By K. J. Devlin

Keith Devlin - usual nationwide Public Radio commentator and member of the Stanford collage employees - writes in regards to the genetic development of mathematical pondering and the main head-scratching math difficulties of the day. And he in some way manages to make it enjoyable for the lay reader.

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Be c l o s e d and n o w h e r e d e n s e s u b s e t s o f ~. Then t h e r e is a proper Dedekind cut which avoids each K . n Proof: This is really of the sets irrational just a special case of the ~ire category K i n l~ a r e c l o s e d a n d n o w h e r e d e n s e s u b s e t s n x which is not in the closure of~, of any of the sets t h e o r e m . The c l o s u r e s so there is an K . Take D t o be t h e n cut defined by x. S i n c e ( ~ , < ~ ) H ~, 2 . 3 w i l l apply to this then is which closed and nowhere dense subsets w h e r e we make u s e o f V = L.

Where it in conjunction plays a fundamental We did not explicitly list the axiom of power set existence as a basic principle of ZF set theory, a l t h o ~ h it certainly is so, because the principle (F1) already conveys the essential structure of the system. But what is the situation when we append the system ZF by the additional assumption V = L ? Could this not invalidate the axiom of power sets (and thus lead to an inconsistent theory)? The answer is no. indeed, assuming the axiom of constructibility as an additional axiom of set theory on top of the principles of ZF is totally harmless, in view of the follcwing classical result of GSdel: 2 .

I s some f o r m u l a o f LST. Then quantifiers range over all of the form ~v sets. or n More p r e c i s e l y , ~ will ~v n in general contain, . And i n g e n e r a l , amongst the variable t h e i n t e n d e d domain o f t h e q u a n t i f i e r is V. Suppose now that M is some set which contains all of the sets which any variables in ~ may denote. By ~ M we shall mean the formula restricted to N. Of course, in a strict sense What is different is the way we interpret M T M ~ with all quantifiers is exactly the same fo~nula as T • and T .