A Simple Proof of Godel’s¨ Incompleteness Theorems Arindama Singh Department of Mathematics, IIT Madras, Chennai 600 036, India E-mail:
[email protected] ··· = g(σ1) × g(σ2) ×···× g(σm) 1. Introduction g(σ1σ2 σm) 2 3 pm where pi is the i-th prime number. This defines g of terms and Godel’s¨ incompleteness theorems are considered as achieve- formulas. Next, extend g to proofs of formulas by ments of twentieth century mathematics. The theorems say that g(X X ···X ) = 2g(X1) × 3g(X2) ×···×pg(Xm). the natural number system, or arithmetic, has a true sentence 1 2 m m which cannot be proved and the consistency of arithmetic where again pi is the i-th prime number. cannot be proved by using its own proof system; see [1]. Due to prime factorization theorem in N , the function g has Though the ideas involved in their proofs are very complex, the following properties: they can be presented in a simple and comprehensible way. (a) g is a computable function. (b) g(uv) can be computed from those of g(u) and g(v). 2. Background (c) Given n ∈ N,ifn = g(X) and X is known to be a symbol, or a formula, or a proof, then X can be computed N = N + × We assume a theory of arithmetic, say ( , , ) to be from n. consistent. Write X for “X is a theorem in N .” The usual Proof theorems or laws of logic hold true in this theory. We will be Let (x, y) be a binary predicate that translates “x is using explicitly the laws of Double Negation, Contradiction, the Godel¨ number of a proof of a formula whose Godel¨ number Distribution of implication, Contraposition, Modus Ponens is y.