week1.pdf - SKETCHY NOTES FOR WEEK 1 OF BASIC LOGIC A...

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SKETCHY NOTES FOR WEEK 1 OF BASIC LOGIC A function F is n -ary (has arity n ) if its domain is a set of n -tuples and finitary if it is n -ary for some n . If F is n -ary and C is a set, we say that C is closed under F if F ( a 0 , . . . a n - 1 ) C for all n -tuples ( a 0 , . . . a n - 1 ) C n dom( F ). A basic idea in the course is to form the closure of a set B under functions in some set K of finitary functions, that is to say the least set that contains B and is closed under all functions F in K . Theorem 1. If B is a set and K is a set of finitary functions then there is a unique set C such that: (1) B C . (2) C is closed under F for all F K . (3) For any set C 0 with B C 0 and C 0 closed under F for all F K , we have C C 0 . Proof. We show first that there is such a set C and then argue that it is unique. To construct a suitable C , let C 0 = B and then define C n +1 = C n ∪ { F ( a 0 , . . . a k - 1 ) : F K is k -ary and ( a 0 , . . . a k - 1 ) C k n dom( F ) } . Then we set C = S n N C n . Clearly B = C 0 C . To prove closure we use the fact that the sets C n are increasing in the sense that m < n implies C m C n . So suppose that F K is k -ary and ( a 0 , . . . a k - 1 ) C k dom( F ). For each i < k we choose n i such that a i C n i and then set n * = max { a 0 , . . . a k - 1 } . Since the
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