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GodelLect060202 - 1 ISSUES IN THE FOUNDATIONS OF...

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1 ISSUES IN THE FOUNDATIONS OF MATHEMATICS Harvey M. Friedman Gödel Lecture, ASL Las Vegas, Nevada http://www.math.ohio-state.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 mathematicians & philosophers. 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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2 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 one-one continuous f:A B or a one-one 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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