[FOM] 198:Strong Thematic Propositions

[Harvey Friedman]

In BRT, our independent statements are presented as one in a very natural
large finite set of statements. Presently, as one in the 6561 classification
result, although that is expected to be considerably expanded.

There remains the problem of giving a natural single statement of a thematic
nature, which is independent and concrete.

Let Q be the set of all rational numbers. We use the lexicographic ordering
<lex on each Qk. We say that A is k-dimensional iff A is a subset of Q^k.

Let A containedin Q^k. We write fld(A) for the set of all coordinates of
elements of A. We say that A is bounded iff there is a rational greater than
all elements of fld(A). We write A<= for the set of all increasing (<=)
elements of A, and A>= for the set of all decreasing (>=) elements of A.

We use forward imaging notation for relations. I.e., we write A[X] = {y:
(therexists x in X)(A(x,y))}. The cross sections of A are the forward images
A[{c}]. If k = 1 then we take the cross sections of A to be empty.

We say that A is order invariant iff membership in A depends only on the
relative order of the coordinates.

The nonnegative shift of (x1,...,xk) is obtained by adding 1 to all
nonnegative coordinates, and leaving the negative coordinates unchanged. The
nonnegative shift of A is the forward image of A under the nonnegative
shift.

THEOREM 1. Let R containedin Qk x Qk x Qk = Q^3k be order invariant, where
R(x,y,z) implies x,y <lex z. There exists infinite k-dimensional A =
fld(A)^k\R[A^2] properly containing its nonnegative shift.

PROPOSITION 2. Let R containedin Qk x Qk x Qk = Q^3k be order invariant,
where R(x,y,z) implies x,y <lex z. There exists k-dimensional A =
fld(A)^k\R[A^2] properly containing its nonnegative shift and 0.

Proposition 2 is provably equivalent, over WKL0, to the consistency (not
1-consistency!) of SUB = ZFC + {there exists an n-subtle cardinal}_n. In
particular, it is independent of ZFC, provided SUB is consistent.

PROPOSITION 3. Let R containedin Qk x Qk x Qk = Q^3k be order invariant,
where R(x,y,z) implies x,y <lex z. There exists k-dimensional A = A>= U
(fld(A)^k<=\R[A^2]) properly containing its nonnegative shift, where the
nonnegative shift of every bounded cross section is a cross section.

Proposition 3 is provably equivalent, in WKL0, to the consistency of ZFC +
{there exists an n-huge cardinal}_n.

In both Propositions, the set A can be taken to be Delta-0-2, resulting in
an arithmetical sentence, but any arithmetic definition seems to be ugly. So
there are problems giving an explicitly arithmetical statement in this
approach. Nevertheless, the statements have very strong absoluteness
properties.

In BRT, we do know how to give various explicitly Pi-0-2 and Pi-0-1 forms
that are at least reasonable.

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This is the 198th in a series of self contained numbered postings to
FOM covering a wide range of topics in f.o.m. The list of previous
numbered postings #1-149 can be found at
http://www.cs.nyu.edu/pipermail/fom/2003-May/006563.html  in the FOM
archives, 5/8/03 8:46AM. Previous ones counting from #150 are:

150:Finite obstruction/statistics  8:55AM  6/1/02
151:Finite forms by bounding  4:35AM  6/5/02
152:sin  10:35PM  6/8/02
153:Large cardinals as general algebra  1:21PM  6/17/02
154:Orderings on theories  5:28AM  6/25/02
155:A way out  8/13/02  6:56PM
156:Societies  8/13/02  6:56PM
157:Finite Societies  8/13/02  6:56PM
158:Sentential Reflection  3/31/03  12:17AM
159.Elemental Sentential Reflection  3/31/03  12:17AM
160.Similar Subclasses  3/31/03  12:17AM
161:Restrictions and Extensions  3/31/03  12:18AM
162:Two Quantifier Blocks  3/31/03  12:28PM
163:Ouch!  4/20/03  3:08AM
164:Foundations with (almost) no axioms, 4/22/0  5:31PM
165:Incompleteness Reformulated  4/29/03  1:42PM
166:Clean Godel Incompleteness  5/6/03  11:06AM
167:Incompleteness Reformulated/More  5/6/03  11:57AM
168:Incompleteness Reformulated/Again 5/8/03  12:30PM
169:New PA Independence  5:11PM  8:35PM
170:New Borel Independence  5/18/03  11:53PM
171:Coordinate Free Borel Statements  5/22/03  2:27PM
172:Ordered Fields/Countable DST/PD/Large Cardinals  5/34/03  1:55AM
173:Borel/DST/PD  5/25/03  2:11AM
174:Directly Honest Second Incompleteness  6/3/03  1:39PM
175:Maximal Principle/Hilbert's Program  6/8/03  11:59PM
176:Count Arithmetic  6/10/03  8:54AM
177:Strict Reverse Mathematics 1  6/10/03  8:27PM
178:Diophantine Shift Sequences  6/14/03  6:34PM
179:Polynomial Shift Sequences/Correction  6/15/03  2:24PM
180:Provable Functions of PA  6/16/03  12:42AM
181:Strict Reverse Mathematics 2:06/19/03  2:06AM
182:Ideas in Proof Checking 1  6/21/03 10:50PM
183:Ideas in Proof Checking 2  6/22/03  5:48PM
184:Ideas in Proof Checking 3  6/23/03  5:58PM
185:Ideas in Proof Checking 4  6/25/03  3:25AM
186:Grand Unification 1  7/2/03  10:39AM
187:Grand Unification 2 - saving human lives 7/2/03 10:39AM
188:Applications of Hilbert's 10-th 7/6/03  4:43AM
189:Some Model theoretic Pi-0-1 statements  9/25/03  11:04AM
190:Diagrammatic BRT 10/6/03  8:36PM
191:Boolean Roots 10/7/03  11:03 AM
192:Order Invariant Statement 10/27/03 10:05AM
193:Piecewise Linear Statement  11/2/03  4:42PM
194:PL Statement/clarification  11/2/03  8:10PM
195:The axiom of choice  11/3/03  1:11PM
196:Quantifier complexity in set theory  11/6/03  3:18AM
197:PL and primes 11/12/03  7:46AM

Harvey Friedman