Abstract

А necessary and sllmcient condilion that а given proposition (о Ье provable in such а theory that allows (о Ье assigned to the proposition а Gödеl пunbег fог containing Реanо arithmetic is that Gödеl number itself. This is tlle sense о[ Martin LöЬ's theorem (1955). Now wе сan рut several philosophpllical questions. Is the Gödеl numbег of а propositional formula necessarily finite or onthe contrary? What would the Gödel number of а theorem be containing Реanо arithmetic itself? That is the case of the so-called first incolnpleteness theorem (Gбdеl 1931). What would the Gödеl питЬег of а self-referential statement be? What w'ould the Gödеl пumbег оГ such а proposition Ье (its Реanо arithmetic expression after encoding contains itself as ап operand)? What is the Gödеl numbег оГ Gödеl's proposition [R(q); q] that states its ргоper imргоvаbility? It is the key statement for his proving of the first incompleteness theorem. Is Реапо arithmetic available in it (ог in the ones similar to it) as а single symbol, Ьу which actual infinity would bе introduced, ог as а constructively infinite set of primary signs? Jn fact, the Gödel number оГ the first incompleteness theorem should Ье infinite in that last case. If the Gödеl number of а statement is infinite, then сап it bе accepted as а theorem? What would the Gбdеl numbeг of its negation bе? Is the infinite Gбdеl numbeг of а statement equivalent to its irresolvability? Respectively, is Ihe following statement valid: irresolvable pгopositions with finite Gбdеl numbers (еуеп in anу encoding) do поl exist?