By Andrzej Mostowski

ISBN-10: 0444534210

ISBN-13: 9780444534217

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0' c x. Proof by induction on the number of elements of s. 2), we obtain w' E If s = s'u s" where s' and s" are disjoint and both have fewer elements that s, then each x in ws is a union XI u x" where x' E os',x" E oS". 1 and the inductive assumption we obtain x E X. 4. If a E X , then as EX. If s = 0, the theorem is obvious. If s = {n}, then x. x. as = { {(n, x>}: x E a} = ~ ~ ( { na)}=, A , ( A , ( ~ n), , a) E X . If s has at least two elements and s = s' u s" where s' # 0 # s", then each z in as can be represented as x' u where x' E as', x" E a'", and each such union is in as.

37 ENUMERATION the variable r in the subsequent formulae to make them more readable; sometimes we shall write 65 or 6, instead of 6(r, 6). dA = sup(6,: 6 < A> 6, = 0, 6c+l = 6 , + l + r - S j , ( A is a limit number). 2 = wo. 1. If6 > o ,then 65 is apower of w and 6,+, = 6:. We prove this by induction using the fact that if u < wp, B > 0, then u+ra)p = wp. 2. 6 is an increasing and continuousfunction. It follows from the definition that the set X , = { a : 6, < a < 6 ~ + 1 } has the order type rdj.

N l . We shall call H a definition of F. 1. If F and G are strongly definable functors with n+1 and m+ 1 arguments, then so are the functors W , P I , ... YP,) = A - W , P , , ... ,Pfl), 52 IV. ,p,, and permutations of these arguments yield strongly definable functors. +] , ... ,x,+,) is a definition of P and (x,)H is a definition of Q. Identifications and permutations of variables in H yield formulae which define functors arising from F by identifications and permutations of its variables.

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