The Consistency of Arithmetic Revisited

Abstract

The consistency problem was raised by Hilbert as a main problem in his famous list. Hilbert formulates his second problem in terms of the non-contradiction of the arithmetical axioms which are nothing else than the elementary arithmetical operations plus the axiom of continuity (see Hilbert, 1925, p.300). The last axiom, Hilbert says, can be split into two simpler axioms, the Archimedean axiom and the (syntactic) completeness axiom which he introduced in order to provide an arithmetical model of Euclidean geometry, thus proving its consistency. But consistency of arithmetic needs a direct proof <ein direkter Beweis> that would lead from a proof for the consistency of elementary arithmetic to a (finitist) proof of existence of the continuum, classical analysis and Cantor’s tranfinite ordinals (with the exclusion of the totality of alephs). The direct way <ein direkter Weg> is a progression from elementary arithmetic of natural numbers N to the rationals Q through the integers Z to the real numbers R. The progression is the one that Kronecker in his Über den Zahlbegriff (1887a) had shown to proceed from the concept of number alone in his general arithmetic. This « arithmetic continuation » as I would like to call it, is the core of Kronecker’s programme and Hilbert is seen here to continue it with logical means, i.e. the axiomatic method which Hilbert defines as a finite number of logical inferences from axioms (Hilbert, 1935, p. 301). Where Kronecker used purely arithmetic methods, for example, congruence relations and polynomial equations in his theory of forms, Hilbert introduced logical operations that are supposed to take over and go beyond arithmetic towards analysis and transfinite set theory, but in a finitist metamathematical framework. Thus logic is but a replica of general arithmetic or its continuation by other means. In order to state more fully the problem, one should add the following quotation taken from Hilbert notebooks and dated around 1905 (following M. Hallett, 1995, p. 152)

Though the Archimedean and my completeness axioms [for Euclidean geometry or the reals respectively], the ordinary continuity axiom is divided into two completely different components. Moreover, with my completeness axiom, not one infinite process is demanded, but we have only a finite number of finite axioms, just as Kronecker demands.