Function Chains From Uniform Sigma-1-1 Well Orders

Abstract

Appropriate restrictions may be placed on a Σ 1 1 well-order on V κ where κ is an inaccessible cardinal, so that it gives rise to a function chain when κ is a Mahlo cardinal. Set chains may be defined for all second order formulas. Postulating that the sets in the resulting set chain are stationary yields a powerful new axiom.

Sigma-1-1 Well-Founded Relations and Set Chains

“…Recalling the definition from [7], a WPS on a set S is a binary relation on S satisfying the following axioms: F. For all functions f with domain ω there is an n such that f (n + 1)…”

“…The structure OS κ is defined in [7]; an earlier version, K s κ , may be found in [2]. To review the definition, let L OS be the language with two sorts, Ord and Seq.…”

“…The version in [7] has also the function symbol Rstr(s, α, β) (of sort Seq). The graph of this function has a ∆ 0 definition in the smaller language, namely,…”

“…In a series of papers ( [2], [3], [4], [5], [6], [7]) the author has "constructed" progressively longer chains of stationary sets of inaccessible cardinals. The existence of such chains is independent of ZFC (indeed the existence of an inaccessible cardinal is), so that the construction in fact yields a chain must be postulated.…”

“…In [7] the set chains constructed involve the notion of a Σ 1 1 WPS. A WPS is a well-founded binary relation (WF).…”