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ISSUES IN THE FOUNDATIONS OF MATHEMATICS
Harvey M. Friedman
Gödel Lecture, ASL
Las Vegas, Nevada
http://www.math.ohiostate.edu/~friedman/
June 2, 2002
I discuss my efforts concerning 3 crucial issues in the
foundations of mathematics that are deeply connected with the
great work of Kurt Gödel.
A. To what extent can set theoretic methods be used in an
essential way to further the development of normal
mathematics?
B. Are there fundamental principles of a general
philosophical nature which can be used to give consistency
proofs of set theory, including the so called large cardinal
axioms?
C. To what extent, and in what sense, is the natural
hierarchy of logical strengths rep resented by familiar
systems ranging from exponential function arithmetic to ZF +
j:V
V robust?
Our discussion of A is aimed at mathematicians; B,C at
A1. HIGH SCHOOL SEQUENCES AND COLLEGE CONTINUITY.
There exists a longest sequence in 2 letters, x
1
,...,x
n
, such
that no block x
i
,..., x
2i
is a subsequence of a later block
x
j
,..., x
2j
. The longest length is n(2) = 11.
This was used as a problem for gifted high school students by
Paul Sally at U. Chicago. One student proved n(2) = 11.
There is a longest sequence x
1
,...,x
n
in 3 letters such that
no block x
i
,...,x
2i
is a subsequence of a later block
x
j
,...,x
2j
. Call this longest length n(3).
THEOREM A1.1. n(1) = 3, n(2) = 11, n(3) > A
7198
(158386).
Here A
k
(n) is the kth level of the Ackermann hierarchy
(starting with A
1
= doubling) at n.
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THEOREM A1.2. (
"
k)(n(k) exists) is provable in 3 quantifier
induction but not in 2 quantifier induction.
In fact, the growth rate of n(k) lies just beyond the
multirecursive functions.
In College Continuity, we use the usual notion of pointwise
continuity for functions f:A
B, where A,B are (countable)
sets of real numbers.
The following statement looks like it comes from the era of
classical descriptive set theory, but is new:
THEOREM A1.3. Let A,B be countable sets of real numbers.
There is a oneone continuous f:A
B or a oneone continuous
g:B
A.
This requires a transfinite induction argument of length
w
1
in several senses. From the reverse math point of view:
THEOREM A1.4. A1.3 is provably equivalent to ATR
0
over RCA
0
.
This holds even for countable sets of rational numbers.
A2. FINITE TREES.
The first finite combinatorial theorem shown to be unprovable
in PA appeared in 1977 by Jeff Paris and Leo Harrington.
By 1977, I had obtained some finite statements independent of
even ZFC, but they had nowhere near the simplicity of PH.
At that time, my best contributions concerning independence
lie in the Borel world, and are discussed below.
The ideas in PH were expected to be pushed to get the
ultimate similarly natural finite statements corresponding to
ZFC and beyond.
However, this remained elusive, and the ideas in PH seem
insufficient to move forward significantly.
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This note was uploaded on 08/05/2011 for the course MATH 366 taught by Professor Joshua during the Fall '08 term at Ohio State.
 Fall '08
 JOSHUA
 Math

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