A Classification of Rapidly Growing Ramsey Functions
نویسندگان
چکیده
Let f be a number-theoretic function. A finite set X of natural numbers is called f -large if card(X) ≥ f(min(X)). Let PHf be the Paris Harrington statement where we replace the largeness condition by a corresponding f -largeness condition. We classify those functions f for which the statement PHf is independent of first order (Peano) arithmetic PA. If f is a fixed iteration of the binary length function, then PHf is independent. On the other hand PHlog∗ is provable in PA. More precisely let fα(i) := |i|H−1 α (i) where | i |h denotes the h-times iterated binary length of i and H−1 α denotes the inverse function of the α-th member Hα of the Hardy hierarchy. Then PHfα is independent of PA (for α ≤ ε0) iff α = ε0.
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