# FOM: 89:Model Theoretic Interpretations of Set Theory

Harvey Friedman friedman at math.ohio-state.edu
Wed Jun 14 10:28:40 EDT 2000

```TYPOS IN POSTING #88, Boolean Relation Theory  6/8/00  10:40AM.

V_6. All multivariate piecewise polynomials f from N into N over Z such
that for all x in dom(f), f(x) <= 2|x|.

should be

V_6. All multivariate piecewise polynomials f from N into N over Z such
that for all x in dom(f), f(x) >= 2|x|.

For some reason, in the FOM archives, at several places the letter

S

appears where there should be

...

This didn't happen in the version I was sent as an FOM subscriber.

WARNING: Posting #80 is also titled Boolean Relation Theory. I meant to
title posting #88 Boolean Relation Theory 2. The next numbered posting on
BRT will be titled Boolean Relation Theory 3.

############################

Let Q be the rationals with the usual ordering. Let R containedin Q^2. The
cross sections of R are the sets R_x = {y: R(x,y)}. We say that A
containedin Q is upper bounded if and only if there exists x in Q such that
for all y in A, y <= x.

A special Q system is a quadruple (Q,<,R,j) such that

i) R containedin Q^2;
ii) every upper bounded set of rationals definable in (Q,<,R,j) is a cross
section of R;
iii) j is an elementary embedding from (Q,<,R) into (Q,<,R) which is the
identity below 0 and maps each nonnegative integer to its successor.

THEOREM 1. The existence of a special system is provable in ZF + "there
exists a nontrivial elementary embedding from a rank into itself" but not
in ZF + (for all n)(there exists an n-huge cardinal).

A weak special Q system is a quadruple (Q,<,R,j) such that

i) R containedin Q^2;
ii) every set of rationals definable in any upper bounded restriction of
(Q,<,R,j) is a cross section of R;
iii) j is an elementary embedding from (Q,<,R) into (Q,<,R) which is the
identity below 0 and maps each nonnegative integer to its successor.

THEOREM 2. The existence of a weak special system is provably equivalent,
over RCA_0, to light faced WKL_0 plus the consistency of ZF + {there exists
an n-huge cardinal}_n.

Now that we have stated the main point of this posting, let us retreat a
bit, and drop the elementary embedding.

A Q system is a triple (Q,<,R) such that

i) R containedin Q^2;
ii) every bounded above set of rationals definable in (Q,<,R) is a cross
section of R.

THEOREM 3. The existence of a Q system is provably equivalent, over RCA_0,
to light faced WKL_0 plus the consistency of Peano arithmetic.

In fact, we can restate this result in terms of bi interpretability. Let
T_1 be the following theory in the language <,R.

a. < is a linear ordering on the universe.
b. every bounded above definable set is a cross section of R.

THEOREM 4. T_1 and PA are bi interpretable.

To indicate how uncountable cardinals are used in such simple model
theoretic contexts, let us call a sharp Q system a triple (Q,<,R) such that

i) R containedin Q^2;
ii) every nonempty upper bounded set of rationals definable in (Q,<,R) is a
cross section of R at a rational of distance at most 1 from the set.

THEOREM 5. The existence of a sharp Q system is provably equivalent, over
RCA_0, to light faced WKL_0 plus the consistency of Zermelo set theory.

We can introduce a unary function symbol and consider the following theory T2:

a. < is a linear ordering on the universe.
b. for all x, every definable set bounded by x is a cross section of R at a
point < F(x).

THEOREM 6. T_2 and Zermelo set theory are bi interpretable.

GENERAL PROBLEM: Let j:Q into_p Q. When is there an R containedin Q^2 and
j' extending j such that

i) R containedin Q^2;
ii) every upper bounded set of rationals definable in (Q,<,R,j) is a cross
section of R;
iii) j is an elementary embedding from (Q,<,R) into (Q,<,R) with critical
point 0?

Here iii) means that 0 is moved but all negative rationals are fixed.

YET MORE GENERAL PROBLEM. Let W be a set of pairs (j,c(j)), where j:Q^2
into_p Q and c(j) in Q. When is there an R containedin Q^2 and extensions
j' of j, such that

i) R containedin Q^2;
ii) every upper bounded set of rationals definable in (Q,<,R,j) is a cross
section of R;
iii) each j' is an elementary embedding from (Q,<,R) into (Q,<,R) with
critical point c(j)?

**********

This is the 89th 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

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