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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