The foundations of Suslin logic

Abstract

LetLbe a first order logic andthe infinitary logic (as described in [K, p. 6] overL. Suslin logicis obtained fromby adjoining new propositional operatorsand. Letfrange over elements ofωωandnrange over elements of ω.Seqis the set of all finite sequences of elements of ω. If θ:Seq→is a mapping into formulas ofthenandareformulasofLA. Ifis a structure in which we can interpretandhis an-assignment then we extend the notion ofsatisfactionfromtoby definingwheref∣nis the finite sequence consisting of the firstnvalues off. We assume thathasωsymbols for relations, functions, constants, andω1variables. θ isvalid ifθ ⊧ [h] for everyhand isvalidif-valid for every. We address ourselves to the problem of finding syntactical rules (or nearly so) which characterize validity.