ϕ

ϕ

ψx ∀x ψ x

x

ϕ

ψx ∃x ψ x

ϕ

ϕ

ith this we have absolutely no guarantee that the introduction of assumptions, in symbolic form, to whose interpretation there is no objection, keeps the system of derivable formulas consistent. For example, there is the unanswered question of whether the addition of the axioms of mathematics to our calculus leads to the provability of every formula. The difficulty of this problem, whose solution has a central significance for mathematics, is in no way comparable to that of the problem just handled by us · · · . To successfully mount an attack on this problem, D. Hilbert has developed a special theory.

∃x F x

∀ xF x

It is still an unsolved problem as to whether the axiom system is complete in the sense that all logical formulas which are valid in every domain can be derived. It can only be stated on empirical grounds that this axiom system has always been adequate in the applications. The independence of the axioms has not been inves- tigated.