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Unformatted text preview: ive block xj,...,x2j. THEOREM 14. For all k ≥ 1, n(k) exists. BRAIN DEAD: n(1) = 3. GIFTED HIGH SCHOOL: n(2) = 11. What about n(3)?? n(3) approx. 60? n(3) approx. 100? n(3) approx. 200? n(3) approx. 300? Bad estimates. E.g., n(3) > 2^2^2^2^2^2^2^2^2^2^2^2^2^2^2. In fact, n(3) > an exponential stack of 2's of length the above number. A better lower bound: n(3) > the 7198-th level of the Ackerman hierarchy at 158,386. 12 What about n(4)?? Let A(n) be the n-th level of the Ackerman hierarchy at n. THEOREM 15. n(4) > AA...A(1), where there are A(187196) A's. Now that is a big number. 7. PEANO ARITHMETIC. Peano arithmetic (PA) is a very fundamental system for f.o.m. It is the formal system that Gödel used to cast his incompleteness theorems. Here we give an example of what PA cannot handle. The first appropriate example of a genuinely combinatorial nature is Paris/Harrington 1977. Here is a more state of the art example. Below we use | | for the sup norm. THEROEM 16. Let n >> k ≥ 1 and F:[0,n]k Æ [0,n]k obey |f(x)| £ |x|. There exist x1 < ... < xk+1 such that F(x1,...,xk) £ F(x2,...,xk+1) coordinatewise. This cannot be proved in PA. 8. PREDICATIVITY. Predicativity is the view that it is illegitimate to form a set of integers obeying a property that involves all sets of integers. One is allowed to form sets of integers only through definitions that involve sets of integers that have been previously formed. This view was attrac-tive to Poincare and Weyl and others. A modification of this view: impredicativity is useful for normal mathematics only for the purpose of proving the existence of infinite sets of integers of a problematic character. In this form, the view is demonstrably false, as witnessed by, say, J.B. Kruskal’s tree theorem 1960: KRUSKAL’S THEOREM. Let T1,T2,... be finite trees. There exists i < j such that Ti is continuously embeddable into Tj as topological spaces. 13 (Continuous embeddability of finite trees is a purely combinatorial notion, involving only the vertices and the edge relation.) Kruskal’s proof is blatantly impredicative. Results from mathematical logic show that under the usual formalizations of predicativity, there is no predicative proof of Kr...
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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.

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