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*To*: fom@math.psu.edu*Subject*: FOM: 50:Enormous Integers/Number Theory*From*: Harvey Friedman <friedman@math.ohio-state.edu>*Date*: Sat, 17 Jul 1999 23:39:50 +0100*Sender*: owner-fom@math.psu.edu

This is the 49th 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 This concerns the generation of big numbers in elementary number theory. We consider the following property *(k,A) of integers k >= 1 and sets A of positive integers: 1. No element of A divides any different element of A. 2. For all x in A greater than k, x divides the product of all y in A less than x. THEOREM 1. For all k >= 1, there are finitely many A such that *(k,A), all of which are finite. However, this cannot be proved in RCA_0. It is equivalent to "the Ackerman function exists" over RCA_0. So we have defined a specific finite mathematical structure associated with any given k >= 1: the family of all sets obeying property *(k). Let's look at some small k. We write #(k) for the largest cardinality of any A such that *(k,A). Note that by 2, a bound on the largest numbers appearing in any of the A's and the number of A's can be given in terms of #(k). #(1) = 1. #(2) = 1. #(3) = 2. #(4) = 2. #(5) = 3. #(6) = 3. #(7) = 4. #(8) = 4. #(9) = 5. #(10) = 7. #(11) = 8. #(12) = 8. #(13) = 9. #(14) >= 530. #(22) >= t, where t is an exponential stack of s 2's, where s is an exponential stack of 2^1032 2's. NOTE: These calculations up through #(13) THOEREM 2. Let k >= 14. #(k) is at least the unary Ackerman function at t, where t is the number of primes <= k/2. #(k) is at most the unary Ackerman function at k.

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