The consistency of arithmetic

This chapter opens the part of the book that deals with ordinal proof theory. Here the systems of interest are not purely logical ones, but rather formalized versions of mathematical theories, and in particular the first-order version of classical arithmetic built on top of the sequent calculus. Classical arithmetic goes beyond pure logic in that it contains a number of specific axioms for, among other symbols, 0 and the successor function. In particular, it contains the rule of induction, which is the essential rule characterizing the natural numbers. Proving a cut-elimination theorem for this system is hopeless, but something analogous to the cut-elimination theorem can be obtained. Indeed, one can show that every proof of a sequent containing only atomic formulas can be transformed into a proof that only applies the cut rule to atomic formulas. Such proofs, which do not make use of the induction rule and which only concern sequents consisting of atomic formulas, are called simple. It is shown that simple proofs cannot be proofs of the empty sequent, i.e., of a contradiction. The process of transforming the original proof into a simple proof is quite involved and requires the successive elimination, among other things, of “complex” cuts and applications of the rules of induction. The chapter describes in some detail how this transformation works, working through a number of illustrative examples. However, the transformation on its own does not guarantee that the process will eventually terminate in a simple proof.

Paths and Laws of Motion of a Mechanism with Two Successive Conductive Elements and a Triad

It starts from a structural scheme of a mechanism with a triad and two successive conductive elements, and a kinematic scheme with ternary element and another element with void lengths is made. The relations to calculate the positions by contour method are written and the nonlinear algebraic system is solved by the method of successive elimination of the unknowns. There are determined the successive positions, the paths of some points and the variations of lifts, for different correlations between the laws of motion of the two conductive elements. It appears that there result paths and interesting laws.