Predicative fragments of Frege arithmetic

Bulletin of Symbolic Logic 10 (2):153-174 (2004)
  Copy   BIBTEX

Abstract

Frege Arithmetic (FA) is the second-order theory whose sole non-logical axiom is Hume’s Principle, which says that the number of F s is identical to the number of Gs if and only if the F s and the Gs can be one-to-one correlated. According to Frege’s Theorem, FA and some natural definitions imply all of second-order Peano Arithmetic. This paper distinguishes two dimensions of impredicativity involved in FA—one having to do with Hume’s Principle, the other, with the underlying second-order logic—and investigates how much of Frege’s Theorem goes through in various partially predicative fragments of FA. Theorem 1 shows that almost everything goes through, the most important exception being the axiom that every natural number has a successor. Theorem 2 shows that the Successor Axiom cannot be proved in the theories that are predicative in either dimension.

Links

PhilArchive



    Upload a copy of this work     Papers currently archived: 92,227

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

Ramified Frege Arithmetic.Richard G. Heck - 2011 - Journal of Philosophical Logic 40 (6):715-735.
Finitude and Hume's Principle.Richard G. Heck Jr - 1997 - Journal of Philosophical Logic 26 (6):589 - 617.

Analytics

Added to PP
2009-01-28

Downloads
246 (#83,169)

6 months
35 (#101,816)

Historical graph of downloads
How can I increase my downloads?

Author's Profile

Øystein Linnebo
University of Oslo

Citations of this work

Properties and the Interpretation of Second-Order Logic.B. Hale - 2013 - Philosophia Mathematica 21 (2):133-156.
Impredicative Identity Criteria.Leon Horsten - 2010 - Philosophy and Phenomenological Research 80 (2):411-439.
Which abstraction principles are acceptable? Some limitative results.Øystein Linnebo & Gabriel Uzquiano - 2009 - British Journal for the Philosophy of Science 60 (2):239-252.

View all 36 citations / Add more citations

References found in this work

Frege's conception of numbers as objects.Crispin Wright - 1983 - [Aberdeen]: Aberdeen University Press.
Frege’s Conception of Numbers as Objects.Crispin Wright - 1983 - Critical Philosophy 1 (1):97.
Nominalist platonism.George Boolos - 1985 - Philosophical Review 94 (3):327-344.
Parts of Classes.Michael Potter - 1993 - Philosophical Quarterly 43 (172):362-366.
Logic, Logic and Logic.George Boolos & Richard C. Jeffrey - 1998 - Studia Logica 66 (3):428-432.

View all 34 references / Add more references