Notre Dame Journal of Formal Logic

Abstraction and Set Theory

Bob Hale


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.

Article information

Notre Dame J. Formal Logic, Volume 41, Number 4 (2000), 379-398.

First available in Project Euclid: 26 November 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03A05: Philosophical and critical {For philosophy of mathematics, see also 00A30}
Secondary: 00A30: Philosophy of mathematics [See also 03A05]

abstraction principle set limitation of size sortal concept definiteness


Hale, Bob. Abstraction and Set Theory. Notre Dame J. Formal Logic 41 (2000), no. 4, 379--398. doi:10.1305/ndjfl/1038336882.

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