1
UNPROVABLE THEOREMS
by
Harvey M. Friedman
[email protected]
http://www.math.ohiostate.edu/%7Efriedman/
Cal Tech Math Colloq
April 19, 2005
INTRODUCTION.
We discuss the growing list of examples of simply stated
Theorems where it is known that there are no concrete
proofs.
Here is our agenda.
A. In sufficiently long finite sequences from a finite set,
certain blocks are subsequences of certain later blocks.
B. In sufficiently tall finite trees, certain truncations
are embedded in certain taller truncations.
C. Multivariate functions on the integers have
nonsurjective infinite restrictions.
D. Any two countable sets of reals are pointwise
continously comparable.
E. Any permutation invariant Borel function from infinite
sequences of reals into infinite sequences of reals maps
some sequence into a subsequence.
F. For every symmetric Borel set in the plane, either it or
its complement has a Borel selection.
G. Any Borel set in the plane that has a Borel selection on
every compact set has a Borel selection.
H. For any two multivariate functions on the natural
numbers of expansive linear growth there are three infinite
sets which bear a certain Boolean relation with their
images under the two functions. (Boolean relation theory).
1.
BLOCK SUBSEQUENCE THEOREM.
The block subsequence theorem involves a single finite
string in k letters. The following binary case is
elementary, and a good challenge for gifted high school
students.
THEOREM 1.1. There is a longest finite sequence x
1
,...,x
n
in
two letters such that no consecutive block x
i
,...,x
2i
is a
subsequence of a later consecutive block x
j
,...,x
2j
.
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The longest length is 11, with 12221111111 and 21112222222
as the only examples.
THEOREM 1.2. There is a longest finite sequence x
1
,...,x
n
in
3 letters where no consecutive block x
i
,...,x
2i
is a
subsequence of a later consecutive block x
j
,...,x
2j
.
Theorem 1.2 merely states the existence of a natural number
with a specific testable property.
But the simplest known way to prove this involves not only
infinite sequences but also defining infinite sequences
using all infinite sequences (impredicativity). This is a
weak use of the uncountable.
A logically more down to earth but more involved proof can
be given for any number of letters. The most logically
economical proof uses induction with an
induction
hypothesis that has three alternating quantifiers over the
natural numbers. Two quantifiers does not suffice.
Coming back to the case of three letters, the exotic nature
of all proofs is illustrated by the following lower bound
on the length of the longest such finite string. In the
case of two letters it is 11. In the case of three letters,
it is greater than the 7198th Ackermann function at
158,386.
The 3rd Ackermann function at 158,386 is already an
exponential tower of 2’s of height 158,386.
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 Fall '08
 JOSHUA
 Math, Set Theory, Natural number, Georg Cantor, borel selection, G. Any Borel

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