David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Ezio Di Nucci
Jack Alan Reynolds
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Journal of Philosophical Logic 25 (6):597 - 615 (1996)
In his new introduction to the 1925 second edition of Principia Mathematica, Russell maintained that by adopting Wittgenstein's idea that a logically perfect language should be extensional mathematical induction could be rectified for finite cardinals without the axiom of reducibility. In an Appendix B, Russell set forth a proof. Godel caught a defect in the proof at *89.16, so that the matter of rectification remained open. Myhill later arrived at a negative result: Principia with extensionality principles and without reducibility cannot recover mathematical induction. The finite cardinals are indefinable in it. This paper shows that while Gödel and Myhill are correct, Russell was not wrong. The 1925 system employs a different grammar than the original Principia. A new proof for *89.16 is given and induction is recovered
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A. P. Hazen & J. M. Davoren (2000). Russell's 1925 Logic. Australasian Journal of Philosophy 78 (4):534 – 556.
Gregory Landini (2013). The Evolution of Principia Mathematica; Bertrand Russell's Manuscripts and Notes for the Second Edition. History and Philosophy of Logic 34 (1):79-97.
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