A Note on Closed Degrees of Difficulty of the Medvedev Lattice

Abstract

We consider some nonprincipal filters of the Medvedev lattice. We prove that the filter generated by the nonzero closed degrees of difficulty is not principal and we compare this filter, with respect to inclusion, with some other filters of the lattice. All the filters considered in this paper are disjoint from the prime ideal generated by the dense degrees of difficulty.Mathematics Subject Classification: 03D30.

Topological aspects of the Medvedev lattice

“…That strict inclusion holds in Corollary 4.7 has been shown in [2]. The proof given there uses another interesting filter of the Medvedev lattice, namely the filter generated by the nonzero discrete M-degrees.…”

“…For a different construction see [2,Corollary 3.7]. Theorem 6.3 There exists a countable mass problem A, of non-zero degree, such that A is not dense in any computable perfect tree, and A ≥ C for any closed C unless C contains computable functions.…”

“…Let now M /F denote the quotient lattice obtained dividing M by the filter F. It is known ( [2]) that the cardinality of M /F is 2 2 ℵ 0 . The equivalence class of a degree A will be denoted by A /F .…”

“…Thus if A has nontotal e-degree then E A does not contain any countable mass problem. The degrees of enumerability turn out to be useful to derive easy observations on I and F. We have already recalled that F Cl ⊆ F Di ( [2]). The converse inclusion does not hold either:…”

Comparing the degrees of enumerability and the closed Medvedev degrees

“…As observed in [3], C A is closed, deg s (C A ) ≤ s E A , and, if A is immune (meaning that A has no infinite r.e. subset), then deg s (C A ) = 0.…”