Amending Frege’s Grundgesetze der Arithmetik

Frege’s Grundgesetze der Arithmetik is formally inconsistent. This system is, except for minor differences, second-order logic together with an abstraction operator governed by Frege’s Axiom V. A few years ago, Richard Heck showed that the ramified predicative second-order fragment of the Grundgesetze is consistent. In this paper, we show that the above fragment augmented with the axiom of reducibility for concepts true of only finitely many individuals is still consistent, and that elementary Peano arithmetic (and more) is interpretable in this extended system.

[1]  Fernando Ferreira.  Amending Frege's Grundgesetze der Arithmetik (draft) , 2002 .

[2]  R. Dedekind.  Essays on the theory of numbers , 1963 .

[3]  G. B. M..  Principia Mathematica , 1911, Nature.

[4]  Richard G. Heck.  Finitude and Hume’s Principle , 1997, J. Philos. Log..

[5]  Patrick Suppes,et al.  Introduction To Logic , 1958 .

[6]  S. Feferman,et al.  Challenges to Predicative Foundations of Arithmetic , 2000 .

[7]  Crispin Wright.  Frege's conception of numbers as objects , 1983 .

[8]  John P. Burgess,et al.  Predicative Logic and Formal Arithmetic , 1998, Notre Dame J. Formal Log..

[9]  R. Dedekind,et al.  Was sind und was sollen die Zahlen , 1961 .

[10]  Crispin Wright,et al.  Frege's conception of numbers as objects , 1984 .

[11]  C. Parsons.  Developing arithmetic in set theory without infinity: some historical remarks , 1987 .

[12]  Crispin Wright,et al.  FREGE: PHILOSOPHY OF MATHEMATICS , 1995 .

[13]  Geoffrey Hellman,et al.  Predicative foundations of arithmetic , 1995, J. Philos. Log..

[14]  B. Russell,et al.  Principia Mathematica Vol. I , 1910 .

[15]  Fernando Ferreira,et al.  Groundwork for weak analysis , 2002, Journal of Symbolic Logic.

[16]  Richard G. Heck.  Frege's Theorem: An Introduction , 1999 .

[17]  George Boolos,et al.  Computability and logic , 1974 .

[18]  Richard G. Heck.  The consistency of predicative fragments of Frege's Grundgesetze der Arithmetik , 1996 .

[19]  TERENCE PARSONS.  On the consistency of the first-order portion of Frege's logical system , 1987, Notre Dame J. Formal Log..

[20]  John P. Burgess,et al.  On a Consistent Subsystem of Frege's Grundgesetze , 1998, Notre Dame J. Formal Log..

[21]  Crispin Wright,et al.  Is Hume's Principle Analytic? , 1999, Notre Dame J. Formal Log..

[22]  Richard G. Heck Jnr.  The Consistency of predicative fragments of frege’s grundgesetze der arithmetik , 1996 .

[23]  I. Levi,et al.  Making It Explicit , 1994 .

[24]  Solomon Feferman,et al.  Why a Little Bit Goes a Long Way: Logical Foundations of Scientifically Applicable Mathematics , 1992, PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association.

[25]  Stephen G. Simpson,et al.  Subsystems of second order arithmetic , 1999, Perspectives in mathematical logic.

[26]  R. Goodstein,et al.  The Basic Laws of Arithmetic , 1966, The Mathematical Gazette.

[27]  Fernando Ferreira,et al.  On the Consistency of the Δ11-CA Fragment of Frege's Grundgesetze , 2002, J. Philos. Log..

[28]  Solomon Feferman,et al.  Infinity in Mathematics: Is Cantor Necessary? , 1989 .

[29]  Jack Kaminsky.  Making it Explicit , 1999 .