FOM: 51:Enormous Integers/Plane Geometry

Sun Jul 18 10:16:09 EDT 1999

This is the 51st 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

NOTE: Somebody was confused with regard to the terminology in #49. Let me
clarify this. I wrote, in the context of countable Abelian groups, that

>THEOREM 1. For reduced p-groups, each of 1-5 are provably equivalent to
ATR_0 over RCA_0. This is also true for any specific prime p. For reduced
torsion groups, each of 2,3 are provably equivalent to ATR_0 over RCA_0.
1,4,5 are false for reduced torsion groups.

>THEOREM 2. For p-groups, each of 1-5 are provably equivalent to ATR_0 over
RCA_0. This is also true for any specific prime p. For torsion groups, each
of 2,3 are provably equivalent to ATR_0 over RCA_0. 1,4,5 are false for
torsion groups.

In Theorem 1, I mean that for all 1 <= i <= 5, the statement "for all
primes p, i holds for all countable Abelian p-groups" is provably
equivalent to ATR_0 over RCA_0. And for all 1 <= i <= 5 and primes p, the
statement "i holds for all countable Abelian p-groups" is provably
equivalent to ATR_0 over RCA_0. Similarly in Theorem 2. This should clear
up any confusion.

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

This posting concerns the emergence of big numbers in some elementary plane
geometry. This scratches the surface. As usual, I expect more and better.

A circle is defined to be a circumference of a nondegenerate circle in the
Euclidean plane.

THEOREM 1. For all k >= 1 there exists n >= 1 such that the following
holds. Let C1,C2,...,Cn be pairwise disjoint circles. There exists k <= i <
j <= n/2 and a homeomorphism of the plane mapping Ci union ... union C2i
into Cj union ... union C2j.

THEOREM 2. Theorem 1 is provably equivalent to the 1-consistency of Peano
Arithmetic within EFA (exponential function arithmetic). The growth rate of
n in terms of k dominates all <epsilon_0 recursive functions, but is
epilson_0 recursive.

A p-circle is the union of p circles. (Some of the p circles may be
identical).

THEOREM 3. For all k >= 1 there exists n >= 1 such that the following
holds. Let C1,C2,...,Cn be pairwise disjoint k-circles. There exists 1 <= i
< j <= n/2 and a homeomorphism of the plane mapping Ci union ... union C2i
into Cj union ... union C2j.

THEOREM 4. Theorem 3 is provably equivalent to the 1-consistency of
Pi-1-2-TI_0, and hence is not provable in ATR_0 or the usual formalizations
of predicativity.

THEOREM 5. For all k >= 1 there exists n >= 1 such that the following
holds. Let C1,C2,...,Cn be pairwise disjoint 2-circles. There exists k <= i
< j <= n/2 and a homeomorphism of the plane mapping Ci union ... union C2i
into Cj union ... union C2j.

THEOREM 6. Theorem 5 implies the 1-consistency of ATR_0.

The corresponding growth rates display the usual pheenomena.