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4.3exist071110

# 4.3exist071110 - 1 4.3 Some Existential Sentences In this...

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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 λ (k,n,m,R 1 ,...,R n-1 ) = ( ∃ infinite B 1 ,...,B n ⊆ N k ) ( ∀ i ∈ {1,...,n-1})( ∀ 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 n-1 ⊆ 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 λ (k,n,m,R 1 ,...,R n-1 ). 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 λ (k,n,m,R 1 ,...,R n-1 ). 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 λ (k,n,m,R 1 ,...,R n-1 ) 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 λ (k,n,m,R 1 ,...,R n-1 ), and prove that the criterion is correct, within the system ACA’. We first put the sentences λ (k,n,m,R 1 ,...,R n-1 ) in substantially simpler form. 2 DEFINITION 4.3.2. Define λ ’(k,n,R 1 ,...,R n-1 ) = ( ∃ infinite B 1 ,...,B n ⊆ N k ) ( ∀ i ∈ {1,...,n-1}) ( ∀ x,y,z ∈ B i )( ∃ w ∈ B i+1 )(R i (x,y,z,w)) where k,n ≥ 1, and R 1 ,...,R n-1 ⊆ N 4k are order invariant relations. LEMMA 4.3.1. There is a primitive recursive procedure for converting any sentence λ (k,n,m,R 1 ,...,R n-1 ) to a sentence λ ’(k’,n’,S 1 ,...,S n’-1 ) with the same truth value. In fact, ACA’ proves that any λ (k,n,m,R 1 ,...,R n-1 ) has the same truth value as its conversion λ ’(k’,n’,S 1 ,...,S n’-1 ). Proof: Start with *) ( ∃ infinite B 1 ,...,B n ⊆ N k )( ∀ i ∈ {1,...,n-1}) ( ∀ 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 m-tuples from N k . We write C# ⊆ N k for the set of all k-tuple 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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4.3exist071110 - 1 4.3 Some Existential Sentences In this...

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