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4.4use1-Con071110

# 4.4use1-Con071110 - 1 4.4 Proof using 1-consistency In this...

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1 4.4. Proof using 1-consistency. In this section we show that Propositions A,B can be proved in ACA’ + 1-Con(SMAH). Here 1-Con(T) is the 1-consistency of T, which asserts that “every Σ 0 1 sentence provable in T is true”. 1-Con(T) is also equivalent to “every Π 0 2 sentence provable in T is true”. By Lemma 4.2.1, Proposition B implies Proposition A in RCA 0 . Hence it suffices to show that Proposition B can be proved in ACA’ + 1-Con(SMAH). DEFINITION 4.4.1. We write ELG(p,b) for the set of all f ELG of arity p satisfying the following conditions. For all x N p , i. if |x| > b then (1 + 1/b)|x| f(x) b|x|. ii. if |x| b then f(x) b 2 . Note that from Definition 2.1, f ELG if and only if there exist positive integers p,b such that f ELG(p,b). Also note that each ELG(p,b) forms a compact subspace of the Baire space of functions from N k into N. DEFINITION 4.4.2. Let p,q,b 1. A p,q,b-structure is a system of the form M* = (N*,0*,1*,<*,+*,f*,g*,c 0 *,...) such that 1. N* is countable. For specificity, we can assume that N* is N. 2. (N*,0*,1*,<*,+*) is a discretely ordered commutative semigroup (see definition below). 3. +*:N* 2 N*, f*:N* p N*, g*:N* q N*. 4. f* obeys the above two inequalities for membership in ELG(p,b), internally in M*. 5. g* obeys the above two inequalities for membership in ELG(q,b), internally in M*. 6. Let i 0. The sum of any finite number of copies of c i * is < c i+1 *. 7. The c*’s form a strictly increasing set of indiscernibles for the atomic sentences of M*. Note that the conditions under clauses 4-7 are all universal sentences.

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2 Note that we do not require every element of N* to be the value of a closed term. DEFINITION 4.4.3. A discretely ordered commutative semigroup is a system (G,0,1,<,+) such that i. < is a linear ordering of G. ii. 0,1 are the first two elements of G. iii. x+0 = x. iv. x+y = y+x. v. (x+y)+z = x+(y+z). vi. x < y x+z < y+z. vii. x+1 is the immediate successor of x. Note that the cancellation law x+z = y+z x = y holds in any discretely ordered commutative semigroup (in this sense), since assuming x+z = y+z, the cases x < y and y < x are impossible. In any p,q,b-structure, the c n * have an important inaccessibility condition: any closed term whose value is c n * is a sum consisting of c n * and zero or more 0*’s. To see this, write c n * = t, and write t as a sum, t = s 1 + ... + s k , k 1, where each s i is either a constant or starts with f or g. By 7, c n * is infinite, and so all s i that begin with f or g must have immediate subterms < c n * (using 4,5). Hence all s i that begin with f or g must be < c n * (using 4,5,6). Hence all s i are either < c n * or are a constant. If no s i is c i * then all s i are < c n *, violating 6. Hence some s i is c n *. By 2, the remaining s i must be 0. We can follow the development of section 4.2 starting right after the proof of Lemma 4.2.7. In this rerun, we do not fix f ELG(p,b), and g ELG(q,b).
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