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phasetrans092610 - 1 FINITE PHASE TRANSITIONS Harvey M...

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1 FINITE PHASE TRANSITIONS Harvey M. Friedman* Ohio State University September 26, 2010 DRAFT This topic has been discussed earlier on the FOM email list in various guises. The common theme is: big numbers and long sequences associated with mathematical objects. See http://www.cs.nyu.edu/pipermail/fom/1998-July/001921.html http://www.cs.nyu.edu/pipermail/fom/1998- October/002332.html http://www.cs.nyu.edu/pipermail/fom/1998- October/002339.html http://www.cs.nyu.edu/pipermail/fom/1998- October/002356.html http://www.cs.nyu.edu/pipermail/fom/1998- October/002365.html http://www.cs.nyu.edu/pipermail/fom/1998- October/002383.html http://www.cs.nyu.edu/pipermail/fom/1998- October/002395.html http://www.cs.nyu.edu/pipermail/fom/1998- October/002407.html http://www.cs.nyu.edu/pipermail/fom/1998- October/002409.html http://www.cs.nyu.edu/pipermail/fom/1998- October/002410.html http://www.cs.nyu.edu/pipermail/fom/1998- November/002439.html http://www.cs.nyu.edu/pipermail/fom/1998- November/002443.html http://www.cs.nyu.edu/pipermail/fom/1999-March/002752.html http://www.cs.nyu.edu/pipermail/fom/1999-May/003134.html http://www.cs.nyu.edu/pipermail/fom/1999-July/003251.html http://www.cs.nyu.edu/pipermail/fom/1999-July/003253.html http://www.cs.nyu.edu/pipermail/fom/2006-March/010292.html http://www.cs.nyu.edu/pipermail/fom/2006-March/010293.html http://www.cs.nyu.edu/pipermail/fom/2006-March/010290.html http://www.cs.nyu.edu/pipermail/fom/2006-March/010279.html http://www.cs.nyu.edu/pipermail/fom/2006-March/010281.html http://www.cs.nyu.edu/pipermail/fom/2006-April/010305.html Suppose we have a Π 0 2 theorem ( k)( n)(A(k,n)). We then get
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2 a recursive function F(k) = the least n such that A(k,n) holds. In the intended cases, we have F(0) < F(1) < . .. . We want to look at F(0),F(1),. .. and determine where there is a "qualitative jump" in size. I.e., a "phase transition". In the cases we focus on, after the first few terms - say about 16 or less - we simply get qualitatively indistinguishable very large integers. There are a number of forms that such results can take. Some are more descriptive than quantitative and others are more quantitative than descriptive. QUANTITATIVE APPROACH. In the quantitative approach, we simply provide upper and lower bounds on some of the terms in F(0),F(1),. .. using a notation system for integers. We try to make exact calculations, if possible. But what notation system to use for the integers? In case the numbers involved are less than, say, 10 100 , the usual base 10 notation is the clear choice. But if the numbers involved are greater than, say, 10 10^100 , base 10 notation is generally of no use whatsoever since it cannot even be presented. There are many special approaches that may be particularly
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