Download Sets of Reals with an Application to Minimal Covers

A Basis Result for ∑0 3 Sets of Reals with an Application to Minimal Covers
Author(s): Leo A. Harrington and Alexander S. Kechris
Source: Proceedings of the American Mathematical Society, Vol. 53, No. 2 (Dec., 1975), pp. 445448
Published by: American Mathematical Society
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PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 53, Number 2, December 1975
A BASIS RESULT FOR E3 SETS OF REALS
WITH AN APPLICATION TO MINIMALCOVERS
LEO A. HARRINGTON
AND ALEXANDER
S. KECHRIS
ABSTRACT.
It is shown that every Y3 set of reals which contains
reals of arbitrarily
high Turing degree in the hyperarithmetic
hierarchy
contains
reals of every Turing degree above the degree of Kleene's
0.
As an application
it is shown that every Turing degrec above the Turing
degree of Kleene's
0 is a minimal cover.
In this paper we consider a particular verification of the following nebulously stated and tenuously held principle:
Every easily definable set of
reals with enough complicated members contains members from any sufficiently large degree of complexity. Let ( be the Turing degree of Kleene's
(9
Our result is
Theorem.
Any E3 set of reals, with the property that every hyperarithmetic real is recursive in some member of it, contains reals of every Turing
degree above C.
As the first author noticed, an application of this theorem gives a new
proof of Jockusch's result that there is a cone of minimal covers and estimates the base of that cone to be C.
1. Preliminaries.
Let c)
10, 1, 2,...
I be the set of natural numbers
the set of all functions from co to co or (for simplicity) reals,
Letters i, j, k, 1, m, ,.
will denote elements of c and a, 3, y, 5 a, T,...
elements of R. We shall use, without explicit reference, standard facts of
and R =c
recursion theory, which can be found, for example, in [5] or [7].
The basic ingredient in the proof of our main theorem is the use of the
well-known determinacy of closed games in its effective form (see, for example, [4]), which we proceed to state. For a general explanation of the
connection between degrees of unsolvability and determinacy of games, see
Martin [3]. Let (a1, ".o,
ad, where ai 6 o or ai eR be a trivial recursive
coding of n-tuples by reals. For A C R consider the game in which players
Is II alternatively choose natural numbers a(0), 83(0), a(l), /3(1),...
and I
/3)
e
a
e
wins iff (a-,
A. If
R is a strategy for player I, let a * /3 be the
Received
by the editors
September 6, 1974.
AMS (MOS) subject
April
3, 1974 and, in revised
classifications
(1970).
form, July
19, 1974
and
Primary 02F30, 04A15.
Copyright( 1975, AmericanMathematicalSociety
445
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446
L. A. HARRINGTON AND A. S. KECHRIS
result of I's moves when he follows a against II playing /8. More formally
(a * f3)(n) = a(o,(n)). Similarly let a o r be the result of II's moves when he
follows
r against I playing a, so that (a o r)(n)
= r(0(n
+
i)).
Then we
have
Fact (Determinacy of open games). Given a fl set of reals B, either
(I) there is a real a recursive in C such that for all reals /, (o * /3, 8)
e B, or
(II)
there is a hyperarithmetic real
t
such that for all reals a, (a, a
o
r)
v B.
2. Proof of the theorem,
Theorem.
Any `0
metic real is recursive
We are now ready to prove our main result.
set of reals, with the property that every hyperarith.
in some member of it, contains reals of every Turing
degree >0.
Proof.
We first prove this for Ill sets of reals. Let thus A C R be a
set satisfying the hypotheses of the theorem. Let JeVl, where e e w,
y e R, denote the eth function from co into (0) partial recursive in y andl put
fl
B = ((e, r-, y), /6): e e
& y e A & (Vn)(leVP(n) converges
u'
in exactly
?](n) steps) & /3 = (e, r;, y) o IeVl.
Clearly B C R is I1H, so by the fact in ?1 either (I) or (HI)holds. If (II) holds,
let r be a hyperarithmetic real such that for all a C R, (a, a o r) f B. By
assumption, there is a y E A such that t is recursive in y. So for some e C
co, {e'l = r. Let 7(n)- number of steps in which JeV7(n) converges. Let
a (e. 7/, y). Then (a, a o r) ' B. But clearly (a, a o r) C B, a contradiction, so case (I) must hold. Thus there is a recursive in () such that for all
/3, (a *83, /3) C B. Let C be recursive in /8. We shall prove that a real of
the same Turing degree as /3 belongs to 4. Since (o * /, /3) e B, clearly
a */3 -= (e, r, y) for some e C co, r,, y e R such that y E A, Iel' is total and
77(n)= the number of steps in which leV7(n) converges. Also /3= (e,
y)
Thus P is recursive in y. But also a is recursive in /3, thus y is
ole}7.
recursive in /3 i.e., y and /8 have the same Turing degree. Since y C A, we
are done.
The following observation will now complete the proof of the theorem:
For any 10 set of reals B there is a 110 set of reals A such that for
3
~~~~~~~~~~~~~~1
all Turing degrees d, d n B
d nA
0.
To prove this, find a recursive set R such that for all a,
a
t
B 4-*3JiVi3kR(i,
i,
a(k))
and let
A = (i, a, /): V/(/(j)
=LkR(i,
j, a(k)))}.
Q.E.D.
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A BASIS RESULT FOR
0 SETS OF REALS
3
Remarks.
(1) The above theorem is best possible
cannot be improved to include all Hf sets of reals-the
3
A = la: (Ve)(Iel
is total
-*
447
in the sense that it
set
leV,4 F)},
where F is a nonempty n? 1set with no hyperarithmetic member, is a Hf
~~~~~~~~~~~~~~
set of reals which is closed under Turing reducibility and which contains
all hyperarithmetic reals and yet which fails to contain Kleene's 0.
(2) The proof of the theorem can be easily modified to show that
where for a collection of
Determinacy(X0)nn 4: Turing Determinacy(?+l1),
sets of reals F, Determinacy () - VA 6 F (A is determined) and
Turing Determinacy (F) -VA 6 F (A is invariant under Turing equivalence =* A is
determined). This result has been also proved independently (and a little
earlier than us) by Ramez Sami (private communication). In fact the proof
above shows that if every 1? set is determined, then given a
?+ set of
reals A which is cofinal (i.e., for every a 6 R there is /3 e A s.t. a is
recursive in 03), there is a Turing degree d such that A contains reals of
every degree > d. (To see this we note first that A may be assumed to be
fl0 since if a 6 A - 3mA*((m, a)) where A* e 110, clearly for any degree
n
n
A* n d 0. Then we define B exactly as in the proof of
d, A n d 04
the theorem above and argue again that II cannot have a winning strategy
in the game determined by B since B is cofinal. Then I has a winning
strategy, since every z
game is determined, so by the argument given
there, A contains reals of every Turing degree above the degree of a winning
strategy or for player I.) Using this fact, Martin [2] showed that
* Turing Determinacy(A0+2)
which is best possible
analysis, since he also proved that Analysis -*. Turing Determinacy(lo)
Determinacy(X0)
in
(see [2]).
3. An application
to minimal covers.
Using X?-determinacy, Jockusch
[1] has shown that there is a cone of minimal covers. Harrington noticed
that this result follows from our main theorem above (and hence follows from
just open determinacy and so, in particular, is a theorem of analysis).
This
also allows for computing that Kleene's C can be taken as a basis of the
cone.
Here is Harrington's result.
Theorem.
Proof.
Every Turing degree >
(Ccontains
a minimal cover,
By Sacks [6], for every real a, there is a minimal cover of a
which is AO in a, uniformly.
that
VaV/3(Pa,
/3) -
Thus there is a Hfl predicate
(,/3 is a minimal cover of a)
Now A = }(a, /3): P(a,
and
P(a,
Va3!/3P(a,
3) such
83).
/3)1 is clearly a H0l
2 set of reals with the property
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L. A. HARRINGTON AND A. S. KECHRIS
448
that every hyperarithmetic real is recursive in some member of it.
It is also
By the theorem in ?2 every Turing degree
> @ contains a member of AP thus every Turing degree > (9 is a minimal
a collection
of minimal covers-
Q.E.D.
cover.
We conclude with an open problem raised by Jockusch [1]. Is there a
In particular, is @O'in
hyperarithmetic real in a cone of minimal covers?
such a cone?
REFERENCES
C. G. Jockusch,
Jr., An application of : determinacy to the degrees of
J. Symbolic
Logic 38 (1973), 293-294.
2. D. A. Martin, Two theorems on Turing determinacy, Mimeographed
notes,
June 1974.
1.
unsolvability,
, The axiom of determinateness
3.
and reduction principles in the anaMR 37 #2607.
lytical hierarchy, Bull. Amer. Math. Soc. 74 (1968), 687-689.
4. Y. N. Moschovakis,
Elementary induction in abstract structures, North-Ilol-
land,
Amsterdam,
1974.
5. H. Rogers,
Jr., Theory of recursive functions and effective computability,
McGraw-lill,
New York, 1967.
MR 37 #61.
6. G. E. Sacks, Degrees of unsolvability,
Ann. of Math. Studies,
no. 55, Princeton Univ. Press,
N. J., 1963.
MR 32 #4013.
Princeton,
7. J. R. Shoenfield,
Mathematical logic, Addison-Wesley,
Reading,
Mass., 1967.
MR 37 #1224.
DEPARTMENT OF MATHEMATICS, STATE UNIVERSITY OF NEW YORK AT BUFFALO,
AMHERST, NEW YORK 14226
DEPARTMENT OF MATHEMATICS, MASSACHUSETTS INSTITUTE OF TECHNOLOGY,
CAMBRIDGE, MASSACHUSETTS 02139
Current address (Leo A. Harrington):
California,
Berkeley,
California
Current address (Alexander
Institute
of Technology,
Pasadena,
Department
of Mathematics,
University
94720
A. Kechris):
California
Department
of Mathematics,
91125
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California
of