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By Ilijas Farah

This publication is meant for graduate scholars and examine mathematicians drawn to set thought.

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3). * As an example of the application of these criteria, consider the following relation : "the sequence of real-valued functions (/,J converges uniformly to 0 in [0, I]". This means "for each e> 0 there exists an integer n such that for each x e [0, 1] and each integer m ~ n we have 11m (x) I ~ e". Suppose we wish to take the negation of this relation (for example, to obtain a proof by contradiction); the criterion e38 shows that this negation is equivalent to the following relation : " there exists an e > 0 such that for each integer n there exists an x e [0, 1] and an m ~ n for which I/m(x) I > e".

1. SIGNS AND WORDS * Let S be a non-empty set, whose elements will be called signs (this terminology being appropriate to the metamathematical application we have in mind). o(S) are called words and are identified with finite sequences A = (Si)O~I~1I of elements of S. The law of composition in Lo(S) will be written multiplicatively, so that AB is the sequence obtained by juxtaposition of A and B. o(S). o(S) is the number of elements in the sequence A; thus I(AB) = leA) I(B), and the words of length 1 are the signs.

Lowing three theorems are theorems in 'lOo. Let '(0 be of 'lO and S7. The The fol45 DESCRIPTION OF FORMAL MATHEMATICS THEOREM 1. X = x. Let S denote the relation x = x in '00 • By 027 (§4, no. 1), for every relation R in '00 , (Vx)(R ~ R) is a theorem in '00 , and therefore, by S7, 't:z:(R) = 't:z:(R), that is to say ('t:z:(R)lx)S, is a theorem in '00 • Taking R to be the relation "not S" and considering 026 (§4, no. 1), we see that (Vx)S is a theorem in '00 • By 030 (§4, no. 3), S is therefore a theorem in '00 • The relation (Vx)(x = x) is also a theorem in Go; and if T is a term in Go, then T = T is a theorem in Go (cf.

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