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6.2. Effectivity.
We begin with a straightforward effectivity result
concerning Propositions AH. Specifically, we show that
Propositions AH hold in the arithmetic sets. Later we will
show that Propositions C,EH 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 AH. 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. 1Con(MAH).
vi. 1Con(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 1Con(MAH),
1Con(SMAH). Hence i,v,vi are equivalent. It suffices to
prove vi
→
iii
→
ii
→
iv
→
vi.
Since ACA' proves X is equivalent to 1Con(SMAH), we see
that in any
ω
model of ACA
0
, X is equivalent to 1Con(SMAH).
For vi
→
iii, suppose 1Con(SMAH). Then 1Con(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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iii
→
ii
→
iv is trivial.
For iv
→
vi, suppose X is true in some countable
ω
model of
ACA
0
. Then 1Con(SMAH) is true in some
ω
model of ACA
0
.
Hence 1Con(SMAH). QED
We are now going to show that Propositions C,EH hold in
the recursive subsets of N. Propositions C,EH, when stated
in the recursive sets, become
Π
0
4
statements.
We shall see that Propositions C,EH hold in the smaller
family of infinite sets with primitive recursive
enumeration functions.
We also show that all of these variants of C,EH are
provably equivalent to 1Con(SMAH) in ACA'.
We conjecture that a more careful argument will show that
Propositions C,EH 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,EH,
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.
 Fall '08
 JOSHUA
 Math, Sets

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