Comments on: There’s Something about Gödel, Ch. 9 http://www.logicmatters.net/2009/12/theres-something-about-godel-ch-9/ Logic, enthusiasms, sceptical thoughts, and a little LaTeX geekery Mon, 06 Feb 2012 19:23:02 +0000 hourly 1 http://wordpress.org/?v=3.1 By: Aldo Antonelli http://www.logicmatters.net/2009/12/theres-something-about-godel-ch-9/comment-page-1/#comment-1053 Aldo Antonelli Tue, 29 Dec 2009 22:16:50 +0000 http://www.logicmatters.net/?p=836#comment-1053 "The First Theorem doesn’t have specifically Platonistic implications." That should come as no surprise at all, since the First Theorem is about PA and other formalized theories, and as such cannot imply anything about the existence (or non-existence) of mathematical entities. That's just Craig interpolation: unless you have principles connecting claims about the elementary properties of the natural numbers to claims about the separate existence of mathematical entities, nothing at all about the latter can follow from the former. And any such principles would have to be supported by substantial philosophical argumentation -- a possibility about which one would best be advised to be skeptical. “The First Theorem doesn’t have specifically Platonistic implications.”

That should come as no surprise at all, since the First Theorem is about PA and other formalized theories, and as such cannot imply anything about the existence (or non-existence) of mathematical entities. That’s just Craig interpolation: unless you have principles connecting claims about the elementary properties of the natural numbers to claims about the separate existence of mathematical entities, nothing at all about the latter can follow from the former. And any such principles would have to be supported by substantial philosophical argumentation — a possibility about which one would best be advised to be skeptical.

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