V. A. Uspensky  Post’s Machine (Little Mathematics Library) This booklet is intended first of all for school children. The first two chapters are comprehensible even for junior schoolchildren. The book deals with a certain “toy” (“abstract” in scientific terms) computing machine – the so called Post machine – on which calculations involve many important features inherent in the computations on real electronic computers. By means of the simplest examples the students are taught the fundamentals of programming for the Post machine, and the machine, though extremely simple, is found to possess quite high potentialities.The reader is not expected to have any knowledge of mathematics beyond the primary school curriculum. The book was translated from the Russian by R. Alavina and was first published by Mir in 1983. Contents of the book are as under: 1. How the Post Machine Works 9 Sec. 1.1. “An Outward Appearance” of the Post Machine Sec. 1.2. The Program for the Post Machine Sec. 1.3. The Operation of the Post Machine Sec. 1.4. Examples of Performing Programs Sec. 1.5. Methodological Notes2. Addition of Unity on the Post Machine 23 Sec. 2.1. Recording umbers on the Post Machine and the Statement of the Problem on Addition of Unity Sec. 2.2. Addition of Unity in the Simplest Case Sec. 2.3. Addition of Unity in More Complicated Cases Sec. 2.4. Addition of Unity in Yet More Complicated Case Sec. 2.5. Addition of Unity in the Most General Case3. Analysis and Synthesis of Programs for the Post Machine 39 Sec. 3.1. Diagrams and Block Diagrams Sec. 3.2. Analysis of the Program of Adding Unity Sec. 3.3. Again on the Problem on Addition of Unity Sec. 3.4. Addition of Numbers in Simple Cases Sec. 3.5. Addition of Numbers in More Complicated Cases4. The Post Machine Potentialities 60 Sec. 4.1. On the Problem of Addition of Numbers at Arbitrary Distances Sec. 4.2. Post’s Proposal Sec. 4.3. The Post Machine and Algorithms Sec. 4.4. Additional Comments on Post’s Hypothesis (Post’s Thesis and Post’s Principle) Sec. 4.5. The Post Machine and Electronic Computers Supplement 82 Finite Combinatory Processes Formulation 1 84