By A. R. D. Mathias, H. Rogers

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On ~ Dn) Vn (an ¢ 0 ~ P ( l m . a ( n , m)) I m . a ( n • m) has the p r o p e r t y to P by the c l o s u r e 4 side h o l d s Vn The f u n c t i o n Hyp 9x (a[x ~ Q) a ( n . ()) ¢ @, so it b e l o n g s properties. Vn (Vx D(n • x) ~ Dn) Vn (Vx P ( I m . a ( n • x • m)) ~ P ( I m . a ( n • m)) 53 54 C o n s i d e r b = I m . m). If b0 ~ 0, t h e n h is a c o n s t a n t positive the o t h e r hand, ( P ( l m . b ( x , m)) ~ Pb) holds the c l o s u r e b0 = 0, t h e n Vx properties. N o w BI allows T h e r e f o r e Hyp us to c o n c l u d e function so Pb holds.

ON As w e given • m ^ am ~ AC-NF: b we that : ^ Va E n * u Vm(a Vn[Ko(b) x AVe From ~ • n. Q(n) R(~,am Let ~ our ^ Va K and from holds Q0 h o l d s , KQ the for Vm(a E m A am ~ can be identified definition the assumption so V~ V~ 3 x of particular a e ~ 3e 55 we • R(~,am under a(~) ~x R(~,x), R(~,x) 0 ~ it the = assumption follows K. now i)I]. conclude K V~ R ( e , e ( ~ ) ) . 5. Choice sequences Between lawless notions of c h o i c e exposition sequences of the to [ T r o e l s t r a , [Troelstra, sequence general 69A], 69].

Then conclude y ~ V~ ~ but V~ ~ not Vx(Sx Vx(~x : ~x+l) : yx+l or : ~x+l). ~ ~ Suppose by hence contradiction Vx(Bx we assump- a contradiction. V~ ~ 3 8 Bx (~x then (~ ~ x ( ~ x 2. Suppose functionals special sequences: V~ ~ S x ~ 3x By m a k i n g of type principles. 1~ Suppose ~ is a f u n c t i o n a l 3a ~a Vx(ex : ax). 6 e n • a (ith (~ = a) Apply n)+l; (a is LS3 then ~n(a we lawlike). E obtain n A VB e n Vx($x a contradiction. = ax) Hence (~ : a). g. relations); they are not c l o s e d for this reason under the n o t i o n very s i m p l e of lawless sequence anti-social.