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*To*: fom@math.psu.edu*Subject*: FOM: 51:Enormous Integers/Plane Geometry*From*: Harvey Friedman <friedman@math.ohio-state.edu>*Date*: Sun, 18 Jul 1999 15:16:09 +0100*Sender*: owner-fom@math.psu.edu

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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