[FOM] 199:Radical Polynomial Behavior Theorems

Mon Dec 22 13:22:54 EST 2003

A radical polynomial in k variables x1,...,xk is a finite sum of radical
monomials with real coefficients. A radical monomial is a finite product of
nonnegative rational powers of variables. 1 is the trivial radical monomial.

We prove a Radical Polynomial Behavior Theorem to the effect that for any
function f:Z+^k< into Z+ there is a radical polynomial P:[0,infinity)^k<
into R+ such that f and P exhibit the same behavior when restricted to some
common infinite A^k<. An obvious finite form also holds. This obvious finite
form is, however, 

*independent of Peano Arithmetic.*

This may raise the known incompleteness of PA to a new thematic level.

Let Z be the set of all positive integers, Q be the set of all rational
numbers, and R be the set of all real numbers. For A containedin R, let A+
be the set of all positive elements of A. Let A^k< be the set of all
strictly increasing k-tuples from A.

Let f,g:A^k< into R. We say that f,g have the same behavior if and only if
for all x1 < ... < xk from A, y1 < ... < yk from A, and 1 <= i <= k, we have

f(x1,...,xk) < xi iff g(x1,...,xk) < xi.
f(x1,...,xk) < f(y1,...,yk) iff g(x1,...,xk) < g(y1,...,yk).

I.e., we preserve the truth values of all unnested atomic relations.

THEOREM 1. For all f:Z+^k< into Z+ there exists a radical polynomial P:R+^k<
into R+ and infinite A containedin Z+ such that f,P restricted to A^k< have
the same behavior. In fact, we may require that P have rational
coefficients.

THEOREM 2. For all k,p in Z+ there exists n in Z+ such that the following
holds. For all f:{1,...,n}^k< into Z+ there exists a radical polynomial
P:R+^k< into R+ and p element A containedin {1,...,n} such that f,P
restricted to A^k< have the same behavior.

Using the decision procedure for real closed fields, Theorem 2 is in Pi-0-3
form.

THEOREM 3. Theorem 1 is provably equivalent, over RCA0, to "the jump
operator can be iterated any finite number of times starting with any set".
Theorems 2 is provably equivalent, over EFA, to the 1-consistency of Peano
Arithmetic. 

*********************************************

I use http://www.mathpreprints.com/math/Preprint/show/ for manuscripts with
proofs. Type Harvey Friedman in the window.
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
198:Strong Thematic Propositions 12/18/03 10:54AM

Harvey Friedman