Some of the most important developments of symbolic logic took place in the 1920s. Foremost among them are the distinction between syntax and semantics and the formulation of questions of completeness and decidability of logical systems. David Hilbert and his students played a very important part in these developments. Their contributions can be traced to unpublished lecture notes and other manuscripts by Hilbert and Bernays dating to the period 1917-1923. The aim of this paper is to describe these results, focussing (...) primarily on propositional logic, and to put them in their historical context. It is argued that truth-value semantics, syntactic ("Post-") and semantic completeness, decidability, and other results were first obtained by Hilbert and Bernays in 1918, and that Bernays's role in their discovery and the subsequent development of mathematical logic is much greater than has so far been acknowledged. (shrink)

Introduction. The system of axioms for set theory to be exhibited in this paper is a modification of the axiom system due to von Neumann. In particular it adopts the principal idea of von Neumann, that the elimination of the undefined notion of a property (“definite Eigenschaft”), which occurs in the original axiom system of Zermelo, can be accomplished in such a way as to make the resulting axiom system elementary, in the sense of being formalizable in the logical calculus (...) of first order, which contains no other bound variables than individual variables and no accessory rule of inference (as, for instance, a scheme of complete induction).The purpose of modifying the von Neumann system is to remain nearer to the structure of the original Zermelo system and to utilize at the same time some of the set-theoretic concepts of the Schröder logic and of Principia mathematica which have become familiar to logicians. As will be seen, a considerable simplification results from this arrangement.The theory is not set up as a pure formalism, but rather in the usual manner of elementary axiom theory, where we have to deal with propositions which are understood to have a meaning, and where the reference to the domain of facts to be axiomatized is suggested by the names for the kinds of individuals and for the fundamental predicates.On the other hand, from the formulation of the axioms and the methods used in making inferences from them, it will be obvious that the theory can be formalized by means of the logical calculus of first order (“Prädikatenkalkul” or “engere Funktionenkalkül”) with the addition of the formalism of equality and the ι-symbol for “descriptions” (in the sense of Whitehead and Russell). (shrink)

Stephen George Simpson. with definition 1.2.3 and the discussion following it. For example, taking 90(n) to be the formula n §E Y, we have an instance of comprehension, VYEIXVn(n€X<—>n¢Y), asserting that for any given set Y there exists a ...

The reader of Part VI will have noticed that among the set-theoretic models considered there some models were missing which were announced in Part II for certain proofs of independence. These models will be supplied now.Mainly two models have to be constructed: one with the property that there exists a set which is its own only element, and another in which the axioms I–III and VII, but not Va, are satisfied. In either case we need not satisfy the axiom of (...) infinity. Thereby it becomes possible to set up the models on the basis of only I–III, and either VII or Va, a basis from which number theory can be obtained as we saw in Part II.On both these bases the Π0-system of Part VI, which satisfies the axioms I–V and VII, but not VI, can be constructed, as we stated there. An isomorphic model can also be obtained on that basis, by first setting up number theory as in Part II, and then proceeding as Ackermann did.Let us recall the main points of this procedure.For the sake of clarity in the discussion of this and the subsequent models, it will be necessary to distinguish precisely between the concepts which are relative to the basic set-theoretic system, and those which are relative to the model to be defined. (shrink)