Ramified Frege Arithmetic

AbstractØystein Linnebo has recently shown that the existence of successors cannot be proven in predicative Frege arithmetic, using Frege’s definitions of arithmetical notions. By contrast, it is shown here that the existence of successor can be proven in ramified predicative Frege arithmetic.

[1]  Richard G. Heck,et al.  The development of arithmetic in Frege's Grundgesetze der arithmetik , 1993, Journal of Symbolic Logic.

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

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

[4]  Richard G. Heck,et al.  Cardinality, Counting, and Equinumerosity , 2000, Notre Dame J. Formal Log..

[5]  Petr Hájek,et al.  Metamathematics of First-Order Arithmetic , 1993, Perspectives in mathematical logic.

[6]  Richard G. Heck.  Language, thought, and logic : essays in honour of Michael Dummett , 1997 .

[7]  Richard G. Heck.  Definition by Induction in Frege's Grundgesetze der Arithmetik , 1995 .

[8]  Wooster Woodruff Beman,et al.  Essays on the theory of mumbers : I. Continuity and irrational numbers : II. The nature and meaning of numbers : Authorized translation by Wooster Woodruff Beman , 1963 .

[9]  Richard G. Heck,et al.  Die Grundlagen der Arithmetik §§82–83 , 2010 .

[10]  William Demopoulos,et al.  Frege’s Philosophy of Mathematics , 1997 .

[11]  George Boolos,et al.  Logic, Logic, and Logic , 2000 .

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

[13]  George Boolos,et al.  Frege's Theorem and the Peano Postulates , 1995, Bulletin of Symbolic Logic.

[14]  Fernando Ferreira,et al.  Amending Frege’s Grundgesetze der Arithmetik , 2005, Synthese.

[15]  Richard G. Heck.  The Julius Caesar objection , 1997 .

[16]  Øystein Linnebo,et al.  Predicative fragments of Frege Arithmetic , 2004, Bull. Symb. Log..

[17]  J. Heijenoort.  Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought GOTTLOB FREGE(1879) , 1970 .

[18]  Bob Hale,et al.  The Reason's Proper Study , 2001 .

[19]  G. Frege.  Grundgesetze der Arithmetik , 1893 .

[20]  George Boolos,et al.  Reading the Begriffsschrift , 1985 .

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

[22]  J. E. Tiles,et al.  Frege's Conception of Numbers as Objects , 1984 .