Kolmogorov's Logic of Problems and a Provability Interpretation of Intuitionistic Logic

In 1932 A.N.Kolmogorov suggested an interpretation of intui tlonls tic logic Int. as a "logic of problems". Then K.G~del in 1933 offered a "provability" understanding of problems, thus, providing an abstract "provability" interpretation for Int via a modal logic $4. Later papers by J.C.C.McKinsey & A.Tarski, A.Grzegorczyk, R.Solovay, A.V.Kuznet sov & A.Yu.Muravit skii, R.Goldblat t, G.Boolos imply that this provabllity interpretation of Int is complete if one decodes G~del modality ~ for an "abstract provability" in t h e f o l l o w i n g way: oQ =Q^Pr[Q]~ w h e r e P~[Q] i s t h e s t a n d a r d p r o v a b i l i t y p r e d i c a t e f o r P e a n o a r i t h m e t i c . The p a p e r s h o w s t h a t t h e d e f i n i t i o n o f oQ a s Q ^ P r [ Q ] i s (in a c e r t a i n s e n s e ) t h e o n l y p o s s i b l e one . The U n i f o r m C o m p l e t e n e s s T h e o r e m f o r p r o v a b l l i t y l o g i c s i s e x t e n d e d t o I n t and o t h e r l o g i c s h a v i n g G/Sdelian provability interpretation. The first order logics having provability interpretation are considered. t I N T R O D U C T I O N A.N.Kolmogorov in [Kol] suggested an informal interpretation of sentences of Intuitionistic logic Int as statements about the possibility of solving certain general problems; propositional variables were supposed to denote "problems", logical connectives were given a natural interpretation as operators over "problems": a formula A^B denotes a problem "to solve both A and B", a formula AvB denotes "to solve either A or B", an implication A --,B is interpreted as a problem "to reduce a solution of B to any solution of A"~ -~A is A--~± that means a problem "to demonstrate an unsolvability of A". Kolmogorov hadn't given a precise definition of "problems", just appealing to the common sense of a working mathematician but had conjectured that his interpretation of Int was complete. In [G~d] K.G~Sdel offered an interpretation of Int. close to that in [Kol], where intuitionistic propositions were