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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/1998July/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/1999March/002752.html
http://www.cs.nyu.edu/pipermail/fom/1999May/003134.html
http://www.cs.nyu.edu/pipermail/fom/1999July/003251.html
http://www.cs.nyu.edu/pipermail/fom/1999July/003253.html
http://www.cs.nyu.edu/pipermail/fom/2006March/010292.html
http://www.cs.nyu.edu/pipermail/fom/2006March/010293.html
http://www.cs.nyu.edu/pipermail/fom/2006March/010290.html
http://www.cs.nyu.edu/pipermail/fom/2006March/010279.html
http://www.cs.nyu.edu/pipermail/fom/2006March/010281.html
http://www.cs.nyu.edu/pipermail/fom/2006April/010305.html
Suppose we have a
Π
0
2
theorem (
∀
k)(
∃
n)(A(k,n)). We then get
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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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 Fall '08
 JOSHUA
 Math, Recursion, Exponentiation, Natural number, Ackermann, proof theoretic integer

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