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39- Lemma 22 Let a ~ ~. There is an L -definable function <-, ->e on L such that: e (i) L is closed under <-, ->~; (ii) for all x, y, x', y' e L , e = ~ ~-+ x = x' ~ y = y'. ) (-)O' u (y ×~+I{i}). We call <-, -> the e-pairing Thus, Le-definable of Le+ I. (w)~ = x, (w)0, ~ (w)~ = y. <-, ->e is defined. e x ^ v e y}. function. For successor e, Thus, So, for x, y e L +i, we can The lemma follows if w = ~ , then from w) and Suppose We can then define, = (x ×~+i{0}) (-)I • on ~.

Theorem 3 (Jensen) Assume V = L. Then O + holds. 5, M s = LB(s) let M e be the smallest M ~ LWl such that a c M. for some B(s) < w 1 . w h i c h is Lwl-definable Thus Ss q; ~(s) and that the sequence Note that as there is a map f : We show that < S is (by the above definition) L 2-definable. subset of Wl such that for all unbounded sets B ~ el' there is a limit Lw -definable. 6, define a chain < N 2 models of L and a continuous, cofinal sequence < s ~2 ordinals, by induction, as follows.

T* = ~a<~ I T"f(a+l). Set Then T*~ Is an (~I , ~l)-tree with no cofinal branch and no uncountable antichain, such that T* satisfies (i) - (iii) in the normality definition. ). One simply picks (by induction on the levels) one limit extension of each limit branch which has an extension. The lemma is proved. Remark By examining the above proof, one can easily amend the definition of the function f so that in the resulting tree, every point has infinitely many successors on the next level. Such a tree is said to be prolific.