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DoesMathNeed102601

DoesMathNeed102601 - 1 DOES NORMAL MATHEMATICS NEED NEW...

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1 DOES NORMAL MATHEMATICS NEED NEW AXIOMS? by Harvey M. Friedman* Department of Mathematics Ohio State University [email protected] http://www.math.ohio-state.edu/~friedman/ October 26, 2001 Lecture Notes Abstract. We present a range of mathematical theorems whose proofs require unexpectedly strong logical methods, which in some cases go well beyond the usual axioms for mathematics. ************************************************ There are a variety of mathematical results that can only be obtained by using more than the usual axioms for mathematics. For several decades there has been a gradual accumulation of such results that are more and more concrete, more and more connected with standard mathematical contexts, and more and more relevant to ongoing mathematical activity. Probably the most well known mathematical problem that cannot be proved or refuted with the usual axioms (ZFC) is the continuum hypothesis - that every set of real numbers is either countable or of cardinality the continuum (Kurt Gödel and Paul Cohen). But mathematicians have instinctively learned to hide from this kind of problem by focusing on relatively “concrete” subsets of complete separable metric spaces. In particular, the Borel measurable sets and functions in and between complete separable metric spaces proves to be a natural boundary. By way of illustration, “every Borel set of real numbers is either countable or of cardinality the continuum via a Borel measurable function” is a well known theorem of descriptive set theory.
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2 All problems discussed here live within this Borel measurable universe. Some are even further down in sets and functions on the integers. 1. Exotic High School Math. Warmup. 2. Ordinals in Freshman Calculus. Warmup. 3. Borel Measurable Selection. 4. Thin Set Theorem. Warmup. 5. Complementation Theorem. Warmup. 6. Disjoint Covers. 7. Boolean Relation Theory. 8. A Sketch. 1. EXOTIC HIGH SCHOOL MATH. This problem was used a few years ago in Paul Sally’s program for gifted high school students at U Chicago. THEOREM 1.1. There is a longest sequence of 1’s and 2’s in which no block x i ,...,x 2i is a subsequence of a later block x j ,...,x 2j . The longest length is 11. Ex: 12221111111. The relevant blocks are 12, 222, 2211, 21111, 111111. None is a subsequence of any later one. One of the students was able to give a correct proof of this. THEOREM 1.2. There is a longest finite sequence of 1’s, 2’s, 3’s, in which no block x i ,...,x 2i is a subsequence of a later block x j ,...,x 2j . The students couldn’t prove this. The natural proof is very infinitary, and the longest length is gigantic. More detailed work gives a less infinitary proof, but all proofs must be somewhat exotic. More generally: THEOREM 1.3. For all k 1 there exists a longest finite sequence from {1,...,k} in which no block x i ,...,x 2i is a subsequence of any later block x j ,...,x 2j .
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