6.2.Effect072710

6.2.Effect072710 - 1 6.2. Effectivity. We begin with a...

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1 6.2. Effectivity. We begin with a straightforward effectivity result concerning Propositions A-H. Specifically, we show that Propositions A-H hold in the arithmetic sets. Later we will show that Propositions C,E-H hold in the recursive sets. We don’t know if any or all of Propositions A,B,D hold in the recursive sets. We conjecture that i. None of Propositions A,B,D hold in the recursive sets. ii. This fact can be proved in ACA’. DEFINITION 6.2.1. Let ACA + be the formal system consisting of ACA 0 and “for all x ω , the ω -th Turing jump of x exists”. See [Si99,09], p. 404, where ACA + is written as ACA 0 + . ACA + is a proper extension of ACA’ that allows us to handle ω models of ACA 0 . Note that the countable ω models of ACA 0 , ACA', ACA are the same as the countable families of subsets of N that are closed under relative arithmeticity, as induction is automatic in ω models. (Here ACA is ACA 0 with induction for all formulas, and is a proper extension of ACA'). THEOREM 6.2.1. Let X be any of Propositions A-H. The following are provably equivalent in ACA + . i. X is true. ii. X is true in the arithmetic sets. iii. X is true in every countable ω model of ACA 0 . iv. X is true in some countable ω model of ACA 0 . v. 1-Con(MAH). vi. 1-Con(SMAH). Proof: Let X be as given. We argue in ACA + . By Theorems 5.9.11, 6.1.2, and 6.1.10, X is equivalent to 1-Con(MAH), 1-Con(SMAH). Hence i,v,vi are equivalent. It suffices to prove vi iii ii iv vi. Since ACA' proves X is equivalent to 1-Con(SMAH), we see that in any ω model of ACA 0 , X is equivalent to 1-Con(SMAH). For vi iii, suppose 1-Con(SMAH). Then 1-Con(SMAH) is true in any ω model of ACA 0 . Hence X is true in every ω model of ACA 0 , and therefore iii,ii,iv.

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2 iii ii iv is trivial. For iv vi, suppose X is true in some countable ω model of ACA 0 . Then 1-Con(SMAH) is true in some ω model of ACA 0 . Hence 1-Con(SMAH). QED We are now going to show that Propositions C,E-H hold in the recursive subsets of N. Propositions C,E-H, when stated in the recursive sets, become Π 0 4 statements. We shall see that Propositions C,E-H hold in the smaller family of infinite sets with primitive recursive enumeration functions. We also show that all of these variants of C,E-H are provably equivalent to 1-Con(SMAH) in ACA'. We conjecture that a more careful argument will show that Propositions C,E-H hold in the yet smaller family of infinite superexponentially Presburger sets. In light of the primitive recursive decision procedure for superexponential Presburger arithmetic, Propositions C,E-H, when stated in the superexponentially Presburger sets, become Π 0 2 statements. This topic will be discussed at the end of this section. Recall TM(0,1,+,-,•,
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This note was uploaded on 08/05/2011 for the course MATH 366 taught by Professor Joshua during the Fall '08 term at Ohio State.

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6.2.Effect072710 - 1 6.2. Effectivity. We begin with a...

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