Turing jumps through provability

0.19 MB·English

by  Joost J. Joosten

Turing jumps through provability
5
1 Joost J. Joosten∗
0
2
University of Barcelona
n
a
J
January 23, 2015
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2

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Abstract
L
FixingsomecomputablyenumerabletheoryT,theFriedman-Goldfarb-
.
h Harrington(FGH)theoremsaysthatoverelementaryarithmetic,eachΣ1
at formula is equivalent to some formula of the form (cid:3)Tϕ provided that T
m is consistent. In this paper we give various generalizations of the FGH
theorem. In particular, for n > 1 we relate Σn formulas to provability
[ statements [n]TTrueϕ which are a formalization of “provable in T together
1 withalltrueΣn+1 sentences”. Asacorollaryweconcludethateach[n]TTrue
v is Σn+1-complete.
7 This observation yields us to consider a recursively defined hierarchy
2 of provability predicates [n+1](cid:3)T which look a lot like [n+1]TTrue except
3 that where [n+1]TTrue calls upon the oracle of all true Σn+2 sentences,
5 the [n+1](cid:3)T recursively calls upon the oracle of all true sentences of the
.0 form hni(cid:3)Tφ. As such we obtain a ‘syntax-light’ characterization of Σn+1
1 definability whence of Turing jumps which is readily extended beyond
0 thefinite. Moreover, weobservethat thecorresponding provabilitypred-
5 icates [n+1](cid:3)T are well behaved in that together they provide a sound
1 interpretation of thepolymodal provability logic GLPω.
:
v
i
X 1 Introduction
r
a
Infirstorderarithmetic we havenaturalsyntacticaldefinitions that correspond
to finite iterations of the Turing jump. Recall that a sentence in first order
logic in the standard language of arithmetic is Σ if it starts with a block of
n+1
alternating quantifiers of length n+1 where the leftmost quantifier is existen-
tial and where the block of quantifiers is followed by a decidable formula only
containing bounded quantification. There are various results known that relate
these formula classes to computational complexity classes.
For example, a set of natural numbers is many-one reducible to the n-th
Turing jump of the empty set if and only if it is Σ definable (on the standard
n
∗[email protected]
1
model of the natural numbers). Likewise, a set of natural numbers is Turing-
reducible to the n-th Turing jump of the empty set if and only if it and its
complement can be defined on the standard model of the natural numbers by
a Σ formula. Similarly, a set of natural numbers is computably enumerable
n+1
relativetothen-thTuringjumpoftheemptysetifandonlyifitcanbedefined
by a Σ formula.
n+1
In this paper we shall use the fact that various provability predicates are
Turing complete in a certain sense so that we can give alternative characteri-
zations for the finite Turing jumps. A central ingredient in proving our results
come from generalizations of the so-called FGH Theorem.
TheFGHTheorem(forFriedman-Goldfarb-Harrington)tellsusthatforany
c.e.theoryT wehaveprovablyinElementaryArithmeticthateachΣ sentence
1
σ is equivalent to a provability statement of T, provided T is consistent. In
symbols,
∀σ∈Σ ∃ψ EA⊢ ♦ ⊤→ σ ↔(cid:3) ψ .
1 T T
(cid:0) (cid:1)
Here, as usual (cid:3) denotes a natural formalization of provability in T and the
T
♦ stands for the dual consistency assertion. In this paper we give various
T
generalizations of the FGH theorem. In particular we prove that the theorem
holds for the provabilitynotion [n] : provable in T together with all true Σ
T n+1
formulas. As a corollary we conclude that each [n]True is Σ -complete.
T n+1
Thisobservationyieldsustoconsiderarecursivelydefinedhierarchyofprov-
ability predicates [n+1](cid:3) which look a lot like [n+1]True except that where
T T
[n+1]TruecallsupontheoracleofalltrueΣ sentences,the[n+1](cid:3)recursively
T n+2 T
(cid:3)
calls upon the oracle of all true sentences of the form hni φ.
T
As such we obtain a ‘syntax-light’ characterization of Σ definability
n+1
whence of Turing jumps which is readily extended beyond the finite. More-
(cid:3)
over,we observethat the correspondingprovability predicates [n+1] are well
T
behaved in that together they provide a sound interpretation of the polymodal
provability logic GLP .
ω
2 Preliminaries
We shall work with theories with identity in the language {0,1,exp,+,·,<} of
arithmetic where exp denotes the unary function x 7→ 2x. We define ∆ =
0
Σ = Π formulas as those where all quantifiers occur bounded, that is, we
0 0
only allow quantifiers of the form ∀x<t or ∃x<t where t is some term not
containing x. We inductively define Σ ,Π ⊂ Π and Σ ,Π ⊂ Σ ; if
n n n+1 n n n+1
φ,ψ ∈Π , then ∀x φ,φ∧ψ,φ∨ψ ∈Π and likewise, if φ,ψ ∈Σ , then
n+1 n+1 n+1
∃x φ,φ∧ψ,φ∨ψ ∈Σ .
n+1
We shall write Σ ! for formulas ϕ of the form ∃x ϕ with ϕ ∈ Π . We
n+1 0 0 n
will work in the absence of strong versions of (bounded) collection B which is
ϕ
defined as
B := ∀z∀~u ∀x<z∃y ϕ(x,y,~u)→∃y′∀x<z∃y<y′ ϕ(x,y,~u) .
ϕ (cid:16) (cid:17)
2
Therefore, we will consider the formula class Σ consisting of existentially
n+1,1
quantified disjunctions and conjunctions of Σ formulas with bounded quan-
n+1
tifiers over them. To be more precise, we first inductively define
Σ :=Σ |(Σ ◦Σ )|(Qx<y Σ )
n+1,b n+1 n+1,b n+1,b n+1,b
with ◦ ∈ {∧,∨} and Q ∈ {∀,∃}. Next we define the Σ formulas to be of
n+1,1
the form ∃x φ with φ∈Σ .
n+1,b
The theory of elementary arithmetic, EA, is axiomatized by the defining
axioms for {0,1,exp,+,·,<} together with induction for all ∆ formulas. The
0
theory Peano Arithmetic, PA, is as EA but now allowing induction axioms for
any first order formula.
It is well known that PA proves any instance B of collection so that in
ϕ
particular each Σ sentence is equivalent to some Σ sentence. Clearly
n+1,1 n+1
wehavethatΣ !⊂Σ . Usingcodingtechniques,itisclearthateachΣ
n+1 n+1 n+1
formula is within EA equivalent to a Σ ! formula.
n+1
For us, a computably enumerable (c.e.) theory T is understood to be given
by a ∆ formula that defines the set of codes of the primitive recursive set of
0
axioms of T. We will employ standard formalizations of meta-mathematical
properties like Proof (x,y) for “x is the G¨odel number of a proof from the
T
axioms of T of the formula whose G¨odel number is y”. We shall often refrain
from distinguishing a syntactical object ϕ from its G¨odel number pϕq or from
a syntactical representation of its G¨odel number.
Wewillwrite(cid:3) ϕfortheΣ !formula∃xProof (x,ϕ)and♦ ϕfor¬(cid:3) ¬ϕ.
T 1 T T T
By (cid:3) ϕ(x˙) we will denote a formula which contains the free variable x, that
T
expresses that for each value of x the formula ϕ(x) is provable in T. Here, x
denotes a syntactical representation of the number x.
By Σ completeness we refer to the fact that for any true Σ sentence σ we
1 1
havethat EA⊢σ. It is well-knownthatEA provesa formalizedversionof this:
for any Σ formula σ(x) and any c.e. theory T we have EA⊢σ(x)→(cid:3) σ(x˙).
1 T
3 The FGH theorem and generalizations
In this section we shall be dealing with various so-called witness-comparison
arguments where the order of (least) witnesses to existential sentences is im-
portant. The first and most emblematic such argument occurred in the proof
of Rosser’s theorem which is a strengthening of G¨odel’s first incompleteness
theorem.
Theorem 3.1 (Rosser’s Theorem). Let T be a consistent c.e. theory extending
EA. There is some ρ∈Σ which is undecidable in T. That is,
1
T 0ρ and,
T 0¬ρ.
Forrhetoricreasonsweshallbelowincludeastandardproofofthiscelebrated
result. Before doing so, we first need some notation.
3
Definition 3.2. For φ:=∃x φ (x) and ψ :=∃x ψ (x) we define
0 0
φ≤ψ := ∃x φ (x)∧∀y<x ¬ψ (y) and,
0 0 0
(cid:0) (cid:1)
φ<ψ := ∃x φ (x)∧∀y≤x ¬ψ (x) .
0 0
(cid:0) (cid:1)
Statements ofthe formφ≤ψ orφ<ψ withφ,ψ ∈Σ arecalledwitness-
n+1
comparisonstatements. Let us now collect some easy principles about witness-
comparison statements whose elementary proofs we leave as an exercise.
Lemma 3.3. For A and B in Σ we have
n+1
1. EA⊢(A<B)→(A≤B);
2. EA⊢(A<B)∧(B ≤C)→(A<C);
3. EA⊢(A≤B)∧(B <C)→(A<C);
4. EA⊢(A≤B)∧(B ≤C)→(A≤C);
5. EA⊢(A≤B)→¬(B <A) and consequently;
6. EA⊢(A<B)→¬(B ≤A);
7. EA⊢[(B ≤B)∨(A≤A)] → [(A≤B)∨(B <A)];
8. EA⊢(A≤B)→A;
9. EA⊢A∧¬B →(A<B) ;
10. EA⊢A∧¬(A≤B)→B.
11. Both C <D and C ≤D are of complexity Σ if C,D∈Σ .
n+1,1 n+1,1
We can now present a concise proof of Rosser’s theorem.
Proof. We consider a fixpoint ρ so that T ⊢ρ↔((cid:3) ¬ρ≤(cid:3) ρ).
T T
IfT ⊢ρ,thenforsomenumbernwehaveProof (n,ρ). SinceT isconsistent
T
we also have ∀m≤n¬Proof (m,¬ρ). Thus, by Σ completeness we have T ⊢
T 1
(cid:3) ρ<(cid:3) ¬ρ whence T ⊢¬ρ; a contradiction.
T T
Likewise, if T ⊢¬ρ we may conclude (cid:3) ¬ρ≤(cid:3) ρ so that T ⊢ρ.
T T
We would like to stress that it is actually quite remarkable that witness
comparison arguments on statements involving G¨odel numbering can be used
to yield anysensible informationatall, since by tweaking the G¨odel numbering
in a primitive recursive fashion we can always flip the order of the codes of
any two syntactical objects. Of course, as we use fixpoints, after tweaking
the G¨odel numbering, the corresponding fixpoint also changes. But still, it
is remarkable that the useful witness comparison information of the fixpoint
cannot be destroyed by tweaking the underlying G¨odel numbering.
We now turn our attention to another theorem, a proof of which can suc-
cinctly be given using witness comparison arguments: the FGH theorem. The
4
initials FGH refer to Friedman, Goldfarb and Harrington who all substantially
contributed to the theorem and we refer to [6] for historical details.
Basically, the FGH theorem says that given any c.e. theory T, any Σ sen-
1
tenceisprovablyequivalenttoaprovabilitystatementoftheform(cid:3) ϕ,modulo
T
the consistency of T. The proof we give here is a slight modification of the one
presented in [6]. The most important improvement is that we avoid the use of
the least-number principle so that the proof becomes amenable for generaliza-
tions without a need to increase the strength of the base theory.
Theorem 3.4 (FGH theorem). Let T be any computably enumerable theory
extending EA. For each σ ∈Σ we have that there is some ρ∈Σ so that
1 1
EA⊢♦ ⊤→ σ ↔(cid:3) ρ .
T T
(cid:0) (cid:1)
Proof. Asin[6]weconsiderthefixpointρ∈Σ forwhichEA⊢ρ ↔ (σ ≤(cid:3) ρ).
1 T
Without loss of generality we may assume that σ ∈ Σ ! so that both σ ≤ (cid:3) ρ
1 T
and (cid:3) ρ<σ are Σ . We now reason in EA, assume ♦ ⊤ and set out to prove
T 1 T
σ ↔(cid:3) ρ.
T
(→): assume for a contradiction that σ and ¬(cid:3) ρ. By Lemma 3.3.9 we
T
conclude σ ≤(cid:3) ρ, i.e., ρ. By provable Σ completeness we get (cid:3) ρ.
T 1 T
(←): assume for a contradiction that ¬σ and (cid:3) ρ. Again, we conclude
T
(cid:3) ρ < σ so that (cid:3) ((cid:3) ρ < σ) whence (cid:3) ¬ρ so that (cid:3) ⊥ contradicting the
T T T T T
assumption ♦ ⊤.
T
A very special feature of ρ from the above proof is that it is of complexity
Σ and that¬ρ is implied a by a relatedΣ formula. Thus,we clearlyprovably
1 1
have ρ → (cid:3) ρ but in general we do not have ¬ρ → (cid:3) ¬ρ. But, due to the
T T
nature of ρ we have that (cid:3) ¬ρ follows from the Σ statement that is slightly
T 1
strongerthan¬ρ,namely(cid:3) ρ<σ: sinceprovably((cid:3) ρ<σ)→(cid:3) ((cid:3) ρ<σ)
T T T T
and
((cid:3) ρ<σ) → ¬(σ ≤(cid:3) ρ)
T T
→ ¬ρ.
The least number principle for Σ formulas, LΣ , says that for any ψ ∈Σ we
n n n
have ∃xψ(x)→∃x(ψ(x)∧∀y<x ¬ψ(y)). Of course,using LΣ and by Lemma
0
3.3.7,¬ρand(cid:3)ρ<σareprovablyequivalentundertheassumptionthat(cid:3)ρ∨σ.
We willbeinterestedingeneralizingtheFGHtheoremtoΣ formulasusing
n
ever stronger notions of provability. Visser’s proof of the FGH theorem as
presentedin[6]usedanapplicationoftheleastnumberprinciplefor∆ formulas
0
intheguiseofA→(A≤A). Thus,adirectgeneralizationofVisser’sargument
tostrongerprovabilitynotionswouldcallforstrongerandstrongerarithmetical
principles:
Lemma 3.5. The schema A → (A ≤ A) for A ∈ Σ ! is over EA provably
n+1
equivalent to the least-number principle for Π formulas.
n
However,since ourproofoftheFGHtheoremdidnotuse theminimal num-
ber principle, we shall now see how the above argument generalizes to other
5
provability predicates. By [n+1]True we will denote the formalization of the
T
predicate “provable in T together with all true Σ sentences”. For conve-
n+2
nience, we set [0] := (cid:3) . Basically, for n > 0, the predicate [n]Trueϕ will be
T T T
a formalization of “there is a sequence π ,...,π so that each π is either an
0 m i
axiom of T, or a true Σ sentence, or a propositional logical tautology, or
n+1
a consequence of some rule of T using earlier elements in the sequence as an-
tecedents”. Thus, it is clear that for recursive theories T we can write [n]True
T
by a Σ -formula. Also, we have provable Σ completeness for these
n+1,1 n+1,1
predicates, that is:
Lemma3.6. LetT beac.e.theoryextendingEAandletφbeaΣ formula.
n+1,1
We have that
EA⊢φ(x)→[n]Trueφ(x˙).
T
Proof. Given φ(z,y ,...,y ,x) ∈ Σ , reason in EA, fix x ,...,x ,x and,
1 k n+1 1 k
assume
∃zQ y <x ...Q y <x φ(z,y ,...,y ,x)
1 1 1 k k k 1 k
where Q y <x ...Q y <x is some block of bounded quantifiers. Thus, for
1 1 1 k k k
some a we have Q y <x ...Q y <x φ(a,y ,...,y ,x). Under the box, we
1 1 1 k k k 1 k
can now replace each ∀y <x by and each ∃y <x by so that
i i Vyi<xi i i Wyi<xi
by applying distributivity we see that Q y <x ...Q y <x φ(a,y ,...,y ,x)
1 1 1 k k k 1 k
is equivalent to a disjunctive normal form of bounded substitution instances of
φ(a,y ,...,y ,x). Note that this operationin available within EA since it only
1 k
requires the totality of exponentiation.
Outsidetheboxweknowthatforsomeofthesebigconjunctionsofbounded
substitutioninstancesofφ(a,y ,...,y ,x)actuallyalloftheconjunctsaretrue.
1 k
SinceeachofthoseconjunctsisatrueΣ sentence,eachconjunctisanaxiom
n+1
whence holds under the box. Thus, the whole conjunct is provable under the
box whereby we obtain [n]TrueQ y <x ...Q y <x φ(a,y ,...,y ,x) whence
T 1 1 1 k k k 1 k
[n]True∃zQ y <x ...Q y <x φ(z,y ,...,y ,x)aswastobeshown. Itisclear
T 1 1 1 k k k 1 k
that this case suffices for the more general form of Σ formulas.
n+1,1
Itiseasytocheckthatthepredicate[n]True iswellbehaved. Inparticularone
T
can check that all the axioms of the standard provability logic GL as defined
in the last section hold for it. Over EA, the notion of [n]True can be related to
T
regular provability (cid:3) by the lemma below.
T
By FinSeq(f) we denote a predicate that only holds on numbers that are
codes of a finite sequence of G¨odel numbers and by |f| we denote the length of
sucha sequence. Moreover,f will denote the ith elementofsucha sequence f.
i
With True we will denote a partial truth predicate for Σ formulas and
Σn+1 n+1
Tr will denote the set of true Σ sentences.
Σn+1 n+1
Lemma 3.7. For any c.e. theory T, we have that
EA⊢[n]Trueϕ↔∃f FinSeq(f)∧∀i<|f| True (f )∧(cid:3) (∧ f )→ϕ .
T (cid:16) Σn+1 i T(cid:0) i<|f| i (cid:1)(cid:17)
6
Proof. We reason in EA fixing some ϕ. The ← direction follows directly from
the formalized deduction theorem. For the other direction fix some p with
ProofT+TrΣn+1(p,ϕ). We can express that pi is a true Σn+1 sentence in a ∆0
fashion simply by saying that it is not a propositional tautology, nor an axiom
ofT,northe resultofapplyingaruletoearlierelementsinthesequence. Thus,
by ∆ induction on the length of p we can prove that there is a sequence that
0
collects all the true Σ sentences.
n+1
Note that our definition of [n]True is slightly non-standard since in the lit-
T
erature (e.g. [1]) it is more common to define [n]True using a Π oracle rather
T n
thanaΣ oracle. WithaΠ oracleonegetsprovableΣ completenessbut
n+1 n n+1
in the absence of BΣ not necessarily provable Σ completeness. With
n+1 n+1,1
our definition of [n]True, since A ≤ B ∈ Σ for A,B ∈ Σ , and since
T n+1,1 n+1,1
we provided a proof where the minimal number principle is avoided, the FGH
theorem smoothly generalizes to the new setting.
Theorem 3.8. Let T be any computably enumerable theory extending EA and
let n<ω. For each σ ∈Σ we have that there is some ρ ∈Σ so that
n+1,1 n n+1,1
EA⊢hniTrue⊤→ σ ↔[n]Trueρ .
T (cid:0) T n(cid:1)
Proof. The proofruns entirely analogue to the proof ofTheorem3.4. Thus, for
each number n we consider the fixpoint ρ so that EA⊢ ρ ↔(σ ≤[n]Trueρ ).
n n T n
Note that both ρ and [n]Trueρ < σ are Σ whence by Lemma 3.6 we can
n T n+1,1
apply provable Σ completeness to them.
n+1,1
As an easy corollary we get that [n]True formulas are closed not only under
T
conjunction, as is well know, but also under disjunctions. Note that the FGH
theorem yields that provably hniTrue⊤ → (σ ↔ [n]Trueρ ). Using the proposi-
T T n
tional tautology
(¬A→C)→ A→(B ↔C) ↔ (¬A∨B)↔C
h(cid:0) (cid:1) (cid:0) (cid:1)i
and [n]True⊥ → [n]Trueρ we see that this is equivalent to [n]True⊥ ∨ σ ↔
T T n (cid:0) T (cid:1)
[n]Trueρ .
T n
Corollary 3.9. Let T be a c.e. theory extending EA and let n ∈ N. For each
formulas ϕ,ψ there is some σ ∈Σ so that
n+1
T ⊢([n]Trueϕ ∨[n]Trueψ) ↔ [n]Trueσ.
T T T
Proof. Weconsidersomeσ ∈Σ sothatprovablyσ ↔([n]Trueϕ ∨[n]Trueψ).
n+1,1 T T
By Theorem 3.8 applied to this σ we provably have that ([n]Trueϕ ∨[n]Trueψ)∨
T T
[n]True⊥ is equivalent to [n]Trueϕ ∨[n]Trueψ.
T T T
As another corollary of Theorem 3.8 we see that, in a sense, the notion of
n-provabilityis Σ complete. Toestablishthis, wewillfirstneedaparticular
n+1
version of the fixpoint lemma.
7
Lemma3.10. Letψ(x,y)beaformulawhosefreevariables areamongst{x,y}.
There is a formula ϕ(y) so that
EA⊢ϕ(y)↔ψ(pϕ(y˙)q,y).
Proof. The lemma easily follows from the well-known version of the fixpoint
lemma by which for each formula ψ(x,y) there is a formula ϕ(y) so that
EA⊢ϕ(y)↔ψ(pϕ(y)q,y) (1)
(e.g., the generalized diagonal lemma from Boolos’ [2], Chapter 3).
Byz =sub(u,v,w)wedenotethe formulathatexpressesthatz istheG¨odel
number of the result of substituting the numeral of w for the variable whose
G¨odel number is v into the formula whose G¨odel number is u. By ψ(x/z,y) we
denotetheresultofsubstitutingz forxinψ(x,y). Wenowconsidertheformula
∃z z =sub(x,pyq,y) ∧ ψ(x/z,y)
(cid:0) (cid:1)
and apply Equation (1) to it to obtain the required fixpoint. Note that
EA⊢ ∃z z =sub(pϕ(y)q,pyq,y) ∧ ψ(x/z,y) ←→ ψ(pϕ(y˙)q,y).
h (cid:0) (cid:1)i
With this lemma we can now prove Σ -completeness of the [n]True prov-
n+1 T
ability predicate.
Lemma 3.11. Let T be any sound c.e. theory and let A ⊆ N. The following
are equivalent
1. A is c.e. in ∅(n);
2. A is many-one reducible to ∅(n+1);
3. A is definable on the standard model by a Σ formula;
n+1
4. A is definable on the standard model by a formula of the form [n]Trueρ(x˙);
T
5. A is definable on the standard model by a formula of the form [n]Trueρ(x˙)
T
where ρ(x)∈Σ ;
n+1,1
Proof. The equivalence of 1, 2, and 3 is just Post’s theorem.
The implication 5 ⇒ 4 is trivial, and implication 4 ⇒ 3 holds in virtue of
[n]True beingaΣ predicatewhichonNisequivalenttosomeΣ formula,
T n+1,1 n+1
so it suffices to prove 3 ⇒ 5.
Thus, let the number n be fixed and, let A be a set of natural numbers so
that for some σ(x)∈Σ we have m∈A ⇐⇒ N|=σ(m). Using lemma 3.10
n+1
we find ρ (x)∈Σ so that
n n+1,1
EA⊢∀x ρ (x) ↔ [σ(x)≤[n]Trueρ (x˙)] .
(cid:0) n T n (cid:1)
8
Reasoning in EA, we pick an arbitrary x and repeat the reasoning as in the
proof of Theorem 3.8 to see that
EA⊢hniTrue⊤→∀x σ(x) ↔ [n]Trueρ (x˙) .
T (cid:16) (cid:0) T n (cid:1) (cid:17)
Since EAis sound andby assumptionof T alsobeing sound we have for eachn
that N|=hniTrue⊤, we may conclude that for any number m,
T
N|=σ(m) ⇐⇒ N|= [n]Trueρ (m)
T n
which was to be proven.
Feferman showed in [3] that for each unsolvable Turing degree d there is a
theoryU sothattheTuringdegreeof{pϕq|U ⊢ϕ}isd. However,thetheories
that Feferman considered were formulated in the language of identity and in
particulardidnotcontainarithmetic. Itisnothardtoseethatfortheoriesthat
do contain arithmetic we can only attain degrees that arise as Turing jumps.
Lemma 3.12. Let A ⊆ N be definable on N by α(x) with Turing degree a.
Then, the
Turing degree of {ψ|EA+{α(n)|N|=α(n)}⊢ψ} equals a′.
Proof. Let us denote the Turing degree of a set B by |B|. We see that
|{ψ |EA+{α(n)|N|=α(n)}⊢ψ}|≤ a′
T
since a Turing machine can enumerate all oracle proofs thereby reducing prov-
ability to the halting problem. Similarly, we see that
a′ ≤ |{ψ|EA+{α(n)|N|=α(n)}⊢ψ}|
T
since we can code Turing machine computations in arithmetic.
WethusseethattheTuringdegreesoftheoriesthataredefinedbyaminimal
amountofarithmetic(EA)togetherwithsomeoracle,areentirelydeterminedby
the Turing degree of the corresponding oracle by means of the jump operator.
By Friedberg’s jump inversion theorem ([5]) we may thus conclude that any
Turing degree above 0′ can be attained as the decision problem of a theory
containing arithmetic.
In this light, Lemma 3.11 should not come as a surprise. However, in the
lemmawehaveprovidedanaturalsubsequenceoftheoriessothatmoreover,all
the necessary reasoning for the reductions can be formalized in EA.
Lemma3.11isstatedentirelyintermsofdefinabilityandcomputabilitybut
the proof tells us actually a bit more, namely that the FGH theorem is easily
formalizable within EA.
Lemma 3.13. For any c.e. theory T we have that
EA⊢∀σ∈Σ ∃ρ∈Σ hniTrue⊤ → True (σ)↔[n+1]Trueρ .
n+1 n+1 (cid:16) T (cid:0) Σn+1 T (cid:1)(cid:17)
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Proof. One can simply formalize the proof of Theorem 3.8 which is easy since
mapping the G¨odel number of a Σ sentence σ to the G¨odel number of its
n+1
corresponding fixpoint ρ is elementary.
n
Alternatively, using Lemma 3.10 we find ρ (x)∈Σ so that
n n+1
EA⊢∀x ρ (x) ↔ [True (x)≤[n]Trueρ (x˙)] .
(cid:0) n Σn+1 T n (cid:1)
so that by reasoning as in the proof of Lemma 3.11 we see that
EA⊢hniTrue⊤ → ∀x True (x) ↔ [n]Trueρ (x˙) .
T (cid:16) Σn+1 (cid:0) T n (cid:1) (cid:17)
For other notions of provability we get similar generalizations of the FGH
theorem. In particular, let [n]Omega denote the formalization of the predicate
T
“provable in T using at most n nestings of the omega rule”. Following the
recursive scheme [0]Omegaϕ:=(cid:3) ϕ and,
T T
[n+1]Omegaϕ:=∃ψ ∀x [n]Omegaψ(x˙) ∧ (cid:3) (∀x ψ(x)→ϕ)
T (cid:16) T T (cid:17)
we see that for c.e. theories T we can write [n]Omega by a Σ -formula. Also,
T 2n+1
we have provable Σ completeness for these predicates, that is:
2n+1
Proposition 3.14. Let T be a computable theory extending EA and let φ be a
Σ formula. We have that
2n+1
EA⊢φ→[n]Omegaφ.
T
Proof. By an external induction on n where each inductive step requires the
application of an additional omega-rule.
Thispropositionistheomega-ruleanalogueofprovableΣ completeness
n+1,1
for the [n]True predicate. As a corollary we get an FGH Theorem for omega-
T
provability.
Corollary 3.15. Let T be any sound computably enumerable theory extending
EA and let n<ω. For each σ ∈Σ we have that there is some ρ ∈Σ
2n+1 n 2n+1,1
so that
PA⊢hniOmega⊤→ σ ↔[n]Omegaρ .
T (cid:0) T n(cid:1)
We have formulated this corollary over PA so that Σ sentences are
2n+1,1
provably equivalent to Σ sentences using collection. Consequently, we can
2n+1
now also prove a definability result for the [n]Omega predicate.
T
Lemma 3.16. Let T be any c.e. theory, let n be a natural number, and let
A⊆N. The following are equivalent
1. A is c.e. in ∅(2n);
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