# What is the “intended project” in Appendix B of Principia Mathematica?

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. Nov 21 '15 at 7:32
• @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. Nov 21 '15 at 10:34
• Can we try to frame the headline here as a question? Jan 5 '16 at 16:40
• @JosephWeissman: I don't understand. Can you clarify?
– user13627
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) Jan 5 '16 at 17:26