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Unformatted text preview: 1 4.3. Some Existential Sentences. In this section, we prove a crucial Lemma needed for section 4.4. We consider existential sentences of the following special form. DEFINITION 4.3.1. Define (k,n,m,R 1 ,...,R n1 ) = ( infinite B 1 ,...,B n N k ) ( i {1,...,n1})( x 1 ,...,x m B i ) ( y 1 ,...,y m B i+1 )(R i (x 1 ,...,x m ,y 1 ,...,y m )) where k,n,m 1, and R 1 ,...,R n1 N 2km are order invariant relations. Recall that order invariant sets of tuples are sets of tuples where membership depends only on the order type of a tuple. Note the stratified structure of (k,n,m,R 1 ,...,R n1 ). It asserts that there are n infinite sets such that for all elements of the first there are elements of the second with a property, and for all elements of the second there are elements of the third with a property, etcetera. It is evident that even RCA suffices to define truth for the sentences of the form (k,n,m,R 1 ,...,R n1 ). For in RCA , we can i. Appropriately code finite sequences of subsets of N k as subsets of N. ii. Appropriately code finite sequences of elements of N as elements of N. iii. Appropriately treat order invariant sets of tuples from N. This does not mean that we can form the set of all true sentences of the form (k,n,m,R 1 ,...,R n1 ) in RCA or even ACA. However, we will show that this is in fact the case for ACA. See Definition 1.4.1. Specifically, we will present a primitive recursive criterion for the truth of sentences (k,n,m,R 1 ,...,R n1 ), and prove that the criterion is correct, within the system ACA. We first put the sentences (k,n,m,R 1 ,...,R n1 ) in substantially simpler form. 2 DEFINITION 4.3.2. Define (k,n,R 1 ,...,R n1 ) = ( infinite B 1 ,...,B n N k ) ( i {1,...,n1}) ( x,y,z B i )( w B i+1 )(R i (x,y,z,w)) where k,n 1, and R 1 ,...,R n1 N 4k are order invariant relations. LEMMA 4.3.1. There is a primitive recursive procedure for converting any sentence (k,n,m,R 1 ,...,R n1 ) to a sentence (k,n,S 1 ,...,S n1 ) with the same truth value. In fact, ACA proves that any (k,n,m,R 1 ,...,R n1 ) has the same truth value as its conversion (k,n,S 1 ,...,S n1 ). Proof: Start with *) ( infinite B 1 ,...,B n N k )( i {1,...,n1}) ( x 1 ,...,x m B i )( y 1 ,...,y m B i+1 )(R i (x 1 ,...,x m ,y 1 ,...,y m )). Let C,D N km . We think of C,D as sets of mtuples from N k . We write C# N k for the set of all ktuple components of elements of C. We write C D if and only if C,D N km , and for all (x 1 ,...,x m ),(y 1 ,...,y m ),(z 1 ,...,z m ) C, i. If (x 1 ,...,x m ) = (y 1 ,...,y m ) = (z 1 ,...,z m ) then (x 1 ,...,x m ) D....
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 Fall '08
 JOSHUA
 Math

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