By Semen Gindikin

Trans. R.H. Silverman

The renowned literature on mathematical good judgment is very huge and written for the main various different types of readers. students or adults who learn it of their unfastened time may perhaps locate the following an enormous variety of thought-provoking logical difficulties. The reader who needs to complement his mathematical historical past within the desire that this may support him in his way of life can notice certain descriptions of functional (and commonly -- now not so practical!) functions of good judgment. the big variety of well known books on common sense has given upward thrust to the wish that via using mathematical common sense, scholars will ultimately find out how to distinguish among invaluable and enough stipulations and different issues of good judgment within the university path in arithmetic. however the behavior of lecturers of mathematical research, for instance, to stay to difficulties facing sequences with no restrict, uniformly non-stop capabilities, and so on. has, regrettably, ended in the writing of textbooks that current prescriptions for the mechanical building of definitions of adverse options which appear to obviate the necessity for any pondering at the reader's half. we're almost definitely unable to enumerate every little thing the reader may well draw out of latest books on mathematical common sense, despite the fact that.

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1. Let Λ4 be a ppm which is passive or active of types I or II. M), pt(Af), and pk(M) by induction on k. k = 0: Let Po(M) = ORM,

E. ΰ (^, /(«)) 6 Th^(α)) iff It follows that Vγ < sup π"(tf, Vp € |Λi|, Th^u(γ U {π(p)» € Ult. Now fix p such that Ή^(p^ U {p}) = Λί. Let α < sup τt"p^ and r 6 Ult; we must see that Th^"(α U {r}) e Ult. Fix γ < sup ir" tf? ,π(p)]. Then Th^lt(7U{τr(p)» is in Ult, and from it we can compute Th""(α U {r}) inside Ult. FINE STRUCTURE AND ITERATION TREES 39 Thus pj^1* = sup π" p*f. e. ΰ , f^(/(ϋ), y(ϋ)) . <= is easy since Th^(α) = 6 is uniformly Πi over rΣ£, and we have the Los Theorem for rΣn formulae.

As M ^= 04, we see that F^ = F is a pre-extender over ΛΓ. Suppose that Ult(^,F) is wellfounded, and that i^Γ(crit F) > OR^. Let α= , G = the (μ*, a) extender derived from i$ : tf -* Ult(^, FINE STRUCTURE AND ITERATION TREES 31 Note that α exists since ip(μ^) > ι/. We claim M is an active type III ppm, and v = VM . For this, note iF(E^) \ a is good at all β < α, since E* is good at all β < μ* . So it is enough to check \ α~G is good at α. Clearly Λί is strongly acceptable. e. jίF(j|Λr) = J*") as j\f |= 04 so Ult(M,G) agrees with Ult(-V, G) up to i^f(ι/) = t^1 (i/), so that α = ι/+ in Ult(Λί, G).