Many years after the publication of “A Logical Calculus of the Ideas Immanent in Nervous Activity,” Warren McCulloch gave Walter Pitts credit for contributing his knowledge of modular mathematics to their joint project. In 1941 I presented my notions on the flow of information through ranks of neurons to Rashevsky’s seminar in the Committee on Mathematical Biology of the University of Chicago and met Walter Pitts, who then was about seventeen years old. He was working on a mathematical theory of (...) learning and I was much impressed. He was interested in problems of circularity, how to handle regenerative nervous activity in closed loops....For two years Walter and I worked on these problems whose solution depended upon modular mathematics of which I knew nothing, but Walter did.. In this paper, we will fill in some of the details regarding Pitts’s interest in problems of circularity, regenerative activity in closed loops of neurons, and modular mathematics, and the way in which they relate to “A Logical Calculus.”. (shrink)

We extend Morley's Theorem to show that if a theory is κ-p-categorical for some uncountable cardinal κ, it is uncountably categorical. We then discuss ω-p-categoricity and provide examples to show that similar extensions for the Baldwin-Lachlan and Lachlan Theorems are not possible.

We extend Shelah's first many model result to show that an unstable theory has 2 κ many non-permutation group isomorphic models of size κ, where κ is an uncountable regular cardinal.

We extend Shelah's first many model result to show that an unstable theory has 2κmany non-permutation group isomorphic models of size κ, where κ is an uncountable regular cardinal.