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greedytrees - 1 FINITE TREES AND THE NECESSARY USE OF LARGE...

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1 FINITE TREES AND THE NECESSARY USE OF LARGE CARDINALS by Harvey M. Friedman 1 Department of Mathematics Ohio State University [email protected] www.math.ohio-state.edu/~friedman/ March 22, 1998 We introduce insertion domains that support the placement of new, higher, vertices into finite trees. We prove that every nonincreasing insertion domain has an element with simple structural properties in the style of classical Ramsey theory. This result is proved using standard large cardinal axioms that go well beyond the usual axioms for mathematics. We also establish that this result cannot be proved without these large cardinal axioms. We also introduce insertion rules that specify the placement of new, higher, vertices into finite trees. We prove that every insertion rule greedily generates a tree with these same structural properties; and every decreasing insertion rule generates (or admits) a tree with these same structural properties. It is also necessary and sufficient to use the same large cardinals (in the precise sense of Corollary D.25). The results suggest new areas of research in discrete mathematics called "Ramsey tree theory" and "greedy Ramsey theory" which demonstrably require more than the usual axioms for mathematics. TABLE OF CONTENTS A. Statement of results. 1. Trees and insertion domains. 2. Greedy trees and insertion rules. 3. Decreasing insertion rules. 4. Finite forms. B. Logical relationships. 1. Basic implications and equivalences. 2. Additional implications and equivalences. C. Proofs in Z; proofs using large cardinals. D. Necessity of large cardinals. 1 This research was partially supported by the NSF. AMS subject classification 03, 04, 05.
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2 A. STATEMENT OF RESULTS A1. TREES AND INSERTION DOMAINS We begin with the concrete representation of finite trees that is used throughout the paper. A partial ordering is a pair (X, £ ), where X is a nonempty set, and £ is reflexive, transitive, and antisymmetric. The ancestors of x in X are just the y < x. For the purposes of this paper, a tree T = (V, £ ) is a partial ordering with a minimum element, where V is finite, and the ancestors of any x ˛ V are linearly ordered under £ . The minimum element of T is called the root of T, and is written r(T). A tree is said to be trivial if and only if it has exactly one vertex, which must be its root. V = V(T) represents the set of all vertices of the tree T = (V, £ ). In a tree T, if x < y and for no z is x < z < y, then we say that y is a child of x and x is the parent of y. Every vertex has at most one parent. However, vertices may have zero or more children. We write p(x,T) for the parent of x in T. We use Ch(T) = V(T)\{r(T)} for the set of all children of T. We write T 1 ˝ T 2 if and only if i) r(T 1 ) = r(T 2 ); ii) for all x ˛ Ch(T 1 ), p(x,T 1 ) = p(x,T 2 ).
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