We study the complexity of the isomorphism relation on classes of computable structures. We use the notion of FF-reducibility introduced in [9] to show completeness of the isomorphism relation on many familiar classes in the context of all ${\mathrm{\Sigma }}_{1}^{1}$ equivalence relations on hyperarithmetical subsets of ω.

We exploit properties of certain directed graphs, obtained from the families of sets with special effective enumeration properties, to generalize several results in computable model theory to higher levels of the hyperarithmetical hierarchy. Families of sets with such enumeration features were previously built by Selivanov, Goncharov, and Wehner. For a computable successor ordinal α, we transform a countable directed graph into a structure such that has a isomorphic copy if and only if has a computable isomorphic copy.A computable structure is (...) categorical if for all computable isomorphic copies of , there is an isomorphism from onto , which is . We prove that for every computable successor ordinal α, there is a computable, categorical structure, which is not relatively categorical. This generalizes the result of Goncharov that there is a computable, computably categorical structure, which is not relatively computably categorical.An additional relation R on the domain of a computable structure is intrinsically on if in all computable isomorphic copies of , the image of R is . We prove that for every computable successor ordinal α, there is an intrinsically relation on a computable structure, which not relatively intrinsically . This generalizes the result of Manasse that there is an intrinsically computably enumerable relation on a computable structure, which is not relatively intrinsically computably enumerable.The dimension of a structure is the number of computable isomorphic copies, up to isomorphisms. We prove that for every computable successor ordinal α and every n≥1, there is a computable structure with dimension n. This generalizes the result of Goncharov that there is a structure of computable dimension n for every n≥1.Finally, we prove that for every computable successor ordinal α, there is a countable structure with isomorphic copies in just the Turing degrees of sets X such that relative to X is not . In particular, for every finite n, there is a structure with isomorphic copies in exactly the non- Turing degrees. This generalizes the result obtained by Wehner, and independently by Slaman, that there is a structure with isomorphic copies in exactly the nonzero Turing degrees. (shrink)

We characterize Δ20-categoricity in Boolean algebras and linear orderings under some extra effectiveness conditions. We begin with a study of the relativized notion in these structures.

We characterize the structure of computably categorical trees of finite height, and prove that our criterion is both necessary and sufficient. Intuitively, the characterization is easiest to express in terms of isomorphisms of (possibly infinite) trees, but in fact it is equivalent to a Σ03-condition. We show that all trees which are not computably categorical have computable dimension ω. Finally, we prove that for every n≥ 1 in ω, there exists a computable tree of finite height which is δ0n+1-categorical but (...) not δ0n-categorical. (shrink)

In many classes of structures, each computable structure has computable dimension 1 or $\omega$. Nevertheless, Goncharov showed that for each $n < \omega$, there exists a computable structure with computable dimension $n$. In this paper we show that, under one natural definition of relativized computable dimension, no computable structure has finite relativized computable dimension greater than 1.

Let ???? be a computable structure and let R be a new relation on its domain. We establish a necessary and sufficient condition for the existence of a copy ℬ of ???? in which the image of R (¬R, resp.) is simple (immune, resp.) relative to ℬ. We also establish, under certain effectiveness conditions on ???? and R, a necessary and sufficient condition for the existence of a computable copy ℬ of ???? in which the image of R (¬R, resp.) (...) is simple (immune, resp.). (shrink)

This brief essay comments on the several preceding essays analyzing Charles S. McCoy’s thought.

We establish that the isomorphism problem for the classes of computable nilpotent rings, distributive lattices, nilpotent groups, and nilpotent semigroups is \-complete, which is as complicated as possible. The method we use is based on uniform effective interpretations of computable binary relations into computable structures from the corresponding algebraic classes.

This is a previously unpublished manuscript and is the last entry in the annotated bibliography above. It is related to the counter culture articles, the liberation theology and Heidegger articles, and, indeed, to the whole corpus. It is offered here with the intention, and hope, that it will stimulate the reader to look closely at the related articles and by that to turn to the entire corpus. Any one of his articles should have a similar result.

This essay treats Michael Polanyi’s post-critical philosophy and the contributions of post-critical thought to ethics. It discusses the from/to structure of human knowing and heurism and ethics. It argues that virtue, viewed post-critically, is an achievement in community; post-critical thought calls for movement beyond specialization.

This review essay on R. Melvin Keiser's Roots of Relational Ethics: Responsibility in Origin and Maturity in H. Richard Niebuhr surveys selected works about Niebuhr, examines the strengths of Keiser's post-critical treatment of Niebuhr and raises questions about Keiser's views and about Niebuhr.

This is a brief reflection published in the now extinct Oratre Fratres. The consequences of the turning from common Fatherhood and the resulting loss of common brotherhood are as evident today as when this was first written. McCoy was in St. Paul Seminary at the time and was to be ordained in May 1941. He had earned his Ph.D. in Political Science from the University of Chicago in 1938.

As noted in the above bibliography the essay below was originally published in now extinct Continuum, Vol.3, No. 4,. Permission to republish was graciously granted by the original publisher and copyright holder, Justus George Lawler. It is reproduced below in the exact format as the original.But the ideal society is not more real than the ideal gas of physics. Not that the true and the good are to be denied; rather, on the contrary, from the errors and evils that must (...) inevitably arise, we ought to draw lessons in the ways of acting with greater prudence and wisdom. That is the meaning of “ideal” in politics, its meaning from the point of view of action. (shrink)

Paul Tillich and Michael Polanyi had their only face-to-face meeting in Berkeley, in February, 1963. The author reports the circumstances of this conversation, which he arranged and in which he participated, and, on the basis of his participation, offers refelections on the postcritical and fiduciary dimensions in the work of Polanyi and Tillich as a means of identifying similarities and differences in the thought of each.