Series Incompleteness : A Personal Perspective

After proving the completeness of the system of predicate logic in his doctoral dissertation (1929), Gödel has continued the investigation of the completeness problem for more comprehensive formal systems, especially for systems encompassing all known methods of mathematical proof. In 1931 (see [28, 25]) Gödel proved his famous (first) incompleteness theorem, which in modern terms reads: Every computably enumerable, consistent axiomatic system containing elementary arithmetic is incomplete, that is, there exist true sentences unprovable by the system. Our aim is to present a personal view of the incompleteness phenomenon. We will focus on interesting/natural concrete independent sentences, on the source of incompleteness, and on how common the incompleteness phenomenon is. Some open questions will be briefly stated. 1 The incompleteness theorem In modern language the incompleteness theorem [28] can be expressed in the following form: Theorem 1 Every axiomatic system F which is (1) finitely specified, (2) rich enough to include the arithmetic, and (3) arithmetically sound, is incomplete; that is, there exists (and can be effectively constructed) a sentence of arithmetic which (A) can be expressed in F , (B) is true, and (C) is unprovable by F . Our main example of an axiomatic theory is the Zermelo–Fraenkel set theory with choice, ZFC. We fix an interpretation of Peano Arithmetic (PA) in ZFC. Each sentence of the language of PA has a translation into a sentence of the language of ZFC, determined by the interpretation of PA in ZFC. A “sentence of arithmetic” indicates a sentence of the language of ZFC that is the translation of some sentence of PA. Condition (1) says that there is a computable function listing all axioms and inference rules (which could be infinite): the axioms and inference rules form a c.e. set. We cannot take as axioms all true arithmetical sentences because this set is not c.e. Condition (2) says that F has all the symbols and axioms used in arithmetic, the symbols for 0 (zero), S (successor), + (plus), × (times), = (equality) and the axioms making them work (as, for example, x + S(y) = S(x + y)). Condition (2) says that F is “infinite”; it cannot be satisfied if we do not have individual terms for the natural numbers. For example, Tarski proved that the Euclidean geometry, which refers to points, circles and lines, is complete. Finally, (3) says that every sentence of arithmetic proved by F is true; in ZFC a true sentence of arithmetic is a sentence whose translation in PA is true in the standard model of PA. Incompleteness shows that there are more true sentences than provable sentences (in a given axiomatic theory with the required properties). A closer analogy is the relation between what is true and what can be proved in court.1 One could replace conditions (B) and (C) in Theorem 1 by the following condition (B’) is neither provable or disprovable by F , which makes no reference to truth. In what follows we will fix an axiomatic theory F satisfying the properties (1), (2), (3) in Theorem 1 and let X be the underlying alphabet of F . A sentence which F cannot prove nor disprove is called independent (of F). A set A ⊆ X∗ is said to be expressible in F (or definable in the language of F) if there is a predicate H of F such that for all x ∈ X∗, H(x) is true iff x ∈ A. Let Arith be the set of true sentences of arithmetic expressible in PA. The Scottish judicial system which admits three forms of verdict, guilty, not-guilty and not-proven, comes closer to the spirit of incompleteness.