FOM: 51:Enormous Integers/Plane Geometry
Harvey Friedman
friedman at math.ohio-state.edu
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.
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