Download The Arithmetical Hierarchy Math 503

The Arithmetical Hierarchy
Math 503
Benjamin Cohen Wallace
December 18, 2011
1
Introduction
It is often the case in the study of the foundations of mathematics that one is led to the classification of
sets according to some property, usually akin to complexity. A prototypical example is the cumulative
hierarchy of set theory, which contains sets of ever-increasing size built up from smaller sets via the
operations of power set and union. In recursion theory, the sets of immediate interest are those generated
effectively, i.e. by the partial recursive functions. These are the recursively enumerable sets (and as a
special case of these, the recursive sets).
These sets can be very limiting, however, and recursion theory—despite its profound importance in
mathematical logic and computer science—may seem to fall short of saying anything about most sets of
natural numbers. Unlike in set theory, the power set operation is of no help to us now, since we are not
concerned with sets of sets. Instead, we can choose to widen our scope through the use of oracles. This
leads to the categorization of sets according to their “degree of unsolvability”.
In the following paper, we follow a different, but equally important route, which generalizes from
the way in which recursively enumerable sets are related to recursive ones. A simple but important fact
in recursion theory is that the recursively enumerable sets are exactly the projections of the recursive
relations. It is also a fact that the complement of any non-recursive recursively enumerable set is not
recursively enumerable. Generalizing from these ideas, we shall build sets of increasing complexity
from simpler ones by the applications of projection and complementation. Since complementation is
self-cancelling and consecutive projections can be effectively combined, we shall really be studying sets
defined via strictly alternating applications of these two operations . This approach is of a descriptional
nature: we will stratify sets according to the complexity of their description. Nevertheless, we shall see
that it is tightly connected to the oracle method of set classification discussed in the previous paragraph.
2
Relativized Recursion Theory
The concept of reducibility allows us to talk about the degree of unsolvability of a set. There are various
ways to formalize the idea of reducibility, some stronger than others. Perhaps the most famous of these is
1
Turing reducibility, which makes use of oracles. Given a function ψ, known as an oracle, we define the
class of ψ-partial recursive functions to be the smallest class of functions satisfying the regular axioms
defining the partial recursive functions but with the addition of an axiom allowing ψ to be an initial
function. Any member ϕ of this class is then said to be Turing reducible to or (partial) recursive in ψ.
Reducibility notions for sets are defined below. The characteristic function for any set S will henceforth
be denoted χS .
Definition. Let A and B be sets. We say that A is Turing reducible to or recursive in B or that A is
B-recursive (A ≤T B) if χA is χB -recursive. If A ≤T B and B ≤T A, then we say that A and B are
Turing equivalent and write A ≡T B.
Most important properties of the partial recursive functions are retained (in relativized form) by the ψpartial recursive functions. For instance, all ψ-partial recursive functions can be effectively enumerated.
That is, not only are there countably many such functions, but it is possible to associate each with a
number (called the index) in an effective manner. From now on, we fix such an enumeration and let
ϕψ
e (or just ϕe when ψ is partial recursive) denote the e-th ψ-partial recursive function in our list. An
important use of this numbering is in the relativized Kleene normal form theorem, which we state below.
Theorem 2.1. There is a recursive function U and a ψ-recursive relation T ψ such that every ψ-partial
recursive function ϕψ
e is expressible in the form
ψ
ϕψ
e (x) = U(µyT (e, x, y)).
In what follows, we reserve the symbols T and U for the purposes they serve above. Intuitively, a
computation is a finite sequence of transformations of an input into an output, and hence can be encoded
as a number. The relation T ψ in the Kleene normal form essentially asserts that y encodes a computation
of ϕψ
e on the input x. The output of this computation—being part of the information encoded in y—is
then “extracted” by U.
We may also define recursive enumerability in the relativized domain. A set is called ψ-recursively
enumerable or recursively enumerable in ψ if it is the domain of some ψ-partial recursive function.
We denote the domain of the e-th ψ-partial recursive function by Weψ and hence obtain an effective
enumeration of all ψ-recursively enumerable sets. It follows immediately from the relativized normal
form theorem that every ψ-recursively enumerable set also has a “normal form”. We state below the
relativized Kleene normal form theorem for recursively enumerable sets.
Corollary 2.2. Every ψ-recursively enumerable set Weψ has the form
Weψ = {x : ∃yT ψ (e, x, y)}.
We will also make use of the Turing jump of a set. Given a set A, its Turing jump is defined as
(n−1)
= K A = {x : x ∈ WxA }. We also define A(n) = K A
, where A(0) = A, to be the n-th Turing
jump of A. By a generalization of the Halting problem (which uses a similar diagonalization proof), A(n)
A0
2
is recursively enumerable in A(n−1) but not the converse. Of particular interest is the n-th “zero jump”
∅(n) , i.e. the n-th Turing jump of the empty set. Note that ∅0 = K since ∅ is recursive.
Most of the important properties of recursively enumerable sets also hold in relativized form, but we
shall only state formally those two properties that we used as motivation for studying the hierarchy in
the introduction. In what follows, we use the standard notation h·i to denote a recursive encoding (with
recursive inverse) of an arbitrary tuple of natural numbers by a single natural number. The decoding
operation will be denoted by (hx1 , . . . xn i)i = xi .
Proposition 2.3.
1. A set is ψ-recursive if and only if it and its complement are ψ-recursively enumerable.
2. A set A is ψ-recursively enumerable if and only if it is the projection of a ψ-recursive relation R,
i.e. if and only if A = {x : ∃yR(x, y)}.
Note. There is no loss of generality above in considering R a binary relation. In recursion theory, it is
common to see sets of tuples as sets of encodings of tuples, and this is the practice we follow here. Any
n-ary ψ-recursive relation R can be seen as an m-ary ψ-recursive relation (m < n) by interpreting every
hx1 , . . . , xn i as, for instance, hhx1 , . . . , xn−m+1 i, xn−m+2 , . . . , xn i.
In certain cases, Turing reducibility is too weak in the sense that it does not discriminate enough
between different sets. For example, given a set A, the equivalence class of all sets Turing-equivalent
to A is known as the Turing degree of A and is denoted dT (A). It may happen that we wish to show
that certain sets in dT (A) have in some sense a more complicated effective content than others, but it is
obvious that this cannot be done via ≤T . One popular alternative to Turing reducibility is m-reducibility.
Another option is given below.
Definition. Let A and B be sets. A is one-one reducible or 1-reducible to B (A ≤1 B) if there is a
one-to-one function f such that x ∈ A if and only if f (x) ∈ B. In this case, we write f : A ≤1 B. We
say that A and B are 1-equivalent (A ≡1 B) if A ≤1 B and B ≤1 A.
It is not hard to see that A ≤1 B ⇒ A ≤m B ⇒ A ≤T B.
Like ≤m , the relation ≤1 is reflexive and transitive. Thus, ≡1 is an equivalence relation, and can in
fact be seen as the natural relation expressing “recursive isomorphism” between recursively enumerable
sets. In fact, this is precisely the content of Myhill’s isomorphism theorem. That is, suppose we define A
and B to be recursively isomorphic (denoted A ≡ B) if there exists a recursive bijection between them.
Then A ≡ B ⇔ A ≡1 B. We shall not prove this here—we mention it only to provide some insight into
the nature of 1-equivalence1 .
Completeness (of recursively enumerable sets) with respect to ≤1 is defined similar as with ≤m . That
is, a recursively enumerable set A is 1-complete if every recursively enumerable set B is 1-reducible to
A. The following gives two interesting characterizations of ≤1 .
1
It is also interesting to note the resemblance between Myhill’s isomorphism theorem and the Schroeder-Bernstein theorem.
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Proposition 2.4. Let A and B be sets.
1. B ≤1 A0 if and only if B is recursively enumerable in A.
2. B 0 ≤1 A0 if and only if B ≤T A.
Proof.
1. This is just the statement of the 1-completeness of the halting set relativized to A.
2. For one direction,
B ≤T A ⇒ B 0 is recursively enumerable in A
⇔ B 0 ≤1 A0 .
(by (1))
Conversely,
B 0 ≤1 A0 ⇒ B, B̄ ≤1 A0
(since B, B̄ ≤1 B 0 )
⇔ B, B̄ are recursively enumerable in A
⇔ B ≤T A.
3
The Arithmetical Hierarchy
As mentioned in the introduction, the arithmetical hierarchy will consist of sets built from simpler sets
by the strictly alternating applications of projection and negation. A more attractive way of presenting
the hierarchy, however, is by alternating applications of existential and universal quantification. That is,
we replace negation by universal quantification (projection is just existential quantification). We may do
this because of the equivalence
∀xφ ←→ ¬∃x¬φ.
In the following definition, we use Q as a notational convenience to denote either ∃ or ∀, whichever is
appropriate given the (strictly alternating) context.
Definition. Let Σ0 = Π0 = ∆0 be the class of recursive sets. For any n ≥ 1, we define Σn to be the
class of sets definable from a recursive relation via n strictly alternating applications of existential and
universal quantification, starting with existential quantification. That is, A ∈ Σn if and only if there
exists R recursive such that
A = {x : ∃y1 ∀y2 . . . Qyn R(x, y1 , . . . , yn )}.
We define Πn similarly, except that we start with universal quantification. Finally, we let ∆n = Σn ∩ Πn .
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Note. As mentioned in the introduction, there is no loss of generality in our use of strictly alternating
quantifiers. This follows from what we may call the quantifier contraction principle: several consecutive
quantifiers of the same type may be effectively combined into a single one. For example, given a recursive
relation R,
∃y1 ∃y2 . . . ∃yn ∀z1 . . . ∀zm R(y1 , . . . , yn , z1 , . . . , zm ) ←→ ∃y∀zR0 (y, z),
where y = hy1 , . . . , yn i, z = hz1 , . . . , zm i and R0 = {hy, zi : R((y)1 , . . . , (y)n , (z)1 , . . . , (z)m )} is
recursive.
A set is said to be arithmetical if it is in the arithmetical hierarchy, i.e. in one of the classes defined
above. We may also refer to the logical expressions defining sets in the hierarchy as having a certain level
(for example, ∃yR(x, y) is Σ1 whenever R is recursive). It is immediate by Proposition 2.3(2), that Σ1
is the class of all recursively enumerable sets. Moreover, by Proposition 2.3(1) and (1) of the following
lemma, ∆1 = ∆0 .
Lemma 3.1.
1. The Σn and Πn are complementary, i.e. Πn consists precisely of the sets whose complements are
in Πn .
2. The hierarchy is cumulative, in the sense that Σn ∪ Πn ⊆ ∆m for m > n.
3. Each class in the hierarchy is closed under union and intersection (i.e. if A, B ∈ Σn , then A∪B ∈
Σn ).
4. The Σn and Πn are closed under existential and universal quantification, respectively, e.g. if
R ∈ Σn , then {x : ∃yR(x, y)} ∈ Σn .
5. The classes of the hierarchy are closed under m-reducibility, e.g. if B ≤m A and A ∈ Σn , then
B ∈ Σn .
6. The classes of the hierarchy are closed under bounded (existential and universal) quantification,
e.g. if R ∈ Σn , then {hx, yi : ∀z < yR(x, y, z)} ∈ Σn .
Proof. Note that the statements above that hold for all classes of the hierarchy (that is, (3), (5), and (6))
need only be proved to hold for the Σn and the Πn (the result for the ∆n then follows immediately).
1. Let A = {x : ∃y1 ∀y2 . . . Qyn R(x, y1 , . . . , yn )} ∈ Σn for some recursive relation R. By negating
quantifiers, this is equivalent to saying that Ā = {x : ∀y1 ∃y2 . . . Qyn R̄(x, y1 , . . . , yn )}. Since the
complement of a recursive relation is still recursive, Ā ∈ Πn .
2. We prove this statement by adding unnecessary quantifiers. Let A ∈ Σn be as above and let
m > n. Then
A = {x : ∃y1 ∀y2 . . . Qym R0 (x, y1 , . . . ym )}
= {x : ∀y0 ∃y1 . . . Qym−1 R0 (x, y1 , . . . , ym−1 , y0 )},
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where R0 = {hx, y1 , . . . , ym i : R(x, y1 , . . . , yn )} is clearly recursive. The first equality shows
that A ∈ Σm and the second that A ∈ Πm . Hence, Σn ⊆ Σm ∩ Πm . The proof for Πn is similar.
3. Let A ∈ Σn be as above and let B = {x : ∃z1 ∀zn . . . Qzn S(x, z1 , . . . , zn )} ∈ Σn . Then
A ∪ B = {x : (∃y1 . . . Qyn R(x, y1 , . . . , yn )) ∨ (∃z1 . . . Qzn S(x, z1 , . . . , zn )}
= {x : ∃y1 ∃z1 . . . Qyn Qzn (R(x, y1 , . . . , yn ) ∨ S(x, z1 , . . . , zn ))}
= {x : ∃u1 . . . Qun T (x, u1 , . . . , un )}
where ui = hyi , zi i and T = {hx, u1 , . . . , un i : R(x, (u1 )1 , . . . , (un )1 ) ∨ S(x, (u1 )2 , . . . , (un )2 )}
is clearly recursive. Hence, A ∪ B ∈ Σn . The proof for A ∩ B is similar.
4. This follows immediately by quantifier contraction.
5. Let A ∈ Σn be as above. Let B be a set and let f : B ≤m A. So x ∈ B if and only if f (x) ∈ A.
Then B = {x : ∃y1 . . . Qyn R0 (x, y1 , . . . , yn )} ∈ Σn since the relation R0 = {hx, y1 , . . . , yn i :
R(f (x), y1 , . . . , yn )} is recursive.
6. We shall prove that the Σn are closed under bounded quantification (the proof for the Πn will
follow directly by complementation). We proceed by induction on n (n = 0 is trivial). The case
of bounded existential quantification follows directly from (4), so we consider bounded universal
quantification. Let R ∈ Σn , so that
R(x, y, z) ⇔ ∃u1 ∀u2 . . . Qun R0 (hx, y, zi, u1 , . . . , un ),
(R0 recursive)
⇔ ∃u1 S(x, y, z, u1 ),
where S = {hx, y, z, ui : ∀u2 ∃u3 . . . Qun R0 (hx, y, zi, u, u2 , . . . , un )} ∈ Πn−1 . But then
{hx, yi : ∀z < yR(x, y, z)} = {hx, yi : ∀z < y∃uS(x, y, z, u)}
= {hx, yi : ∃σS 0 (x, y, z, σ)},
where S 0 = {hx, y, z, σi : ∀z < yS(x, y, z, (σ)z )}, and σ is an encoding of the u corresponding
to each z < y such that S(x, y, z, u) holds. But by induction, S 0 ∈ Πn−1 , so it follows that
{hx, yi : ∀z < yR(x, y, z)} ∈ Σn .
It is generally quite easy to upper bound arithmetical sets A within the hierarchy. For example,
consider the set
Cof = {x : Wx is finite}.
By Rice’s theorem, Cof is non-recursive, but this doesn’t tell us very much. However, it is easy to see
that
x ∈ Cof ⇔ ∃y1 ∀y2 (y2 ≤ y1 ∨ y2 ∈ Wx ).
6
Since the unquantified part of this expression is the disjunction of a Σ0 and a Σ1 expression, it follows
from the cumulative property of the hierarchy and closure under union of Σ1 (hence, closure under
conjunction of Σ1 expressions) that Cof ∈ Σ3 . Note, however, the possibility that Cof is in a lower level
of the hierarchy than Σ3 . A good way to establish a lower bound is by showing completeness of a set for
some level of the hierarchy.
Definition. A set A is said to be Σn -complete (respectively, Πn -complete) if every set in Σn (respectively, Πn ) is 1-reducible to A.
The parallels between the Σn and recursive enumerability (as in Proposition 2.3(2)) and the equivalence between recursive enumerability and 1-reducibility (as in Proposition 2.4(1)) can be thought of as
motivation for our use of 1-reducibility rather than m-reducibility above.
The following fundamental theorem vindicates this definition. Post’s theorem characterizes the arithmetical hierarchy recursion-theoretically. That is, it shows how the arithmetical approach to set classification can be translated into the language of effective computation and vice-versa. Here, the manner in
which the Σn and ∆n generalize the class of recursively enumerable and recursive sets (respectively) is
made explicit ((1) and (3) generalize Proposition 2.3). All the results we state refer to the Σn . Complementary results for the Πn are immediate.
Theorem 3.2 (Post’s theorem).
1. A set B is in Σn+1 if and only if B is recursively enumerable in some set in Πn (equivalently, some
set in Σn ).
2. The n-th zero jump ∅(n) is Σn -complete for n ≥ 1.
3. A set B is in ∆n+1 if and only if B is recursive in ∅(n) .
Proof. To prove this theorem, we will break up parts (1) and (2) into two parts each, which we shall call
(1A), (1B) and (2A), (2B), respectively. We will prove (1A), (2A), (1B), and (2B) in that order and then
finish with (3). A diagram of logical dependencies amongst these parts is shown below2 .
/ (3)
<
(1A)
"
/ (1B)
(2A)
/ (2B)
(1A) Let B ∈ Σn+1 . Equivalently, B is the projection of a Πn relation. So B is (trivially) the projection of a Πn -recursive relation and therefore is recursively enumerable in a set within Πn by Proposition
2.3(2). The reason this is equivalent to B being recursively enumerable in a Σn set is that any set is
Turing-equivalent to its own complement.
2
In [3], parts (1)–(3) are proved in sequence, but using some additional notions that we have not introduced and that we
avoid here.
7
(2A) We proceed by induction. The base case n = 1 is simply the halting problem, so assume that
for some n ≥ 1, ∅(n) is Σn -complete. It follows that
B ∈ Σn+1 ⇒ B is recursively enumerable in some Σn set
(by (1A))
⇔ B is recursively enumerable in ∅(n)
(n+1)
⇔ B ≤1 ∅
(by induction)
.
(by Proposition 2.4(1))
This does not yet show that ∅(n) is Σn -complete; we still need to show that ∅(n) ∈ Σn for all n ≥ 1.
This will be done below.
(1B) Notice that if A is a set and C ∈ Πn , then
A ≤T C ⇒ A ≤T ∅(n)
(by (2A)
⇔ A ≤T ∅(n)
⇔ A0 ≤1 ∅(n+1)
(by Proposition 2.4(2))
⇒ A0 ∈ Σn+1
(by closure under ≤m )
⇒ A ∈ Σn+1 .
Now let B be recursively enumerable in some C ∈ Πn . Equivalently, B is the projection of a relation
A ≤T C. So then as we’ve just shown, B is the projection of a Σn+1 relation. Hence, by closure of
Σn+1 under projection, B ∈ Σn+1 .
(n)
(2B) We continue the induction proof from earlier on. Since, ∅(n+1) = {x : ∃yT ∅ (x, x, y)}, and
(n)
since, by induction, T ∅ is a ∅(n) -recursive relation, we have that ∅(n+1) is recursively enumerable in
∅(n) . It follows from (1B) that ∅(n+1) ∈ Σn+1 and so ∅(n+1) is Σn+1 -complete.
(3) Finally, we prove (3) with the following:
B ≤T ∅(n) ⇔ B, B̄ are recursively enumerable in ∅(n)
⇔ B, B̄ ∈ Σn+1
(by (1))
⇔ B ∈ ∆n+1 .
A natural question now arises: are the sets in Σn+1 but not in Σn ∪ Πn precisely those that are equal
to the n-th zero degree (the Turing degree of ∅(n) )? In fact, this is quite clearly a generalization of Post’s
problem, which is the case n = 0, and so we can expect it to be very difficult.
An easy corollary of Post’s theorem is that the inclusion relationships in the arithmetical hierarchy
are all strict.
Corollary 3.3 (Hierarchy Theorem). For all n ≥ 1, ∆n ⊂ Σn and ∆n ⊂ Πn .
Proof. We claim that ∅(n) ∈ Σn \ ∆n . That ∅(n) ∈ Σn follows from (2) of Post’s theorem. But by (3), if
n ≥ 1, then ∅(n) cannot be in ∆n , for then it would be Turing-reducible to ∅(n−1) , and we know this to
be false by the relativized Halting problem. Similarly, ∅(n) ∈ Πn \ ∆n .
8
The following diagram depicts the structure we have uncovered.
Σ0
∆0
= Σ2
!
∆1
Π0
4
= Σ1
!
!
= ∆2
!
Π1
= ∆3
Π2
Complete Sets
In this section, we give some examples of sets complete for certain levels of the arithmetical hierarchy. A
natural class of non-recursive sets is the class of non-trivial index sets. Given a class C of partial recursive
functions, the index set of C is the set {x : ϕx ∈ C} of indices of functions in C. Indeed, our first two
examples shall be Fin = {x : Wx is finite} and Tot = {x : ϕx is total}.
It is clear that x ∈ Fin ⇔ ∃y∀z(z ≤ y ∨ z ∈
/ Wx ). Since z ≤ y is Σ0 (recursive) and z ∈
/ Wx is Π1 ,
we get Fin ∈ Σ2 by quantifier contraction. Next, since x ∈ Tot ⇔ ∀y(y ∈ Wx ) and since y ∈ Wx is
a recursively enumerable relation3 , Tot ∈ Π2 . We show that each of these sets is also complete for the
level we have placed them in.
Proposition 4.1. Fin is Σ2 -complete and Tot is Π2 -complete.
Proof. We shall use the same function to reduce Σ2 sets to Fin and Π2 sets to Tot. Let A ∈ Σ2
(equivalently, Ā ∈ Π2 ), so that A = {x : ∃y∀zR(x, y, z)} for some recursive relation R. By the s-m-n
theorem, there is a one-to-one recursive function f such that4
(
0
∀y ≤ u∃zR(x, y, z)
ϕf (x) (u) =
.
undefined otherwise
It can now be seen that
x ∈ A ⇒ Wf (x) is finite
⇔ f (x) ∈ Fin .
and that
x ∈ Ā ⇒ Wf (x) = N
⇔ f (x) ∈ Tot
3
The set {hx, yi : y ∈ Wx } is equivalent (under ≤1 ) to the Halting set {y : y ∈ Wy }. It is obvious that the latter
is reducible to the former. The converse follows from completeness of the Halting problem and the fact that the first set is
recursively enumerable.
4
We are actually using the one-to-one version of the s-m-n theorem, in which the function sm
n is one-to-one. This theorem
is not much harder to prove than the ordinary s-m-n theorem and simply takes advantages of the padding lemma, which states
that any given partial recursive function has infinitely many indices.
9
But since Fin ∩ Tot = ∅, we get x ∈ A ⇔ f (x) ∈ Fin and x ∈ Ā ⇔ f (x) ∈ Tot.
It is interesting to note the complementarity between the set Fin of indices of “minimal” (i.e. finite)
recursively enumerable sets and the set Tot of indices of “maximal” recursively enumerable sets (those
equal to N). It is also immediate from the above that Inf = Fin is also Π2 -complete, hence at the same
level of complexity as Tot.
Some sets are harder to lower bound. For example, consider the index set of the recursive sets and
the index set of sets equivalent to the Halting problem:
Rec = {x : Wx is recursive}
Halt = {x : Wx ≡T K}.
We can write
Wx is recursive ⇔ ∃y1 Wx = Wy1
⇔ ∃y1 ∀y2 ((y2 ∈ Wx ∨ y2 ∈ Wy1 ) ∧ ¬(y2 ∈ Wx ∧ y2 ∈ Wy1 )).
The unquantified part of this expression is the conjunction of a Σ1 expression and a Π1 expression. Thus,
it is Π2 . It follows by quantifier contraction that Rec ∈ Σ3 , and so Rec ≤1 ∅(3) .
Now it is shown in [1] that ∅(3) ≤1 Halt, but it should not be surprising that this is very difficult.
For it follows from this that Halt is not 1-reducible to Rec, for then we would have ∅(3) ≤1 ∅(3) , which
we know to be false by closure of Π3 under ≤1 along with the hierarchy theorem. Thus, since ≤1 is a
reflexive relation, we must have Rec 6= Halt. But this solves Post’s problem by showing the existence
of a non-recursive recursively enumerable set of strictly lower degree than K!
We finish this section with a simple way to generate a family of complete sets for the arithmetical
hierarchy. For any A ∈ Πn , we define the weak jump of A to be the set HA = {x : Wx ∩ A 6= ∅}. Since
x ∈ HA ⇔ ∃y(y ∈ Wx ∧ y ∈ A), and since y ∈ Wx is a recursively enumerable relation and y ∈ A is a
Πn relation, we may place HA in Σn+1 .
Proposition 4.2. If A is Πn -complete, then HA is Σn+1 -complete.
Proof. Let C ∈ Σn+1 so that C = {x : ∃yR(x, y)} for some relation R ∈ Πn . Since A is Πn -complete,
there exists a recursive function f reducing R to A so that C = {x : ∃y(f (x, y) ∈ A)}. Now it follows
from the s-m-n theorem that there is a one-to-one recursive function g such that
ϕg(x) (z) = µy(f (x, y) = z).
Thus, we get
x ∈ C ⇔ ∃z∃y(z ∈ A ∧ f (x, y) = z)
⇔ ∃z(z ∈ A ∩ Wg(x) )
⇔ A ∩ Wg(x) 6= ∅
⇔ g(x) ∈ HA .
10
5
Conclusion
We have come to the end of our investigations of the arithmetical hierarchy. We have seen how some
natural generalizations of the properties held by recursive and recursively enumerable sets run in parallel
with the generalizations of partial recursive functions through oracles. As a result, we have uncovered
several ways of studying Turing degrees and understanding recursion theory from a descriptional perspective.
The arithmetical hierarchy has also elucidated some connections between recursion theory and logic,
most notably the effects of logical quantification on effective computability. Post’s theorem made evident
the connection between existential quantification and recursive enumerability. Through the process of
abstraction, we have gained a different—and helpful—perspective of recursion theory.
This is not the end of the story, of course. One can study the arithmetical hierarchy itself in the
relativized domain simply by changing its lowest level to be the class of all A-recursive sets, for some
oracle A. Other hierarchies have also been modelled after the arithmetical hierarchy. The analytical
hierarchy, for instance, extends the arithmetical hierarchy through the use of second-order logic (that is,
it allows quantification over sets) and contains the entire arithmetical hierarchy in its lowest level. On the
opposite end of the spectrum, computational complexity theorists have studied a scaled-down version of
the arithmetical hierarchy, known as the polynomial hierarchy, that classifies recursive sets according to
the time they take to compute on a Turing machine.
The arithmetical hierarchy, however, is of fundamental importance, providing the basis for these
extensions and modifications. The arithmetical hierarchy organizes and sheds light on many ideas from
recursion theory and logic. The elegance of its structure may very well be seen as providing aesthetic
support for the appropriateness of the axioms of recursion theory.
References
[1] Hartley Rogers. Computing degrees of unsolvability. Math. Annalen, 138(2):125–140, 1959.
[2] Hartley Rogers. Theory of Recursive Functions and Effective Computability. ‘McGraw-Hill, Inc.,
1967.
[3] Robert I. Soare. Recursively Enumerable Sets and Degrees. Springer-Verlag, 1987.
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