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In Linsky's The Evolution of Principia Mathematica it is written,

Chapter 6 studies in detail the content of Appendix B, On induction. The appendix consists of a technical proof that even without the axiom of reducibility, a limited form of the principle of mathematical induction can be derived. This proof was found to contain a technical flaw by G¨odel (1944), “Russell’s mathematical logic”. Later John Myhill (1974) followed this up with a proof showing that the project of Appendix B is impossible in principle. The manuscript material presented here allows for a better understanding of these issues. The “mistake” can be traced through the earlier drafts of the appendix, and a sense of the intended project of the appendix can be seen in the manuscript notes “Amended list of propositions.” It appears that the published material in Appendix B was not slipshod or casual, as Myhill suggests, but rather the result of intense efforts, although in the end, the details of the proposed alterations to the logic of the first edition are not clear. A puzzle remains for the reader to solve by careful study of the notes published here.

I am puzzled by the bold portion (my bold) of the quote. My questions are,

  1. What is the “intended project” of which Linsky speaks and how did he get the “sense” of the project from the manuscript notes “Amended list of propositions”?

  2. Is it possible to have some new interpretation of Russell's theory of types so that line 3 of *89.16 is not in error (note that I am not asking whether there is a different proof of *89.16 in the "new" system of Principia Mathematica)? If so, then can some related literature(s) be mentioned?

  • I don't know enough about the Principia Mathematica to give a complete answer, but the "mistake" referred to appears to lie in supposing that mathematical induction is a kind of theorem provable within the formal system of PM. Rather, it is an inference rule, expressible as a second-order axiom. As such, it cannot be proved, which was the project of the appendix, any more than the well-ordering principle (with which it is equivalent) can be proved. – Bumble Nov 21 '15 at 7:32
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    @Bumble - mathematical induction has been proved into the system of Principia Mathematica; the system is a "workable" solution to the foundational problem (as well as e.g.Zermelo set theory). The issue is to prove it without "unacceptable" (form the point of view of the Logicist program axioms, like the Axiom of reducibility. – Mauro ALLEGRANZA Nov 21 '15 at 10:34
  • Can we try to frame the headline here as a question? – Joseph Weissman Jan 5 '16 at 16:40
  • @JosephWeissman: I don't understand. Can you clarify? – user 170039 Jan 5 '16 at 16:44
  • It can help improve the likelihood of getting a great answer if the headline actually states the question (rather than just saying "A question about..." which doesn't really say much about the specific problem you're encountering or what sort of answer you're after) – Joseph Weissman Jan 5 '16 at 17:26
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The intended project of Appendix B of the second edition of Principia Mathematica was to prove the Principle of Induction without using the Axiom of Reducibility. The manuscript of "Amended List of Propositions" consists of repeated attempts at proving this result by considering what notions are needed to define an interval within the series of natural numbers. Presumably, Russell was trying to prove that a counterexample to the Principle of Induction (that is, a case of a property P true of 0 and of n+1 but not of all numbers) can be shown to be impossible. This argument would depend somehow on the interval between 0 and the first number m to which P does not belong.

That was my sense from reading the notes, which repeatedly discuss the definition of intervals. Looking at section 6.8 of my book, you will see that Gregory Landini has described a system of types for the second edition of PM which would allow the proof of 89.16. But I think that system is too strong to be what Russell intended, because the Axiom of Reducibility is easily proved in that system, going against the original project of Appendix B.

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    I could never understand Russell's attitude towards reducibility:"We assume then, that every function is equivalent, for all its values, to some predicative function of the same argument. This assumption seems to be the essence of the usual assumption of classes". That every function, defined by whatever means, is equivalent to a predicative one sounds more like a bold non-trivial conjecture to be proved, not a "usual assumption". It is also reminiscent of the Church thesis. – Conifold May 4 '16 at 2:42

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