-definability at higher cardinals: Thin sets, almost disjoint families and long well-orders

Abstract

Given an uncountable cardinal $\kappa $ , we consider the question of whether subsets of the power set of $\kappa $ that are usually constructed with the help of the axiom of choice are definable by $\Sigma _1$ -formulas that only use the cardinal $\kappa $ and sets of hereditary cardinality less than $\kappa $ as parameters. For limits of measurable cardinals, we prove a perfect set theorem for sets definable in this way and use it to generalize two classical nondefinability results to higher cardinals. First, we show that a classical result of Mathias on the complexity of maximal almost disjoint families of sets of natural numbers can be generalized to measurable limits of measurables. Second, we prove that for a limit of countably many measurable cardinals, the existence of a simply definable well-ordering of subsets of $\kappa $ of length at least $\kappa ^+$ implies the existence of a projective well-ordering of the reals. In addition, we determine the exact consistency strength of the nonexistence of $\Sigma _1$ -definitions of certain objects at singular strong limit cardinals. Finally, we show that both large cardinal assumptions and forcing axioms cause analogs of these statements to hold at the first uncountable cardinal $\omega _1$ .

Reinhardt cardinals and iterates of V

“…Regarding wellorders, first, in §3.1, we consider cardinals κ such that there is a measurable cardinal µ < κ and κ is µ-steady. Under these assumptions, in Theorem 3.2, we establish some restrictions on wellordered subsets of P(κ) of cardinality > κ which are Σ 1 definable in certain parameters; these results are minor refinements of results in [9] and [11]. The main new fact here is that there is no Σ 1 (V µ ∪ {κ}) injection from κ + into P(κ), even if cof(κ) = ω; this was proved under the added assumption that cof(κ) > ω in [9, Theorem 7.1].…”

“…The work relates particularly to that in Lücke and Müller [9], Lücke, Schindler and Schlicht [10], and Lücke and Schlicht [11]; in particular, we answer some of the questions posed in [9].…”

“…In §3.2 we consider cardinals κ which are limits of measurables. Here we answer [9,Questions 10.3,10.4], by showing that in this context, there is no set D ⊆ P(κ) and wellorder < * of D such that D has cardinality > κ and D, < * are Σ 1 (H κ ∪ {κ})-definable. (Lücke and Müller already established some results in this direction in [9, Theorem 1.4,Corollary 7.4]. )…”

“…In §6 we answer [9,Question 10.1], which asked whether, if κ is weakly compact and Σ 1 (H κ ∪ {κ}) has a certain perfect embedding property, there must be a (proper class) inner model satisfying "there is a weakly compact limit of measurable cardinals". In Theorem 6.1 we show the answer is "yes" (the "perfect embedding property" is clarified in the theorem's statement).…”