Pénzes István (szerk.): Műszaki nagyjaink 6. Matematikusok, az oktatás, a gépészet és a villamos vontatás alkotói, kiváló lisztvegyészek (Budapest, 1986)
Dr. Ádám András - Dr. Dömösi Pál: Kalmár László
Another proof of the Gödel—Rosser incompletability theorem. By László Kalmár in Szeged. In a paper1) which became a source of a series of investigations, Gödel has proved a theorem to the effect that for every postulate system satisfying some very general conditions, there is an arithmetical problem unsolvable in chat system. One of the conditions for tne postulate system requires not only its non-contradictoriness, i. e. the absence of two theorems one of which is the negation of the other, but also its ^-consistency, i. e. the absence of an enumerable series of theorems, one stating that some positive integer has a given property while the others state in succession that 0 does not have that property, 1 does not have that property, etc. While non-contradictoriness is a natural condition for in a contradictory system (containing some parts of logic, e. g. those allowing to form indirect proofs) everything can be proved, hence there are no unsolvable problems, w-consistency is regarded a rather sophisticated condition. Hence it was a great progress that Rosser succeeded2) in replacing the condition of co-consistency by non-contradictoriness. In this paper I shall give a simplified proof for Rosser’s theorem using, with appropriate modifications, the method by which I proved Gödel’s theorem3). At the same time, l shall present the proof with the same degree of generality as I did that of Gödel’s theorem in two recent publications4), *) K. Gödel, Über formal unent'cheidbare Sätze der Principia Mathematics und verwandter Systeme I, Monatshefte für Math, und Phys., 38 (1931), p. 173 — 198. 2) B. Rosser, Extensions of some theorems of Gode! and Church, Journal of symbolic logic, 1 (1936), p. 87-91, especially theorem II, p. 89. See also D. Hilbert ani P. Bernays, Grundlagen der Mathematik. II (Berlin, 1939), pi 275—276. 3) L. Kalmár, a) Egyszeríí példa eldönthetetlen aritmetikai problémára, Mat. és fiz. lapok, 50 (1943), p. 1-23; b) Eine e nfache Konstruktion unentscheldbarer Sätze in formalen Systemen, forthcoming in Methodos; and see footnote 4). 4) L. Kalmár, a) Une forme du théoréme de Gödel avec des hypotheses minimales, Comptes Rendus Ac id. Sei. Paris, 229 <1949), p 963-965; b) Quelques formes généi ales du théoréme de Göael, ibidem, 229 (1949, p. j 947-1049. 4. ábra Kalmár László egyik munkájának címlapja [A. 34], amelyben új bizonyítást adott a Gödel—Rosser-féle téréire 62
