Abstract
This chapter discusses various notions of models that appeared in the recent investigations of formal systems. The discussion is applied to the study of the following problem: given a formal system S based on an infinite number of axioms A 1 , A 2 , A 3 ,… it is possible to prove in S the consistency of the system based on a finite number A 1 , A 2 …, A n of these axioms. The chapter considers two systems S and s based on the functional calculus of the first order. The question arises whether the assumptions concerning the form of axioms and rules of proof are general enough to cover the cases of standard formal systems based on a finite number of axioms. The answer is affirmative. To see this, the chapter remarks that the ɛ-rule enables to get rid of quantifiers in the axioms provided that introduce a sufficient number of ɛ-term.
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