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# DisIndResults - 1 DISCRETE INDEPENDENCE RESULTS by Harvey M...

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1 DISCRETE INDEPENDENCE RESULTS by Harvey M. Friedman [email protected] www.math.ohio-state.edu/~friedman/ November 16, 1999 1. APPROXIMATE FIXED POINTS AND LARGE CARDINALS Let P(Z) be the Cantor space of all subsets of N. We are interested in mappings j :P(Z) P(Z). We say that j :P(Z) P(Z) is a contraction if and only if for all n 0 and A,B P(Z), if A,B agree on (-n,n) then j (A), j (B) agree on (-n-1,n+1). THEOREM 1.1. Every contraction on P(Z) has a unique fixed point. Let n,k 1. An approximate fixed point of j of type (n,k) is a chain of sets A 1 Õ ... Õ A n Õ Z such that *for all 1 £ i £ n-1, j (A i+1 ) and A i+1 agree on all sums and products of length k from A i {0, ± 1}.* An infinite approximate fixed point of type (n,k) is an approximate fixed point of type (n,k) whose terms are infinite. A bi-infinite approximate fixed point of type (n,k) is an approximate fixed point of type (n,k) whose terms are bi- infinite; i.e., contain infin-itely many positive and infin- itely many negative elements. We would like to obtain a bi-infinite approximate fixed point theorem. I.e., PROTOTYPE. Every “suitable” j :P(Z) P(Z) has bi-infinite approximate fixed points of every type. We present a weak condition on j for this Prototype to hold. It turns out that we need large cardinals in order to prove that this condition works. We say that j :P(Z) P(Z) is a compression on P(Z) if and only if there exists r 0 and t > 1 such that

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2 *if A,B agree on (-n,n) then j (A), j (B) agree on (-tn- 1,tn+1).* We say that j is a decreasing compression on P(Z) iff j is a compression on P(Z) satisfying *A Õ B implies j (A) j (B).* We say that j is a uniformly decreasing compression on P(Z) iff j is a decreasing compres-sion on P(Z) for which there exists r 0 such that *the value of j at any set is equaled to the value of j at some subset of cardinality £ r.* PROPOSITION 1.2. Every uniformly decreasing compres-sion on P(Z) mapping finite sets to cofinite sets has bi-infinite approximate fixed points of every type. Here is a weak special case of Proposition 1.2. PROPOSITION 1.3. Every uniformly decreasing compression on P(Z) mapping finite sets to cofinite sets has an infinite approximate fixed point of every type with a positive element. We must use t > 1 in the definition of compression. If we use t = 1 then Propositions 1.2, 1.3 are false. PROPOSITION 1.4. Let V be any countable set of infinte subsets of Z. Any uniformly decreasing contraction on P(Z) mapping finite sets to cofinite sets has approximate fixed points of every type whose first set meets every element of V. PROPOSITION 1.5. Let t be a type. Any finite set of uniformly decreasing compressions on P(Z) mapping finite sets to cofinite sets have respective infinite approximate fixed points of type t with the same first set. Moreover, we can replace "infinite" by "bi-infinite." PROPOSITION 1.6. Let t be a type. Any countable set of uniformly decreasing compressions on P(N) mapping finite sets to cofinite sets have respective infinite approximate fixed
3 points of type t with almost equal first sets. Moreover, we can replace "infinite" by "bi-infinite."

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DisIndResults - 1 DISCRETE INDEPENDENCE RESULTS by Harvey M...

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