Successor Cardinals Research Articles

Double jumps of minimal degrees over cardinals

Jockusch and Posner [4] showed that every minimal ω-degree is GL2. This is achieved by exhibiting a function f recursive in 0′ which dominates every function of minimal ω-degree. The function f has the peculiar property that for every s, f(s) is defined after a search (using 0′) over the power set of Ls (Gödel's constructible hierarchy at the level s). It can be seen that a function defined in a similar manner over an infinite successor cardinal k will not be a total function, since for example if k = ρ+, then f(ρ) will not be defined until after all the subsets of ρ have been examined, and this will take at least k steps. The following questions then naturally arise: (i) For successor cardinals k, is there a function dominating every set of minimal k-degree? (ii) For arbitrary cardinals k, is every minimal k-degree GL2 (i.e. b″ = (b ∨ 0′) for b of minimal k-degree)? In this paper we answer (i) in the negative and provide a positive answer to (ii), assuming V = L. We show in fact that if k is a successor cardinal and h ≤k 0′, then there is a function of minimal k-degree below 0′ not dominated by h (Theorem 1). This implies that any refinement of the function f described above will not be able to remove the difficulties encountered. On the other hand, we introduce the notion of ‘strong domination’ to provide a positive answer to (ii) (Theorem 2 and Corollary 1). We end this paper by indicating that for limit cardinals k, there is a function below 0′ dominating every function of minimal k-degree.

Jonsson algebras in successor cardinals

We shall show here that in many successor cardinals λ, there is a Jonsson algebra (in other words Jn(λ), or λ is not a Jonsson cardinal). In connection with this we show that, e.g., for every ultrafilterD over ω, in (ωω, <)ω/D there is no increasing sequence of length\(\aleph _{(2^\aleph 0)^ + } \). On Jonsson algebras see e.g. [1]; for successor λ+ = 2λ there is a Jonsson algebra, (λ)⇒Jn(λ+) (due to Chang, Erdos and Hajnal) and even in\(2^{\aleph _\alpha } = \aleph _{(\alpha + n)} \) ([3]). We give here a method to prove, e.g., (λω+1) when\(2^{\aleph _\alpha } \leqq \aleph _{(\omega + 1)} and Jn(2^{\aleph _0 } ) when 2^{\aleph _0 } = \aleph _{\alpha + 1,} \alpha< \omega _1 \); and similar results for higher cardinals.