FOM: 44:Indiscernible Primes
Harvey Friedman
friedman at math.ohio-state.edu
Thu May 27 07:53:43 EDT 1999
This is the 44th 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
A complete archiving of fom, message by message, is available at
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Also, my series of self contained postings (only) is archived at
http://www.math.ohio-state.edu/foundations/manuscripts.html
FAVORITE SELF CONTAINED POSTINGS: 21, 25, 27, 31, 32, 34, 35', 37, 38, 39,
41, 42, 43, 44.
CLAIRIFICATION OF #42. In #42 I wrote:
>Let k,n >= 1. We define L(F,k,n) to be the longest possible length of any
>decreasing chain of algebraic subsets of F where the presentation degree of
>each succeeding set is exactly one higher than the presentation degree of
>the preceding set.
I meant
Let k,n >= 1. We define L(F,k,n) to be the longest possible length of any
decreasing chain of algebraic subsets of F^k where the presentation degree of
each succeeding set is exactly one higher than the presentation degree of
the preceding set and the first set has presentation degree exactly n
**********
A shift indiscernible set of primes is a finite set of primes E = {p_1 <
... < p_k+1} such that every polynomial over Z whose number of variables,
degree, and magnitudes of coefficients are at most k that achieves the
value p_1...p_k also acheives the value p_2...p_k+1.
THEOREM 1. There are shift indiscernible sets of primes of any given finite
cardinality. However, the smallest possible first term of shift
indiscernible primes of cardinality k is greater than an exponential stack
of 2's of roughly length k, for sufficiently large k.
A stong shift indiscernible set of primes is a finite set of primes E =
{p_1 < ... < p_k} such that for all 1 <= i <= k-2, every polynomial over Z
whose number of variables, degree, and magnitudes of coefficients are at
most p_i that achieves the value p_i+1...p_k-1 also achieves the value
p_i+2...p_k.
THEOREM 2. There are strong shift indiscernible sets of primes of any given
finite cardinality. This is independent of Peano Arithmetic (equivalent to
the 1-consistency of Peano Arithmetic), and the associated growth rate
corresponds to level epsilon_0 in standard hierarchies of functions.
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