## Notre Dame Journal of Formal Logic

### Abstraction and Set Theory

Bob Hale

#### Abstract

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

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

Dates
First available in Project Euclid: 26 November 2002

https://projecteuclid.org/euclid.ndjfl/1038336882

Digital Object Identifier
doi:10.1305/ndjfl/1038336882

Mathematical Reviews number (MathSciNet)
MR1963488

Zentralblatt MATH identifier
1014.03014

#### Citation

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

#### References

• Boolos, G., “Is Hume's Principle analytic?” pp. 301–15 in Logic, Logic, and Logic, edited by R. Jeffrey, Harvard University Press, Cambridge, 1998.
• Boolos, G., "Iteration again", pp. 88–104 in Logic, Logic, and Logic, edited by R. Jeffrey, Harvard University Press, Cambridge, 1998.
• Boolos, G., "Saving F"rege from contradiction, pp. 171–82 in Logic, Logic, and Logic, edited by R. Jeffrey, Harvard University Press, Cambridge, 1998.
• Cook, R. T., "The state of the economy: N"eo-logicism and inflation, Philosophia Mathematica. Series 3, vol. 10 (2002), pp. 43–66.
• Field, H., Realism, Mathematics and Modality, Basil Blackwell Inc., New York, 1989.
• Field, H. H., Science without Numbers. A Defence of Nominalism, Princeton University Press, Princeton, 1980.
• Frege, G., The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number, Harper & Brothers, New York, 1960.
• Hale, B., "Reals by abstraction", Philosophia Mathematica, vol. 8 (2000), pp. 100–123. Reprinted in [hw01?], pp. 399–420.
• Hale, B., and C. Wright, "Implicit definition and the a priori", pp. 286–319 in New Essays on the A Priori, edited by P. Boghossian and C. Peacocke, Oxford University Press, Oxford, 2000. Reprinted in [hw01?], pp. 117–50.
• Hale, B., and C. Wright, The Reason's Proper Study, Oxford University Press, Oxford, 2001.
• Shapiro, S., and A. Weir, "New V", ZF and abstraction, Philosophia Mathematica, vol. 7 (1999), pp. 293–321. The George Boolos Memorial Symposium (Notre Dame, IN, 1998).
• Wright, C., "The philosophical significance of F"rege's Theorem, pp. 201–45 in Language, Thought, and Logic. Essays in honour of Michael Dummett, edited by R. G. Heck, Jr., Oxford University Press, Oxford, 1997. Reprinted in [hw01?], pp. 272–306.
• Wright, C., “Is Hume's Principle analytic?” Notre Dame Journal of Formal Logic, vol. 40 (1999), pp. 6–30. Special issue in honor and memory of George S. Boolos (Notre Dame, IN, 1998). Reprinted in [hw01?], pp. 307–32.