my forty years on his shoulders - Department of Mathematics

49in E depends only on the order type of x ∈ Q k . We say that R ⊆ Q k × Q k is orderinvariant if and only if R is order invariant as a subset of Q 2k . For A ⊆ Q k , wewrite R[A] for the image of A under R.We write SDOI(Q k ) for the family of all strictly dominating order invariantR ⊆ Q k × Q k .The upper shift of x ∈ Q k , is defined as us(x) = x+1 if x ≥ 0; x if x < 0. For x∈ Q k , us(x) is obtained from x by applying us coordinatewise. For A ⊆ Q k , wedefine us(A) = {us(x): x ∈ A}.For A ⊆ Q k , write cube(A,0) for the least set V k such that A ⊆ V k ∧ 0 ∈ V.We use \ for set theoretic difference.UPPER SHIFT FIXED POINT PROPOSITION. For all R ∈ SDOI(Q k ), someA = cube(A,0)\R[A] contains us(A).THEOREM 11.1. The Upper Shift Fixed Point Proposition is provable inSUB+ but not in any consistent fragment of SUB. It is provably equivalent toCon(SUB), in WKL 0 .Here SUB+ = ZFC + "for all k there exists a k-subtle cardinal". SUB = ZFC+ {there exists a k-subtle cardinal} k . For the definition of k-subtle cardinal, due toJ. Baumgartner, see Friedman 2001. WKL0 is one of the principal five systems ofReverse Mathematics. See Simpson 1999.SEQUENTIAL UPPER SHIFT FIXED POINT PROPOSITION. For all R ∈SDOI(Q k ), there exist finite A 1 ,A 2 ,... ⊆ Q k such that for all i ≥ 1, A i ∪ us(A i ) ⊆ A i+1= cube(A i+1 ,0).