Download e-book for kindle: Theory of Sets by Nicolas Bourbaki (auth.)

By Nicolas Bourbaki (auth.)

ISBN-10: 3540225250

ISBN-13: 9783540225256

ISBN-10: 3642593097

ISBN-13: 9783642593093

This is a softcover reprint of the English translation of 1968 of N. Bourbaki's, Théorie des Ensembles (1970).

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Suppose this point has been established for At, A 2 , ••• , Ai - l , and let us prove it for Ai' If Ai is a letter, then Ai is either the same letter or T, and therefore a term. If Ai is of the form 1 A j' where A j is a relation which precedes Ai in construction, then Ai is identical with 1 A j by OS5, and we know already that Ai is a relation, hence Ai is a relation by OF2. The proof is analogous if Ai is of the form V AjAk' or sAj, ... Ajm' Finally, if Ai is of the form 'tz(A j), where A j is a relation which precedes Ai in the construction, there are various cases to be considered : (a) Z is distinct from x and from the letters which appear in T.

Hence (3x)R is identical with (3x')R'. It follows that (Vx)R is identical with (Vx')R'. CS9. Let R and U be assemblies and let x and y be distinct letters. If X does not appear in U, then (Uly)(3x)R and (U/y)(Vx)R are respectively identical with (3x)R' and (Vx)R', where R' is (U/y)R. For, by CS2 (§ 1, no. 2), (U/y)('t'x(R)lx)R is identical with (Tlx)(Uly)R, where Tis (Uly)'t'x(R), that is 'tx(R') , by CS4 (§l, no. 2). Hence (Uly)(3x)R is identical with (3x)R', and consequendy (UIY)(Vx)R is identical with (Vx)R'.

For each integer k such that k < leA) there is exactly one significant segment of A which begins at the I)th place. COROLLARY o~ (k + COROLLARY 2. Every significant word may be written in exactlY one way in the form f AlAs ... • 4. APPLICATION TO ASSEMBLIES IN A MATHEMATICAL THEORY Suppose that the set S is the set of signs ofa mathematical theory iD. We put nCO) = 0, n('t) = n(l) = 1, n( V) = 2, n(x) = 0 for every letter X; 53 DESCRIPTION OF FORMAL MATHEMATICS I and finally, for every specific sign s of is fixed when to is given.

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Theory of Sets by Nicolas Bourbaki (auth.)


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