Extensions of some theorems of Gödel and Church

Abstract

We shall say that a logic is “simply consistent” if there is no formula A such that both A and ∼ A are provable. “ω-consistent” will be used in the sense of Gödel. “General recursive” and “primitive recursive” will be used in the sense of Kleene, so that what Gödel calls “rekursiv” will be called “primitive recursive.” By an “Entscheidungsverfahren” will be meant a general recursive function ϕ(n) such that, if n is the Gödel number of a provable formula, ϕ(n) = 0 and, if n is not the Gödel number of a provable formula, ϕ(n) = 1. In specifying that ϕ must be general recursive we are following Church in identifying “general recursiveness” and “effective calculability.”First, a modification is made in Gödel's proofs of his theorems, Satz VI (Gödel, p. 187—this is the theorem which states that ω-consistency implies the existence of undecidable propositions) and Satz XI (Gödel, p. 196—this is the theorem which states that simple consistency implies that the formula which states simple consistency is not provable). The modifications of the proofs make these theorems hold for a much more general class of logics. Then, by sacrificing some generality, it is proved that simple consistency implies the existence of undecidable propositions (a strengthening of Gödel's Satz VI and Kleene's Theorem XIII) and that simple consistency implies the non-existence of an Entscheidungsverfahren (a strengthening of the result in the last paragraph of Church).

Die Gödelschen Unvollständigkeitssätze

“…Gödel benötigte in seinem Originalbeweis (dem auch die unten angegebene Beweisskizze folgt) die sogenannte ω-Konsistenz als stärkere Voraussetzung. Allerdings konnte J. Barkley Rosser (1907Rosser ( -1989 später zeigen, daß sich der Gödelsche Beweis so modifizieren läßt, daß die (einfache) Konsistenz als Voraussetzung ausreicht, [Ros36].…”

Uniform Density in Lindenbaum Algebras

“…The essential ingredients of the proof that, for every consistent recursively enumerable theory U that interprets R, there exists a sentence R that is independent of U were provided by J. Barkley Rosser in his classical paper [16]. Here R is a very weak arithmetic introduced in [21].…”

Reflection principles and provability algebras in formal arithmetic

“…We shall also mention a well-known theorem due to Rosser, generalizing Gödel's first incompleteness theorem [75].…”