FOM: 93:Orderings on Formulas
Harvey Friedman
friedman at math.ohio-state.edu
Mon Sep 18 03:46:22 EDT 2000
We have discovered some simple recursively enumerable well founded
orderings whose ordinals are the provable ordinals of set theories. This is
a topic I have written about several times earlier, but this appears to be
the simplest yet. See #14, #14', #15, #16, #16',#23, #67.
1. Orderings on formulas.
Let < be a well founded transitive relation. We define the usual mapping
from points to ordinals. The least ordinal greater than all values is
written ord(<).
We look at formulas phi in the language L(<,R) of first order predicate
calculus with equality, with the binary function symbol R and the special
binary relation symbol <. When we write (A,<,R), it is understood that A is
a nonempty set, F:A^2 into A, and < is a strict linear ordering on A.
Let K be a class of structures (A,<,R). We define phi<_K psi if and only if
*phi,psi are formulas L(<,F) with at most the free variable x, such that in
every (A,<,F) lying in K, there exists a solution to phi which is greater
than all solutions to psi.*
We say that (A,<,R) is logically rich if and only if every definable
bounded subset of A is a cross section of R. Here we take bounded to mean
contained in some (-infinity,x). By a limit point we will mean a limit from
the left. We let Q be the rationals with its usual ordering.
THEOREM 1.1. Let K be the class of all logically rich (Q,<,R). Then
ord(<_K) is strictly between the provable ordinal of Z_2 and the provable
ordinal of the extension of Z_2 by a truth predicate with full induction.
<_K is a well founded r.e. ordering. We can alternatively take K to be the
class of all logically rich (A,<,R) with a limit point.
THEOREM 1.2. Let K be the class of all logically rich (A,<,R) of
cardinality beth_omega with Q as an initial segment. Then ord(<_K) is
strictly between the provable ordinal of the theory of types and the
provable ordinal of Z. <_K is a well founded r.e. ordering. We can
alternatively take K to be the class of all logically rich (A,<,R) of
cardinality >= beth_omega with a limit point with countably many
predecessors.
Let MAH = ZFC + {there exists an n-Mahlo cardinal}_n. Let MAH+ = ZFC + "for
all n there exists an n-Mahlo cardinal".
Let SUB = ZFC + {there exists an n-Mahlo cardinal}_n. Let SUB+ = ZFC + "for
all n there exists an n-Mahlo cardinal".
Fix kappa to be the limit over n of the first n-Mahlo cardinal. Fix lambda
to the limit over n of the first n-subtle cardinal.
THEOREM 1.3. Let K be the class of all logically rich (A,<,R), where A is a
dense kappa-like ordering. Then ord(<_K) is strictly between the provable
ordinal of MAH and the provable ordinal of MAH+. <_K is a well founded r.e.
ordering. We can alternatively take K to be the class of all logically rich
(A,<,R), where A is kappa-like and has a limit point.
THEOREM 1.4. Let K be the class of all logically rich (A,<,R), where A is
kappa-like and has a closed unbounded subset. Then ord(<_K) is strictly
between the provable ordinal of SUB and the provable ordinal of SUB+. <_K
is a well founded r.e. ordering. We can alternatively take K to be the
class of logically rich (A,<,R), where A is kappa-like, has a closed
unbounded subset, and is dense.
The constructions can be stratified in obvious ways in order to hit the
provable ordinals of the relevant theories right on the button.
****************************
This is the 93rd 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
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