This paper is a culmination of our new foundations for recursive analysis through recursive topology as reported in Kalantari and Welch 125; 98 87). While in those papers we developed groundwork for an approach to point free analysis and applied recursion theory, in this paper we blend techniques of recursion theory with those of topology to establish new findings. We present several new techniques different from existing ones which yield interesting results. Incidental to our work is a unifying explanation of (...) various schools of study for recursive analysis. (shrink)

In this paper we develop a point-free approach to the study of topological spaces and functions on them, establish platforms for both and present some findings on recursive points. In the first sections of the paper, we obtain conditions under which our approach leads to the generation of ideal objects with which mathematicians work. Next, we apply the effective version of our approach to the real numbers, and make exact connections to the classical approach to recursive reals. In the succeeding (...) sections of the paper, we introduce machinery to produce functions on topological spaces and find succinct conditions which will be effectivized in our sequel. (shrink)

In this paper we continue our work of Kalantari and Welch . There we introduced machinery to produce a point-free approach to points and functions on topological spaces and found conditions for both which lend themselves to effectivization. While we studied recursive points in that paper, here, we present two useful classes of recursive functions on topological spaces, apply them to the reals, and find precise accounting for the nature of the properties of some examples that exist in the literature. (...) We end with a construction of a recursive function on a small subset of the unit interval which is strongly nonextendible. (shrink)

Since James defined emotion as consciousness of bodily reactions and Cannon and others detailed the nature of these reactions, there has been an increasing tendency among behaviorists to equate emotions with visceral reactions and to neglect some of the genetic and adaptive aspects of emotion which had been discussed by Darwin.

We develop the concepts of recursively nowhere dense sets and sets that are recursively of first category and study closed sets of points in light of Baire's Category Theorem. Our theorems are primarily concerned with exdomains of recursive quantum functions and hence with avoidable points . An avoidance function is a recursive function which can be used to expel avoidable points from domains of recursive quantum functions. We define an avoidable set of points to be an arbitrary subset of the (...) avoidable points of a single avoidance function, and we study an effective union of such sets which we call a piecemeal avoidable set. We note that each of ‘the set of recursive points’ and ‘the set of avoidable points’ is of first category but not recursively of first category. We show an exdomain exists which is recursively nowhere dense, as well as one that is nowhere dense but not recursively nowhere dense. After establishing that every exdomain is recursively of first category, we prove that given any fixed exdomain, there is another exdomain, which while dense in the underlying space, is disjoint from the fixed exdomain. Finally, we show how to build a recursive sequence of recursive quantum functions that have mutually disjoint, dense exdomains. (shrink)

In this paper we first give a variant of a theorem of Jockusch–Lewis– Remmel on existence of a computable, degree-preserving homeomorphism between a bounded strong ${\Pi^0_2}$ class and a bounded ${\Pi^0_1}$ class in 2 ω . Namely, we show that for mathematically common and interesting topological spaces, such as computably presented ${\mathbb{R}^n}$ , we can obtain a similar result where the homeomorphism is in fact the identity mapping. Second, we apply this finding to give a new, priority-free proof of existence (...) of a tree of shadow points computable in 0′. (shrink)

This paper continues our study of computable point-free topological spaces and the metamathematical points in them. For us, a point is the intersection of a sequence of basic open sets with compact and nested closures. We call such a sequence a sharp filter. A function fF from points to points is generated by a function F from basic open sets to basic open sets such that sharp filters map to sharp filters. We restrict our study to functions that have at (...) least all computable points in their domains.We follow Turing's approach in stating that a point is computable if it is the limit of a computable sharp filter; we then define the Turing degree Deg of a general point x in an analogous way. Because of the vagaries of the definition, a result of J. Miller applies and we note that not all points in all our spaces have Turing degrees; but we also show a certain class of points do. We further show that in ℝn all points have Turing degrees and that these degrees are the same as the classical Turing degrees of points defined by other researchers.We also prove the following: For a point x that has a Turing degree and lies either on a computable tree T or in the domain of a computable function fF, there is a sharp filter on T or in dom converging to x and with the same Turing degree as x. Furthermore, all possible Turing degrees occur among the degrees of such points for a given computable function fF or a complete, computable, binary tree T. For each x ∈ dom for which x and fF have Turing degrees, Deg) ≤ Deg. Finally, the Turing degrees of the sharp filters convergent to a given x are closed upward in the partial order of all Turing degrees. (shrink)

Recounts the efforts of Orage, an influential London editor, to prepare America for the teachings of Gurdjieff.

In lieu of an abstract, here is a brief excerpt of the content:The Call for New Theological Reflection on the Sacramental Character of Marriage and the Thought of St. ThomasLawrence J. WelchTheologians across the theological spectrum have called attention to the urgent need for a new reflection on the theological and sacramental character of marriage. Peter Hünermann, known for his strong criticism of magisterial teachings on marriage, and the late Cardinal Carlo Caffarra, known for his equally strong defense of them, (...) have both drawn attention to this need. At an international symposium in September 2015, held in Rome prior to the Ordinary Synod of Bishops on Marriage and the Family, Hünermann gave a presentation entitled "On the Specific Theological Character of the Sacrament of Matrimony."1 The symposium aimed at providing a critical and scholarly discussion of new and constructive perspectives for the Church doctrine on marriage and family life in time for the 2015 synod on the family.2 Some of Hünermann's work from this paper also came to be known from an interview in Commonweal magazine where he summarized some of the main points of his symposium presentation.3 In the interview, he also revealed that before the 2015 synod he [End Page 845] was invited to speak with Pope Francis about the theology of marriage. Understandably, this meeting heightened interest in the German theologian's thought.Hünermann argues that "the specific character of the sacrament of marriage, which is very closely connected to the dying of the marriage bond,... has scarcely been the object of theological reflection, [and] is almost unknown in the consciousness of the bishops." 4 Hünermann appeals to St. Thomas as a way forward for understanding the theological character of marriage. This appeal is situated within a wider examination5 that offers an elaboration and new interpretation of "the specific dogmatic characteristics of the sacrament of matrimony that emerge in today's situation from a critical comparison of the encyclical Casti connubii and the Catholic theological tradition, especially the theology of St. Thomas."6 Sharply critical of the encyclical, Hünermann sees it as having saddled the Church with a legalistic concept of marriage elevated above history that is too far removed from the Catholic theological tradition and St. Thomas's theology of marriage.The year before his death, Cardinal Caffarra gave the 2016 graduation address at the John Paul II Institute for Marriage and Family in Washington, DC.7 Among the challenges he identified for the institute to address was the [End Page 846] need for theological reflection on the sacramental character of marriage. Referring to his earlier work, he pointed out that theology after the Council of Trent "lost the awareness of sacramental ontology, of the real transformation produced by the sacrament in the persons of the spouses [res et sacramentum]."8 Furthermore, according to Caffarra, the "theology of the conjugal covenant was supplanted by the notion of a contract, as a human act by which two persons commit to one another." The sacramental character was thought of as consisting of the assurance of divine graces given to fulfill the contract. Caffarra claimed that, "consequently, the words of Jesus 'What God has joined together, let not man put asunder,' lost all their realistic force. Absolute indissolubility, once the sacrament was perfected [=a ratified, consummated marriage], became essentially indefensible."9The primary interest of this paper is to respond to the call for a renewed theological reflection on the specific character of the sacrament of marriage. I wish to take up Caffarra's suggestion to focus on the sacramental character (res et sacramentum) on the one hand, and on the other hand, Hünermann's suggestion that a kind of ressourcement from St. Thomas Aquinas's theology of marriage can be of great help for a fresh theological reflection on the specific character of the sacrament of marriage. Although I agree with Hünermann that a renewed consideration of the thought of St. Thomas can be a way forward, I differ from him on how it can be. Hünermann notices that St. Thomas speaks of the grace of the sacrament of marriage as a helping grace.10 St. Thomas does indeed speak this way of the... (shrink)

This teaching of Vatican II raises the problem of how to know when the ordinary universal magisterium has proposed some doctrine as definitive. Fr. Francis A. Sullivan has put forth some principles that he claims are important for establishing that a doctrine has been definitively taught. Key to Sullivan’s principles is the claim that the constant consensus of theologians is a sign and a necessary condition for establishing whether the ordinary universal magisterium has taught a particular doctrine as definitive. This (...) essay argues against this claim. It maintains that while the consensus of theologians may be sign of definitive teachings, it is not a necessary condition for those teachings. Lastly, the essay suggests that the current discussion about the definitive, infallible teachings of the ordinary universal magisterium would be greatly aided if it took place in the context of the quest for catholicity in time and if it firmly kept in mind the nature of church as a communion. (shrink)

This paper is a sequel to our [7]. In that paper we constructed a Π1 0 tree of avoidable points. Here we construct a Π1 0 tree of shadow points. This tree is a tree of sharp filters, where a sharp filter is a nested sequence of basic open sets converging to a point. In the construction we assign to each basic open set on the tree an address in 2<ω. One interesting fact is that while our Π1 0 tree (...) of sharp filters (a subtree of Δ<ω) is isomorphic to the tree of addresses (a subtree of 2<ω), the tree of addresses is recursively enumerable but not recursive. To achieve this end we use a finite injury priority argument. (shrink)