FOM: 98:Model Theoretic Interpretation of Large Cardinals

Harvey Friedman friedman at
Mon Mar 5 15:08:51 EST 2001

This posting is an improvement on posting #89 at the level of large large

We consider linearly ordered relational structures M in a finite relational
type without function symbols. A relation is atomically definable if and
only if it can be defined by an atomic formula where some of the variables
are assigned values from dom(M) that serve as parameters. This is the same
as the notion of definability (with parameters) except that the formula
used must be atomic.

E containedin dom(M)^k is unbounded if and only if there is no strict upper
bound on the coordinates of elements of E.

An initial segment is a set I containedin dom(M) such that for all x < y in
I, we have x in I.

Let f be a function from a subset of dom(M)^k into dom(M). An initial
segment of f is a function g containedin f such that for some initial
segment I, g[I^k] containedin I.

An involution of f is a one-one function h:fld(f) into fld(f) such that for
all x_1,...,x_k,y in fld(f), f(x_1,...,x_k) = y if and only if
f(hx_1,...,hx_k) = hy. An involution is nontrivial if and only if it is not
an identity function.

Let sigma be a finite relational type that includes at least the linear
ordering <= and a ternary relation symbol.

Let *(sigma) be the class of linearly ordered structures M of type sigma
such that the following holds:

###Every definable function whose domain is an unbounded subset of dom(M)^2
has a shortest initial segment with an atomically definable nontrivial

THEOREM 1. *(sigma) is an elementary class.

THEOREM 2. The class of models of ZFC + "there is a proper class of
cardinals kappa for which there is a nontrivial elementary embedding from
V(kappa) into V(kappa)" is interpretable in *(sigma). And *(sigma) is
interpretable in the class of models of VBC + "there is an elementary
embedding of V into a transitive class M with V(delta) containedin M for
some fixed point delta above the critical point". I.e., *(sigma) interprets
and is interpretable in set theory with large large cardinals.

We can also use this for a new kind of independent sentence.

PROPOSITION 3. There is a countable linearly ordered structure in a finite
relational type without function symbols where every definable function
whose domain is an unbounded set of ordered pairs has a shortest initial
segment with an atomically definable nontrivial involution.

THEOREM 4. Proposition 3 is provably equivalent to a Pi-0-1 sentence in
WKL_0. Proposition 3 is provably equivalent to its relativization to
Delta-0-2 sets in ACA_0. Proposition 3 proves the consistency of ZFC +
"there is a proper class of cardinals kappa for which there is a nontrivial
elementary embedding from V(kappa) into V(kappa)". And VBC + "there is an
elementary embedding of V into a transitive class M with V(delta)
containedin M for some fixed point delta above the critical point" proves
the consistency of ZFC + Proposition 3.


This is the 98th 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

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