Tarski’s Theorem for Arithmetic

Abstract

In the last chapter, we dealt with mathematical languages in considerable generality. We shall now turn to some particular mathematical languages. One of our goals is to reach Gödel’s incompleteness theorem for the particular system known as Peano Arithmetic. We shall give several proofs of this important result; the simplest one is based partly on Tarski’s theorem, to which we first turn. The first concrete language that we will study is the language of first order arithmetic based on addition, multiplication and exponentiation. [We also take as primitive the successor function and the less than or equal to relation, but these are inessential.] We shall formulate the language using only a finite alphabet (mainly for purposes of a convenient Gödel numbering); specifically we use the following 13 symbols. . . . 0’ ( ) f, υ ∽ ⊃ ∀ = ≤ # . . . The expressions 0, 0′, 0″, 0‴, · · · are called numerals and will serve as formal names of the respective natural numbers 0, 1, 2, 3, · · ·. The accent symbol (also called the prime) is serving as a name of the successor function. We also need names for the operations of addition, multiplication and exponentiation; we use the expressions f′, f″, f‴ as respective names of these three functions. We abbreviate f′ by the familiar “+”; we abbreviate f’’ by the familiar dot and f‴ by the symbol “E”. The symbols ~ and ⊃ are the familiar symbols from prepositional logic, standing for negation and material implication, respectively. [For any reader not familiar with the use of the horseshoe symbol, for any propositions p and q, the propositions p ⊃ q is intended to mean nothing more nor less than that either p is false, or p and q are both true.] The symbol ∀ is the universal quantifier and means “for all.” We will be quantifying only over natural numbers not over sets or relations on the natural numbers. [Technically, we are working in first-order arithmetic, not second-order arithmetic.] The symbol “=” is used, as usual, to denote the identity relation, and “≤” is used, as usual, to denote the “less than or equal to” relation.