By Neil H. Williams

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CONTENTS
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Preface
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SubstitutionLemma. Let A be a structure andp: X + A an assignment in A. Ifq: Y+Tm is a substitution, thenfor an arbitrary term t , (t(q))AIPI= t A [ P ( { d[PI/Y),E Y 11. (a> For an arbitrary formula F , A (b) I= F(q)[pl = A I= F[P({qf)[PI/Y),,y)l, provided that q is proper for F. In particular. for any s E Tm and y E X , (f

Y,,)]. Now,each S H ,is~assumed as an axiom. On the basis of the Zermelo theory we can develop the algebra of sets, the theory of functions and relations, and the theory of ordering and power to an extent sufficient for mathematical practice. In particular we can define the set of natural numbers, the set of integers, and the set of rationals and of reals with the ordinary arithmetical operations. We can also build the well-known function and topological spaces, and so on. However, a stronger system of a set theory is more popular namely the so-called Zermelo-Fraenkel theory, [F2].

A,) , , = ~ A ~ ( a. l ,a,) ,. for j E J . Thus, r t J holds (or does not hold) for the elements a l ,. . ,a, simultaneously for all s for which al, . . ,a, E A,. Similarly, the value of the function @ ( a I , . . ,a,) is the same in all the systems As for which a ] ,. . ,a, E A,. Also the corresponding distinguished elements are the same; c i s = c i ' for all s, t E S . Directly from the definition it follows that all the systems A, are subsystems of the sum A = U{As: s E S } ; As C A, for every s E S.

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