Abstraction and set theory. (English) Zbl 1014.03014

Hale, Bob

Notre Dame J. Formal Logic 41, No. 4, 379-398 (2000).

Summary: The neo-Fregean program in the philosophy of mathematics seeks a foundation for a substantial part of mathematics in abstraction principles – for example, Hume’s Principle: The number of \(F\)s = the number of \(G\)s iff the \(F\)s and \(G\)s correspond one-one – which can be regarded as implicitly definitional of fundamental mathematical concepts – for example, cardinal number. This paper considers what kind of abstraction principle might serve as the basis for a neo-Fregean set theory. Following a brief review of the main difficulties confronting the most widely discussed proposal to date – replacing Frege’s inconsistent Basic Law V by Boolos’s New V which restricts concepts whose extensions obey the principle of extensionality to those which are small in the sense of being smaller than the universe – the paper canvasses an alternative way of implementing the limitation of size idea and explores the kind of restrictions which would be required for it to avoid collapse.

MSC:

03A05	Philosophical and critical aspects of logic and foundations
00A30	Philosophy of mathematics
03E99	Set theory

Keywords:

sortal concept; definiteness; neo-Fregean program; abstraction principles; neo-Fregean set theory; limitation of size

References:

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