deserves serious considerations. (B) Russell offered measures against the paradox other than the simple theory of types developed in an appendix to the Principles. (C) Russell avoided the paradox without forbidding self-memberships of classes in general. (D) Though introducing a hierarchy of types into pseudo-entities is a very important step to Russell's later theories (e.g., the substitutional theory and the ramified theory of types), we can find this strategy already in the Principles.
Presentism is the thesis that everything is present, which implies that there are no past (or future) things or events. It is sometimes said to be imcompatible with the claim that every truth must have some ground on being. In this paper, I will examine the efficacy of such an argument in favor of presentism. In my view, however, how to respond to the grounding objection depends on what kinds of past truths we deal with: (i) truths about how present things were, (ii) general truths about things that no longer eixst, and (iii) singular truths about wholly past things. Before discussing this, I will also give an overview of the grounding project.