[FOM] 197:PL and primes

[Harvey Friedman]

This is a slightly new version of posting #194 in which we use prime
numbers. The key idea from number theory that makes this work is Dirichlet's
theorem.

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Let N be the set of all nonnegative integers. T:Nk into N is said to be
strictly dominating if and only if for all x in Nk, T(x) > max(x).

We write A U. B for A U B if A,B are disjoint; undefined otherwise.

THEOREM 1. Let k >= 1 and T:Nk into N be strictly dominating. There exists a
unique A containedin N such that A U. T[Ak] = N.

We say that T:Nk into N is piecewise linear if and only if T is defined by
finitely many cases, each case given by a finite set of linear inequalities,
where in each case T is given by an affine expression. We will be concerned
with the coefficients used throughout this presentation of T.

PROPOSITION 2. Let T:Nk into N be strictly dominating. There exists A
containedin N such that A U. T[Ak] contains the minimum value over Ak of any
piecewise linear function with coefficients among 0,1 and the infinitely
many primes in A.

PROPOSITION 3. Let k >= 1 and T:Nk into N be a strictly dominating piecewise
linear function with integer coefficients. There exists A containedin N such
that A U. T[Ak] contains the minimum value over Ak of any piecewise linear
function with coefficients from 0,1 and the infinitely many primes in A.

When we give an explicitly Pi01 independent statement, we don't see how to
use primes in a simple way. There are many alternatives to the particular
numbers used in Proposition 4, but the ones chosen are convenient and safe.
Of course, they are silly.

PROPOSITION 4. Let k,p >= 1 and T:Nk into N be a strictly dominating
piecewise linear function with coefficients from {-k,...,k}. There exists
finite A containedin N such that A U. T[Ak] contains the minimum value over
Ak of any piecewise linear function with coefficients from 0,1 and the
subset {(8k)!!!,(8k+1)!!!,...,p!!!} of A.

We can obviously bound the largest element of A in terms of k and the
largest element of E. This results in an explicitly Pi01 statement.

Proposition 2 is provably equivalent, over ACA, to the 1-consistency of
Mahlo cardinals of finite order. Propositions 3 and 4 are provably
equivalent, over ACA, to the consistency of Mahlo cardinals of finite order.

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proofs. Type Harvey Friedman in the window.
This is the 197th 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