posted on 2016-01-22, 00:00authored byJ.T. Baldwin
Abstract. We propose a criterion to regard a property of a theory (in first or second
order logic) as virtuous: the property must have significant mathematical consequences for
the theory (or its models). We then rehearse results of Ajtai. Marek. Magidor. H. Friedman
and Solovay to argue that for second order logic, 'categoricity' has little virtue. For first order
logic, categoricity is trivial: but 'categoricity in power' has enormous structural consequences
for any of the theories satisfying it. The stability hierarchy extends this virtue to other com
plete theories. The interaction of model theory and traditional mathematics is examined by
considering the views of such as Bourbaki. Hrushovski. Kazhdan. and Shelah to flesh out the
argument that the main impact of formal methods on mathematics is using formal definability
to obtain results in 'mainstream' mathematics. Moreover, these methods (e.g. the stability
hierarchy) provide an organization for much mathematics which gives specific content to
dreams of Bourbaki about the architecture of mathematics.