By J. N. Crossley, Anil Nerode (auth.), Prof. Dr. Rohit Parikh (eds.)

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Her e t he r eader should not forget that even the notion of heredi tarily finite set has become really familia r only in the second half of this century! Otherwise it is hardly likely that Ackerma~would have gone into as much detail as he did in [AJ to discuss the axioms of general set theory excluding t he axiom of infinity, or that Gadel would have arithmetized syntax instead of coding it in terms of hereditarily finite sets (where the repre s entation of finite sequences of sets i s quite stra ight-forward).

A) ~ ~ no . > However, in 1" Let , this fl and f2 stand for the conjunction *One of us used precisely this phrase 'preserving relations between proofs' [SFT II, p. 100] but 5 pages later got involved in a discussion of the relation between (the proofs expressed by - what corresponds to -)f and pf if p is the normalization operation for systems of natural deduction. sharpening! H2 [Der2 (f 2 ,A)" R(f l ,f 2 )l} is given by: that (except possibly for Repeat) all rules used on on fl , subsets 1'2 where R expresses are also used Clearly this is equivalent to the conjunction,.

V, 3; N. (j < p( i» J - and let be the nodes introduced at stage 21. (The reader who p; paying attention to the fact likes number theoretic functions may compute that at each odd stage several formulae are analyzed and so the depth of increases by > 1 say, and between TA The propositional analysis is standard: If at the node resp. P' I- ~G L u (F}, resp. P v fG} P iF] u P' branching: If at Nj l- I- Land we have Nj 6'. 6' v [F,G]. , F, has just one immediate predecessor: P }- {F v G} u I f we have is • considered , and replaced by ment that t- P' I- I- [F v G} v P' I- L', the immediate L there is a 6.