Download e-book for kindle: Aspects of Mathematical Logic by H. Hermes (auth.), Prof. E. Casari (eds.)

By H. Hermes (auth.), Prof. E. Casari (eds.)

ISBN-10: 3642110789

ISBN-13: 9783642110788

H. Hermes: simple notions and purposes of the idea of decidability.- D. Kurepa: On a number of continuum hypotheses.- A. Mostowski: types of set theory.- A. Robinson: difficulties and strategies of version theory.- S. Sochor, B. Balcar: the overall conception of semisets. Syntactic versions of the set theory.

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Example text

Q- 1 Applying (*k4 we get L n F~ # 0. But 2 ~ : ( ~ ) + 1 , f h e o n l y e l e m e n t of F~ , i s not arl e l e m e n t of L. R E F E R -----E N C E-S : S e e L e c t u r e 4. ------ n c-t i-on? -P-r i-m-i-t i-v-e -R-e-c-u-r s-i -v e- -F-u1. In L e c t u r e 2 , no. I the c l a s s -P of p v i m i t i v e r e c u r s i \ : r t'crn- c t i o n s h a s been i n t r o d u c e d a s t h e s m a l l e s t c l a s s of ( t o t a l ) functions t o which 1 belong S a n d e v e r y vn and w h i c h is c l o s e d u n d e r s u b s t i t u t i o n a n d C: p r i m i t i v e r e c u r s i o n .

T o h a v e t h e n e c e s - D, Do t h e p r o p e r t y of b e i n g good (in t h e d i a g o n a l - undecidable, w e h a v e t o a d m i t , t h a t = ) dl,. , d p ) . Then D 0 has several in p l a c e of ( 8 . 9) . w e c a n u s e t h e f o r m u l a A x ( D xxV. VD xx) which d e s c r i b e s t h e d i a g o n a l 1 P condition. T h i s f o r m u l a c a n b e m e r g e d into A x A u A y CXDlD0 ( c f . ( 8 . 1 5 ) ) , w h i c h s h o w s t h a t s a t i s f i a b i l i t y f o r / l v / j i s undecidable.

A formula y of predicate calculus i s called satisfiable if it has a model. It has been proved by Church (1936) that there i s no algorithm to check the satisfiability for a r b i t r a r y formulae (undecidability of the Deci- sion Problem). One may be interested in the s a m e question, where tJ) i s restricted to a subclass A of the c l a s s of all formulae. Among these subclasses those a r e of special interest which may be described by a prefix. E. 2. t b r e belong to the subclass AAv a l l I\x ten in the form x, y , z /\ y /\ I+ , where z i s quantor-free may be writ- and has a t most a s f r e e variables.

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Aspects of Mathematical Logic by H. Hermes (auth.), Prof. E. Casari (eds.)


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