Orthogonality between two stably embedded definable sets is preserved under the addition of constants.

We prove several results about groups of finite Morley rank without unipotent p-torsion: p-torsion always occurs inside tori, Sylow p-subgroups are conjugate, and p is not the minimal prime divisor of our approximation to the "Weyl group". These results are quickly finding extensive applications within the classification project.

The problem of the existence of a universal structure omitting a finite set of forbidden substructures is reducible to the corresponding problem in the category of graphs with a vertex coloring by two colors. It is not known whether this problem reduces further to the category of ordinary graphs. It is also not known whether these problems are decidable.

We show that all QE rings of prime power characteristic are constructed in a straightforward way out of three components: a filtered Boolean power of a finite field, a nilpotent Jacobson radical, and the ring Z p n or the Witt ring W 2 (F 4 ) (which is the characteristic four analogue of the Galois field with four elements).

We study definable irreducible actions of SL₂(K) on an abelian group of Morley rank ≤ 3rk(K) and prove they are rational representations of the group.

THEOREM. The Jacobson radical of an $\aleph_0$-categorical associative ring is nilpotent.

THEOREM. The Jacobson radical of an ℵ 0 -categorical associative ring is nilpotent.

We analyze the two cardinal properties of definable sets in homogeneous graphs.

We prove the hereditary undecidability of the L t theories of: (1) torsion-free Hausdorff topological abelian groups; (2) locally pure Hausdorff topological abelian groups.

Borovik proposed an axiomatic treatment of Morley rank in groups, later modified by Poizat, who showed that in the context of groups the resulting notion of rank provides a characterization of groups of finite Morley rank [2]. (This result makes use of ideas of Lascar, which it encapsulates in a neat way.) These axioms form the basis of the algebraic treatment of groups of finite Morley rank undertaken in [1].There are, however, ranked structures, i.e., structures on which a Borovik-Poizat rank (...) function is defined, which are not ℵ0-stable [1, p. 376]. In [2, p. 9] Poizat raised the issue of the relationship between this notion of rank and stability theory in the following terms: “… ungroupede Borovik est une structure stable, alors qu'un univers rangé n'a aucune raison de l'être …” (emphasis added). Nonetheless, we will prove the following:Theorem 1.1.A ranked structure is superstable.An example of a non-ℵ0-stable structure with Borovik-Poizat rank 2 is given in [1, p. 376]. Furthermore, it appears that this example can be modified in a straightforward way to give ℵ0-stable structures of Borovik-Poizat rank 2 in which the Morley rank is any countable ordinal (which would refute a claim of [1, p. 373, proof of C.4]). We have not checked the details. This does not leave much room for strenghthenings of our theorem. On the other hand, the proof of Theorem 1.1 does give a finite bound for the heights of certain trees of definable sets related to unsuperstability, as we will see in Section 5. (shrink)

In this paper the following theorem is proved regarding groups of finite Morley rank which are perfect central extensions of quasisimple algebraic groups.Theorem1.Let G be a perfect group of finite Morley rank and let C0be a definable central subgroup of G such that G/C0is a universal linear algebraic group over an algebraically closed field; that is G is a perfect central extension of finite Morley rank of a universal linear algebraic group. Then C0= 1.Contrary to an impression which exists in (...) some circles, the center of the universal extension of a simple algebraic group, as an abstract group, is not finite in general. Thus the finite Morley rank assumption cannot be omitted.Corollary1.Let G be a perfect group of finite Morley rank such that G/Z(G) is a quasisimple algebraic group. Then G is an algebraic group. In particular, Z(G) is finite([4],Section27.5).An understanding of central extensions of quasisimple linear algebraic groups which are groups of finite Morley rank is necessary for the classification oftamesimpleK*-groupsof finite Morley rank, which constitutes an approach to the Cherlin-Zil’ber conjecture. For this reason the theorem above and its corollary were proven in [1] (Theorems 4.1 and 4.2) under the assumption oftameness, which simplifies the argument considerably. The result of the present paper shows that this assumption can be dropped. The main line of argument is parallel to that in [1]; the absence of the tameness assumption will be countered by a model-theoretic result and results fromK-theory. The model-theoretic result places limitations on definability in stable fields, and may possibly be relevant to eliminating certain other uses of tameness. (shrink)

We classify countable metrically homogeneous graphs of diameter 3.

We classify countable metrically homogeneous graphs of diameter 3.

We deal with two forms of the "uniqueness cases" in the classification of large simple K*-groups of finite Morley rank of odd type, where large means the 2-rank m2 is at least three. This substantially extends results known for even larger groups having Prüfer 2-rank at least three, so as to cover the two groups PSp 4 and G 2. With an eye towards more distant developments, we carry out this analysis for L*-groups, a context which is substantially broader than (...) the K* setting. (shrink)

The theory of lower previsions is designed around the principles of coherence and sure-loss avoidance, thus steers clear of all the updating anomalies highlighted in Gong and Meng's "Judicious Judgment Meets Unsettling Updating: Dilation, Sure Loss, and Simpson's Paradox" except dilation. In fact, the traditional problem with the theory of imprecise probability is that coherent inference is too complicated rather than unsettling. Progress has been made simplifying coherent inference by demoting sets of probabilities from fundamental building blocks to secondary representations (...) that are derived or discarded as needed. (shrink)

In this paper, we defend the view that Spinoza is committed to allowing for the existence of a plurality of possible worlds, that his necessitarianism is merely moderate, not strict enough to exclude the possibility of other worlds. To show that evidence for attributing strict necessitarianism to Spinoza is lacking, we shall concentrate on Don Garrett's article, “Spinoza's Necessitarianism,” in the conviction that his case for attributing strict necessitarianism to Spinoza is the strongest one available.

A comprehensive introduction to current research on the psychology of concept formation and use.

Gregory T. Doolan provides here the first detailed consideration of the divine ideas as causal principles. He examines Thomas Aquinas's philosophical doctrine of the divine ideas and convincingly argues that it is an essential element of his metaphysics. According to Thomas, the ideas in the mind of God are not only principles of his knowledge, but they are productive principles as well. In this role, God's ideas act as exemplars for things that he creates. As Doolan shows, this theory (...) of exemplarism is an integral part of Thomas's account of the existence and order of the created universe. It also accounts in part for the freedom of God's creative act. The volume begins with an introduction to Thomas's doctrine of exemplarism and then addresses his arguments for the existence of exemplar ideas within the mind of God. Having established the existence of divine ideas, Doolan considers how Thomas reconciles their multiplicity with God's simplicity. The work identifies the various things for which Thomas posits divine ideas. As Doolan is careful to show, Thomas does not consider all of these ideas to be exemplars because not all of them act as causal principles. After identifying the ideas that do act in this way, Doolan considers how such exemplars can be causes of natural things without compromising the causality of natural agents. The volume culminates with a consideration of the role that the divine ideas play within the structure of Thomas's theory of participation. ABOUT THE AUTHOR: Gregory T. Doolan, assistant professor of philosophy at The Catholic University of America, is a contributor to Wisdom's Apprentice: Thomistic Essays in Honor of Lawrence Dewan, O.P., and Reassessing the Liberal State: Reading Maritain's Man and the State. This is his first book. (shrink)