Aeolian sands and soils of a Wetland Biosphere Reserve: The Tablas de Daimiel

Three soil profiles were characterized from a sandy area situated in a transitional zone in the “Tablas de Daimiel” UNESCO wetland biosphere reserve (TDNP) in the Mancha Plain (Central Spain). The original thickness of the soil layer in this area was as much as 100 cm, but the combined effect of unbalanced cultivation (including irrigation) and an increasingly dry climate has partially eroded their natural properties, almost leading to a desertification process. The main properties of these soils, classified as Xeropsamments (Soil Survey Staff 2006) or Arenosols (IUSS Working Group WRB 2006), are the dominance of sand in the soil matrix, its basic character, and low soil organic matter and carbonate contents. Scanning electron microscopy of the quartz grain surfaces indicated that the superficial textureswere commonly well preserved and characteristic of specific conditions of aeolian formation. Exoscopy revealed a mixture of sub-angular and rounded quartz morphotypes withwell-preserved mechanical impacts on the grain surfaces. These percussion effects are characteristic of aeolian processes and confirmed that mechanical actions were more significant than chemical processes in their pedogenesis.This micromorphological study of the textural sands in the transitional zone of the TDNP revealed that the (red) sands were transported to this area by wind, probably from material of degraded red soils, and deposited on soil horizons previously developed over marls and limestones. The relict character of these materials and specially their susceptibility to degradation should be considered as a priority argument to preserve this area and introduce management measurements to avoid soil erosion.

Σ2 Induction and infinite injury priority argument, Part I: Maximal sets and the jump operator

The study of recursion theory on models of fragments of Peano arithmetic has hitherto been concentrated on recursively enumerable (r.e.) sets and their degrees (with a few exceptions, such as that in [2] on minimal degrees). The reason for such a concerted effort is clear: priority arguments have occupied a central position in post Friedberg-Muchnik recursion theory, and after almost forty years of intensive development in the subject, they are still the essential tools on which investigations of r.e. sets and their degrees depend. There are two possible approaches to the study within fragments of arithmetic: To give a general analysis of strategies, and identify their proof-theoretic strengths (for example in [6] on infinite injury priority methods), or to consider specific theorems in recursion theory, and, if possible, pinpoint the exact levels of induction provably equivalent to the theorems. The work reported in this paper belongs to the second approach. More precisely, we single out two infinitary injury type constructions of r.e. sets—one concerning maximal sets and the other based on the notion of the jump operator—to be the topics of study.

The density of the meet-inaccessible r. e. degrees

In this paper it is shown that the meet-inaccessible degrees are dense in R. The construction uses an 0′-priority argument. As a consequence, the meet-inaccessible degrees and the meet-accessible degrees give a partition of R into two sets, either of which is a nontrivial dense subset of R and generates R − {0} under joins (thus an automorphism base of R).

Index sets and degrees of unsolvability

The characterization of classes of r.e. sets by their index sets has proved valuable in producing new results about the r.e. sets and degrees. The classic example is Yates' proof [5, Theorem 7] of Sacks' density theorem for r.e. degrees using his classification of {e: We ≤TD) as Σ3(D) whenever D is r.e. Theorem 1 of this paper is a refinement of this index set theorem of Yates which has already proved to have interesting consequences about the r.e. degrees. This theorem was originally announced by Kallibekov [1, Theorem 1]. Kallibekov there proposed a new and ingenious method for doing priority arguments which has also since been used by Kinber [2]. Unfortunately his proof to this particular theorem contains an error. We have a totally different proof using standard techniques which is of independent interest.The proof to Theorem 1 is an infinite injury priority argument. In §1 therefore we give a short summary of the infinite injury priority method. We draw heavily on the exposition of Soare [4] where a complete description of the method is given along with many examples. In §2 we prove the main theorem and also give what we think are the most interesting corollaries to this theorem announced by Kallibekov. In §3 we prove a theorem about Σ3 sets of indices of r.e. sets. This theorem is a strengthening of a theorem of Kinber [2, Theorem 1] which was proved using a modification of Kallibekov's technique. As application, we use our theorem to show that an r.e. set A has supersets of every r.e. degree iff A is not simple.

Non-generable formal languages

A set A of words on a finite alphabet Σ is said to be generable if it is the closure of a computable inductive operator; in particular, if S is a semi-Thue system then the set of words derivable in S is generable. An RE set of words (equivalently, a phrase-structure or type 0 language in the sense of Chomsky [Information and Control 2 (1959), 137–167]) which is non-generable is constructed by means of a finite injury priority argument. The construction is refined to obtain a non-generable set of degree 0′ and, for any degree d, a non-generable set of degree ⩽ d.

An α-finite injury method of the unbounded type

Let α be an admissible ordinal. In this paper we study the structure of the upper semilattice of α-recursively enumerable degrees. Various results about the structure which are of fundamental importance had been obtained during the past two years (Sacks-Simpson [7], Lerman [4], Shore [9]). In particular, the method of finite priority argument of Friedberg and Muchnik was successfully generalized in [7] to an α-finite priority argument to give a solution of Post's problem for all admissible ordinals. We refer the reader to [7] for background material, and we also follow closely the notations used there.Whereas [7] and [4] study priority arguments in which the number of injuries inflicted on a proper initial segment of requirements can be effectively bounded (Lemma 2.3 of [7]), we tackle here priority arguments in which no such bounds exist. To this end, we focus our attention on the fine structure of Lα, much in the fashion of Jensen [2], and show that we can still use a priority argument on an indexing set of requirements just short enough to give us the necessary bounds we seek.

A minimal degree not realizing least possible jump

The least possible jump for a degree of unsolvability a is its join a ∪ 0′ with 0′. Friedberg [1] showed that each degree b ≥ 0′ is the jump of a degree a realizing least possible jump (i.e., satisfying the equation a′ = a ∪ 0′). Sacks (cf. Stillwell [8]) showed that most (in the sense of Lebesgue measure) degrees realize least possible jump. Nevertheless, degrees not realizing least possible jump are easily found (e.g., any degree b ≥ 0′) even among the degrees <0′ (cf. Shoenfield [5]) and the recursively enumerable (r.e.) degrees (cf. Sacks [3]).A degree is called minimal if it is minimal in the natural partial ordering of degrees excluding least element 0. The existence of minimal degrees <0” was first shown by Spector [7]; Sacks [3] succeeded in replacing 0” by 0′ using a priority argument. Yates [9] asked whether all minimal degrees <0′ realize least possible jump after showing that some do by exhibiting minimal degrees below each r.e. degree. Cooper [2] subsequently showed that each degree b > 0′ is the jump of a minimal degree which, as corollary to his method of proof, realizes least possible jump. We show with the aid of a simple combinatorial device applied to a minimal degree construction in the manner of Spector [7] that not all minimal degrees realize least possible jump. We have observed in conjunction with S. B. Cooper and R. Epstein that the new combinatorial device may also be applied to minimal degree constructions in the manner of Sacks [3], Shoenfield [6] or [4] in order to construct minimal degrees <0′ not realizing least possible jump. This answers Yates' question in the negative. Yates [10], however, has been able to draw this as an immediate corollary of the weaker result by carrying out the proof in his new system of prioric games.