Abstract
AbstractWe 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 complete 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.
Highlights
We propose a criterion to regard a property of a theory as virtuous: the property must have significant mathematical consequences for the theory
In Subsection 4.6, we provide some examples of the uses of model theory in mathematics, stressing the connections to the hierarchy of proper ties of theories described in Subsection 4.4
We have posited a criterion for evaluating the virtue of a property of theories: whether the property has significant mathematical consequences
Summary
(c) Specify the notion of a formal deduction for these sentences ( b ). I contrast this development with the use offully formalized theories as a tool in mathemat ics. In the latter context the particular axiomatization is irrelevant. I define: a theory T is a collection of sentences in some logic £, which is closed under semantic consequence. This submodel' are to vocabulary is to emphasize firmly Even as late as Shoenfield's classic graduate text [94], the word theory is used for the entire syntactical apparatus: formal language, axioms and rules of inference for the logic, and specific axioms. A logic C is deductively complete if there is a deductive system such that for every
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