Tarski2,052407

# Tarski2,052407 - 1 INTERPRETING SET THEORY IN DISCRETE...

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1 INTERPRETING SET THEORY IN DISCRETE MATHEMATICS: BOOLEAN RELATION THEORY Harvey M. Friedman* Nineteenth Annual Tarski Lectures Interpretations of Set Theory in Discrete Mathematics and Informal Thinking Lecture 2 Delivered, April 11, 2007 Expanded, May 24, 2007 1. Wqo Theory. 2. Countable Pointsets. 3. Borel Selection. 4. Boolean Relation Theory. 5. Finite Graphs. 1. WQO THEORY. Wqo theory = well quasi ordering theory, is a branch of combinatorics which has proved to be a fertile source of deep metamathematical pheneomena. In wqo theory, one freely uses some overtly set theoretic arguments. They are definitely noticeable, although they come fairly early in the interpretation hierarchy. Way before, say, ZFC. A qo (quasi order) is a reflexive transitive relation (A, £ ). A wqo (well quasi order) is a qo (A, £ ) such that for all x 1 ,x 2 ,... from A, there exists i < j such that x i £ x j . Highlights of wqo theory: that certain qo’s are wqo’s. There are many equivalent definitions of wqo. THEROEM 1.1. Let (A, £ ) be a qo. The following are equivalent. i. (A, £ ) is a wqo. ii. Every infinite sequence from A has an infinite subsequence which is increasing ( £ ). iii. For all x 1 ,x 2 ,... A there exists n such that every term is at least one of x 1 ,...,x n . iv. Every infinite subset of A has a two element chain.

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2 v. Every infinite subset of A has an infinite chain of type w . vi. Every subset of A has a finite subset such that every element of A is some element of that finite subset. Proof: i Æ ii. Color (i,j), i < j, according to whether xi £ xj or not. By Ramsey’s theorem for pairs, there is an infinite subsequence which is either increasing ( £ ), or with y i £ y j , i < j. Only the former is possible. ii Æ iii. Suppose not. Define an infinite subsequence y1,y2,. .. so that y1 = x1, and each yi+1 is not y1,. ..,yi. This contradicts wqo. iii Æ iv. Enumerate the set without repetition. iv Æ v. Enumerate the set without repetition, and argue as in i Æ ii. v Æ vi. Enumerate the set without repetition, and argue as in ii Æ iii. vi Æ i. Let x 1 ,x 2 ,... A. Let {x 1 ,x 2 ,...,x n } include the finite set. QED Mainly application of Ramsey’s theorem for pairs. Also simple inductive constructions. QED J.B. Kruskal treats finite trees as finite posets, where there is a root, and the set of strict predecessors of any vertex is linearly ordered, and studies the qo there exists an inf preserving embedding from T 1 into T 2 . I.e., h:T 1 Æ T 2 , where h is one-one, preserves £ , and h(x inf y) = h(x) inf h(y). (Every finite tree has an obvious inf operation on the vertices). THEOREM. (J.B. Kruskal). The above qo of finite trees as posets is a wqo. Let’s see what is set theoretic about the proof. Nash Williams’s introduction of the minimal bad sequence technique seriously simplified the original Kruskal proof, and is also used extensively in wqo theory. See [NW65].
3 Suppose we want to prove that a given qo (A, £ ) is a wqo.

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Tarski2,052407 - 1 INTERPRETING SET THEORY IN DISCRETE...

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