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Thomas William Barrett
University of California at Santa Barbara
  1. Theories Are Not Partially Ordered.Thomas William Barrett & Hans Halvorson - manuscript
    This paper presents a simple example of first-order theories T1 and T2 such that i) T1 can be embedded in T2 and vice versa, ii) T1 posits all of the structure of T2 and vice versa, but iii) T1 and T2 are not equivalent. This shows that theories lack both the Cantor-Bernstein and co-Cantor-Bernstein properties and are neither partially ordered by the relation 'is embeddable in' nor by 'posits all of the structure of'. In addition, these results clarify the overall (...)
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  2. What Do Symmetries Tell Us About Structure?Thomas William Barrett - 2017 - Philosophy of Science (4):617-639.
    Mathematicians, physicists, and philosophers of physics often look to the symmetries of an object for insight into the structure and constitution of the object. My aim in this paper is to explain why this practice is successful. In order to do so, I present a collection of results that are closely related to (and in a sense, generalizations of) Beth’s and Svenonius’ theorems.
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  3. Extension, Translation, and the Cantor-Bernstein Property.Thomas William Barrett & Hans Halvorson - manuscript
    The purpose of this paper is to examine in detail a particularly interesting pair of first-order theories. In addition to clarifying the overall geography of notions of equivalence between theories, this simple example yields two surprising conclusions about the relationships that theories might bear to one another. In brief, we see that theories lack both the Cantor-Bernstein and co-Cantor-Bernstein properties.
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