Markov A. A.. On the representation of recursive functions . English translation of XV 66 (2). American Mathematical Society, translation number 54. Lithoprinted. New York 1951, 13 pp.

H. RASIOWA and R. SIKORSKI. A proof of the completeness theorem of Godel. Fundamenta mathemalicae, vol. 37 (for 1950, pub. 1951), pp. 193-200. Among the proofs of Gddel's completeness theorem known from the literature, the proof supplied here seems closest in spirit to that given by Henkin (XV 68), and hence the latter offers the most convenient basis for indicating the progress achieved by the authors. Consider a functional calculus So , of first order, with individual variables Xi , • • • , Xk , ••• , the signs of negation ', disjunction + , implication —», and existential quantification S i t . Satisfiability having been defined in the domain of positive integers / , Godel's theorem can be stated in the form: If o is any consistent formula of So , then a is satisfiable in /. Henkin's technique lies in enlarging So to a calculus Su by adding a double infinity of individual constants and showing: If o is a consistent formula obtained from a by replacing all its free variables by constants, then there exists a maximal consistent set of sentences in Su , containing o, such that if 2iti8 belongs to this set, then some substitution 0 of a constant for z* in /S also belongs to the set. Now the same construction of the model in / for o, then made by Henkin, can also be carried out if we prove, as the authors do: (i) There exists a maximal consistent set of formulas in So , containing a, such that whenever this set contains Sit|8, it also contains some substitution f}(xk/xn) of a variable x„ for xi, in /8. Define B* to be the Boolean algebra whose elements are the classes E(f3), where y is in E(fi) if, and only if, 7—»18,0 -*7 are provable formulae, with the operations E(fi) + E(y) = E{& + 7), (E(fi))' = E(0'). (The authors refer to the algebra thus constructed as "Lindenbaum's algebra," without giving any reference; it should be noted that an explicit description of the construction of B* can already be found in Tarski (28518, Satz 4).) If we further define 2i«rat as the l.u.b. of the elements at in B*, we have