Ineffability within the limits of abstraction alone

The purpose of this article is to assess the prospects for a Scottish neo-logicist foundation for a set theory. We show how to reformulate a key aspect of our set theory as a neo-logicist abstraction principle. That puts the enterprise on the neo-logicist map, and allows us to assess its prospects, both as a mathematical theory in its own right and in terms of the foundational role that has been advertised for set theory. On the positive side, we show that our abstraction based theory can be modified to yield much of ordinary mathematics, indeed everything needed to recapture all branches of mathematics short of set theory itself. However, our conclusions are mostly negative. The theory will fall far short of the power of ordinary Zermelo-Fraenkel set theory. It is consistent that our set theory has models that are relatively small, smaller than the first cardinal with an uncountable index. More important, there is a strong tension between the idea that the iterative hierarchy is somehow ineffable, or indefinitely extensible, and the neo-logicist theme of capturing mathematical theories with abstraction principles.