FOM: 101:Turing Degrees/1
Harvey Friedman
friedman at math.ohio-state.edu
Mon Apr 2 03:32:30 EDT 2001
This is the first installment about Turing Degrees planned for FOM.
We have been rethinking the work we did in the 80's on Borel
Diagonalization, which were the first independence results from ZFC + V = L
and various fragments and extensions thereof, of a mathematical nature.
We have discovered that there is a large field of statements about the
Turing degrees which correspond to various levels of set theory, including
large cardinals.
By statements about Turing degrees, we mean just the existence of Turing
degrees, or countable sequences of Turing degrees, obeying some interesting
uniformity conditions. By the well known basis theorems, these Turing
degrees or sequences of Turing degrees can be taken to be recursive in the
hyperjump, and in fact what we call hyperlow - i.e., that the hyperjump is
not hyperarithmetic in them.
The conditions on Turing degrees and sequences of Turing degrees fall
squarely into the standard framework of current recursion theory. Thus we
anticipate that this will be viewed as a lively new direction in the local
theory of Turing degrees, which is, moreover, intimately connected with set
theory, including large cardinals.
So what should this theory be called? I tentatively propose
*uniform degree theory*.
This supports the idea of doing this not only for the Turing degrees but
for other kinds of degrees.
************************
Here we make the common conventions that
i) degree always means Turing degree;
ii) real always means a subset of omega.
The fundamental definition that starts the subject is the following.
Let d be a degree and x be a real. We say that x is uniformly arithmetic in
d if and only if there is an arithmetic formula phi(n,y), with only the
free variables shown, such that for ALL y in d,
x = {n: phi(n,y)}.
I.e, x is not only arithmetic in every real of degree d, but x is uniformly
arithmetic in every real of degree d.
Note that under usual terminology, x is arithmetic in d if and only if
there is an arithmetic formula phi(n,y), with only the free variables
shown, such that for SOME y in d,
x = {n: phi(n,y)}.
We also stratify this definition as follows. x is uniformly Sigma-0-k
(Pi-0-k) in d if and only if there is a Sigma-0-k (Pi-0-k) formula
phi(n,y), with only the free variables shown, such that for all y in d,
x = {n: phi(n,y)}.
And the same definition of uniformity can be given with respect to
practically any notion of reducibility. I.e., take a notion of reducibility
to be a countable collection of partial functions from the power set of
omega into itself.
We say that x is uniformly Delta-0-k in d if and only if it and its
complement are uniformly Sigma-0-k in d. We say that x is uniformly
recursive in d if and only if x is uniformly Delta-0-1 in d.
Obviously x is uniformly Delta-0-k in d if and only if x is uniformly
recursive in the k-th jump of d.
THEOREM 1. Let alpha be a recursive ordinal. There is a real in 0^(alpha)
which is uniformly arithmetic in 0^(alpha). As a consequence, every real
arithmetic in 0^(alpha) is uniformly arithmetic in 0^(alpha). In fact,
there is a real in 0^(alpha) which is uniformly Delta-0-3 in 0^(alpha). For
all 3 <= k <= infinity, every real Sigma-0-k in 0^(alpha) is uniformly
Sigma-0-k in 0^(alpha).
How many degrees have uniformly arithmetic elements (i.e., an element of
the degree which is uniformly arithmetic in the degree)?
THEOREM 2. There are continuumly many degrees with a uniformly arithmetic
element.
THEOREM 3. There are arbitrarily high degrees without a uniformly
arithmetic element.
THEOREM 4. There is a cone of degrees without uniformly arithmetic
elements. This can be proved in Zermelo set theory but not in Zermelo set
theory with bounded separation.
We now considerably strengthen the property "no uniformly arithmetic
element" by a positive property.
THEOREM 5. There is a degree d such that every real uniformly arithmetic in
d is recursive in d. This is not provable in second order arithmetic. It is
provable if a satisfaction predicate is added and induction and
comprehension are extended to all formulas in the expanded language. It is
provable in second order arithmetic if "arithmetic" is replaced by
"Sigma-0-k" for any particular k.
THEOREM 6. There is a cone of degrees d such that every real uniformly
arithmetic in d is recursive in d. This can be proved in Zermelo set theory
but not in Zermelo set theory with bounded separation.
The most sensible way to study the reverse mathematics of uniform degree
theory is to use the base theory ACA0' = ACA0 + "for all x containedin
omega and n >= 1, the n-th Turing jump of x exists." This is equivalent,
over RCA0, to the usual classical infinite Ramsey theorem. It would be
interesting to develop this new reverse mathematics area.
******************************
This is the 101st in a series of self contained postings to FOM covering
a wide range of topics in f.o.m. Previous ones are:
1:Foundational Completeness 11/3/97, 10:13AM, 10:26AM.
2:Axioms 11/6/97.
3:Simplicity 11/14/97 10:10AM.
4:Simplicity 11/14/97 4:25PM
5:Constructions 11/15/97 5:24PM
6:Undefinability/Nonstandard Models 11/16/97 12:04AM
7.Undefinability/Nonstandard Models 11/17/97 12:31AM
8.Schemes 11/17/97 12:30AM
9:Nonstandard Arithmetic 11/18/97 11:53AM
10:Pathology 12/8/97 12:37AM
11:F.O.M. & Math Logic 12/14/97 5:47AM
12:Finite trees/large cardinals 3/11/98 11:36AM
13:Min recursion/Provably recursive functions 3/20/98 4:45AM
14:New characterizations of the provable ordinals 4/8/98 2:09AM
14':Errata 4/8/98 9:48AM
15:Structural Independence results and provable ordinals 4/16/98
10:53PM
16:Logical Equations, etc. 4/17/98 1:25PM
16':Errata 4/28/98 10:28AM
17:Very Strong Borel statements 4/26/98 8:06PM
18:Binary Functions and Large Cardinals 4/30/98 12:03PM
19:Long Sequences 7/31/98 9:42AM
20:Proof Theoretic Degrees 8/2/98 9:37PM
21:Long Sequences/Update 10/13/98 3:18AM
22:Finite Trees/Impredicativity 10/20/98 10:13AM
23:Q-Systems and Proof Theoretic Ordinals 11/6/98 3:01AM
24:Predicatively Unfeasible Integers 11/10/98 10:44PM
25:Long Walks 11/16/98 7:05AM
26:Optimized functions/Large Cardinals 1/13/99 12:53PM
27:Finite Trees/Impredicativity:Sketches 1/13/99 12:54PM
28:Optimized Functions/Large Cardinals:more 1/27/99 4:37AM
28':Restatement 1/28/99 5:49AM
29:Large Cardinals/where are we? I 2/22/99 6:11AM
30:Large Cardinals/where are we? II 2/23/99 6:15AM
31:First Free Sets/Large Cardinals 2/27/99 1:43AM
32:Greedy Constructions/Large Cardinals 3/2/99 11:21PM
33:A Variant 3/4/99 1:52PM
34:Walks in N^k 3/7/99 1:43PM
35:Special AE Sentences 3/18/99 4:56AM
35':Restatement 3/21/99 2:20PM
36:Adjacent Ramsey Theory 3/23/99 1:00AM
37:Adjacent Ramsey Theory/more 5:45AM 3/25/99
38:Existential Properties of Numerical Functions 3/26/99 2:21PM
39:Large Cardinals/synthesis 4/7/99 11:43AM
40:Enormous Integers in Algebraic Geometry 5/17/99 11:07AM
41:Strong Philosophical Indiscernibles
42:Mythical Trees 5/25/99 5:11PM
43:More Enormous Integers/AlgGeom 5/25/99 6:00PM
44:Indiscernible Primes 5/27/99 12:53 PM
45:Result #1/Program A 7/14/99 11:07AM
46:Tamism 7/14/99 11:25AM
47:Subalgebras/Reverse Math 7/14/99 11:36AM
48:Continuous Embeddings/Reverse Mathematics 7/15/99 12:24PM
49:Ulm Theory/Reverse Mathematics 7/17/99 3:21PM
50:Enormous Integers/Number Theory 7/17/99 11:39PN
51:Enormous Integers/Plane Geometry 7/18/99 3:16PM
52:Cardinals and Cones 7/18/99 3:33PM
53:Free Sets/Reverse Math 7/19/99 2:11PM
54:Recursion Theory/Dynamics 7/22/99 9:28PM
55:Term Rewriting/Proof Theory 8/27/99 3:00PM
56:Consistency of Algebra/Geometry 8/27/99 3:01PM
57:Fixpoints/Summation/Large Cardinals 9/10/99 3:47AM
57':Restatement 9/11/99 7:06AM
58:Program A/Conjectures 9/12/99 1:03AM
59:Restricted summation:Pi-0-1 sentences 9/17/99 10:41AM
60:Program A/Results 9/17/99 1:32PM
61:Finitist proofs of conservation 9/29/99 11:52AM
62:Approximate fixed points revisited 10/11/99 1:35AM
63:Disjoint Covers/Large Cardinals 10/11/99 1:36AM
64:Finite Posets/Large Cardinals 10/11/99 1:37AM
65:Simplicity of Axioms/Conjectures 10/19/99 9:54AM
66:PA/an approach 10/21/99 8:02PM
67:Nested Min Recursion/Large Cardinals 10/25/99 8:00AM
68:Bad to Worse/Conjectures 10/28/99 10:00PM
69:Baby Real Analysis 11/1/99 6:59AM
70:Efficient Formulas and Schemes 11/1/99 1:46PM
71:Ackerman/Algebraic Geometry/1 12/10/99 1:52PM
72:New finite forms/large cardinals 12/12/99 6:11AM
73:Hilbert's program wide open? 12/20/99 8:28PM
74:Reverse arithmetic beginnings 12/22/99 8:33AM
75:Finite Reverse Mathematics 12/28/99 1:21PM
76: Finite set theories 12/28/99 1:28PM
77:Missing axiom/atonement 1/4/00 3:51PM
78:Qadratic Axioms/Literature Conjectures 1/7/00 11:51AM
79:Axioms for geometry 1/10/00 12:08PM
80.Boolean Relation Theory 3/10/00 9:41AM
81:Finite Distribution 3/13/00 1:44AM
82:Simplified Boolean Relation Theory 3/15/00 9:23AM
83:Tame Boolean Relation Theory 3/20/00 2:19AM
84:BRT/First Major Classification 3/27/00 4:04AM
85:General Framework/BRT 3/29/00 12:58AM
86:Invariant Subspace Problem/fA not= U 3/29/00 9:37AM
87:Programs in Naturalism 5/15/00 2:57AM
88:Boolean Relation Theory 6/8/00 10:40AM
89:Model Theoretic Interpretations of Set Theory 6/14/00 10:28AM
90:Two Universes 6/23/00 1:34PM
91:Counting Theorems 6/24/00 8:22PM
92:Thin Set Theorem 6/25/00 5:42AM
93:Orderings on Formulas 9/18/00 3:46AM
94:Relative Completeness 9/19/00 4:20AM
95:Boolean Relation Theory III 12/19/00 7:29PM
96:Comments on BRT 12/20/00 9:20AM
97.Classification of Set Theories 12/22/00 7:55AM
98:Model Theoretic Interpretation of Large Cardinals 3/5/01 3:08PM
99:Boolean Relation Theory IV 3/8/01 6:08PM
100:Boolean Relation Theory IV corrected 11:29AM 3/21/01
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