Emil L. Post

The two-valued iterative systems of mathematical logic

*Frontmatter, pg. i*CONTENTS, pg. vi*INTRODUCTION, pg. 1*Part I. PRELIMINARIES, pg. 10*PART II. DERIVATION OP CLOSED SYSTEMS, pg. 43*PART III. CO-ORDINATION AND APPLICATION, pg. 96*BIBLIOGRAPHY, pg. 119

A variant of a recursively unsolvable problem

By a string on a, 6 we mean a row of as and 6s such as baabbbab. I t may involve only a, or 6, or be null. If, for example, gi, g2, gz represent strings baby aa, b respectively, string g2gigigzg2 on gi, g2, gz will represent, in obvious fashion, the string aababbabbaa on a, 6. By the correspondence decision problem we mean the problem of determining for an arbitrary finite set (gu g{), (g2, g2), • • • , (gM, gi) of pairs of corresponding non-null strings on a, b whether there is a solution in w, iu ii, • • • , in of equation

Finite Combinatory Processes--Formulation 1

The present formulation should prove significant in the development of symbolic logic along the lines of Godels theorem on the incompleteness of symbolic logics and Churchs results concerning absolutely unsolvable problems. We have in mind a general problem consisting of a class of specific problems . A solution of the general problem will then be one which furnishes an answer to each specific problem. In the following formulation of such a solution two concepts are involved: that of a symbol space in which the work leading from problem to answer is to be carried out, and a fixed unalterable set of directions which will both direct operations in the symbol space and determine the order in which those directions are to be applied. In the present formulation the symbol space is to consist of a two way infinite sequence of spaces or boxes, i.e., ordinally similar to the series of integers …, − 3, − 2, − 1, 0, 1, 2, 3, …. The problem solver or worker is to move and work in this symbol space, being capable of being in, and operating in but one box at a time. And apart from the presence of the worker, a box is to admit of but two possible conditions, i.e., being empty or unmarked, and having a single mark in it, say a vertical stroke.