# 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
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
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
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

```