 
Summary: MOSCOW MATHEMATICAL JOURNAL
Volume 1, Number 4, OctoberDecember 2001, Pages 475490
ON FIRST ORDER LOGIC OF PROOFS
SERGEI ARTEMOV AND TATIANA YAVORSKAYA (SIDON)
To the memory of I. G. Petrovskii on the occasion of his 100th anniversary
Abstract. The Logic of Proofs LP solved long standing GĻodel's prob
lem concerning his provability calculus (cf. [4]). It also opened new lines
of research in proof theory, modal logic, typed programming languages,
knowledge representation, etc. The propositional logic of proofs is de
cidable and admits a complete axiomatization. In this paper we show
that the first order logic of proofs is not recursively axiomatizable.
2000 Math. Subj. Class. 03F45 (primary), 03F30, 03F50.
Key words and phrases. Logic of proofs, provability, recursive axiomatiz
ability.
1. Introduction
The study of provability by means of modal logic was originated by GĻodel in
1930s in [11, 12]. He suggested reading the modality as provability; so the
formula F is interpreted as "F is provable". This GĻodel's proposal led to two
substantially different provability interpretations of F each having its own specific
mathematical model. We will call them model A and model B.
