my forty years on his shoulders - Department of Mathematics

48PROPOSITION A. For all f,g ∈ ELG there exist A,B,C ∈ INF such thatA ∪. fA ⊆ C ∪. gBA ∪. fB ⊆ C ∪. gC.This statement is provably equivalent to the 1-consistency of SMAH, overACA’.If we replace “infinite” by “arbitrarily large finite” then we can carry outthis second classification entirely within RCA 0 .Inspection shows that all of the non exotic cases come out with the sametruth value in the two classifications, and that is of course provable in RCA 0 .Furthermore, the exotic case comes out true in the second classification.THEOREM 10.1. The following is provable in SMAH+ but not in SMAH, evenwith the axiom of constructibility. An instance of the Template holds if and onlyif in that instance, “infinite” is replaced by “arbitrarily large finite”.11. FINITE INCOMPLETENESS.Here we present some Fixed Point Propositions, involving operators onsubsets of Q k , where Q is the rationals. These Propositions cannot be proved inZFC. This development leads to a finite form that is explicitly Π 0 1.For more details, including further results involving much largercardinals, see Friedman 2009.We caution the reader that this is intensively ongoing research, which hasnot been published. We expect a submission for publication by early 2010.We say that R ⊆ Q k × Q k is strictly dominating if and only if R(x,y) →max(x) < max(y). We say that E ⊆ Q k is order invariant if and only if membership