Philosophia Mathematica 24 (1):60-90 (2016)

Roy T. Cook
University of Minnesota
Gottlob Frege defined cardinal numbers in terms of value-ranges governed by the inconsistent Basic Law V. Neo-logicists have revived something like Frege's original project by introducing cardinal numbers as primitive objects, governed by Hume's Principle. A neo-logicist foundation for set theory, however, requires a consistent theory of value-ranges of some sort. Thus, it is natural to ask whether we can reconstruct the cardinal numbers by retaining Frege's definition and adopting an alternative consistent principle governing value-ranges. Given some natural assumptions regarding what an acceptable neo-logicistic theory of value-ranges might look like, successfully implementing this alternative approach is impossible
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DOI 10.1093/philmat/nkv029
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Logic, Logic, and Logic.George Boolos - 1998 - Harvard University Press.
The Limits of Abstraction.Kit Fine - 2002 - Oxford University Press.
Frege's Theorem.Richard G. Heck - 2011 - Clarendon Press.
Frege’s Theorem.Richard Heck - 1999 - The Harvard Review of Philosophy 7 (1):56-73.

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