my forty years on his shoulders - Department of Mathematics

45THEOREM 9.5. (Friedman 2005). Theorem 9.3 is provable in ZFC but not withoutthe axiom scheme of replacement. Theorem 9.4 is neither provable nor refutablein ZFC.We can say more.THEOREM 9.6. (Friedman 2005). The existence of the cumulative hierarchy upthrough every countable ordinal is sufficient to prove Theorems 9.1 and 9.3.However, the existence of the cumulative hierarchy up through any suitablydefined countable ordinal is not sufficient to prove Theorem 9.1 or 9.3.DOM: The f:N → N constructible in any given x ⊆ N are eventually dominatedby some g:N → N.THEOREM 9.7. ZFC + Theorem 9.4 implies DOM (Friedman 2005). ZFC + DOMimplies Theorem 9.4 (Debs, Saint Raymond 2007).10. BOOLEAN RELATION THEORY.The principal reference for this section is the forthcoming book Friedman2010.We begin with two examples of statements in BRT of special importance for thetheory.THIN SET THEOREM. Let k ≥ 1 and f:N k → N. There exists an infinite set A ⊆ Nsuch that f[A k ] ≠ N.COMPLEMENTATION THEOREM. Let k ≥ 1 and f:N k → N. Suppose that for allx ∈ N k , f(x) > max(x). There exists an infinite set A ⊆ N such that f[A k ] = N\A.