FOM: 101:Turing Degrees/1

Harvey Friedman friedman at
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

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
 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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