I think you've misunderstood what I was saying. I am not claiming that we can prove a material equivalence, an internal formal equivalence, between N and Na; that would require induction. The only things which that statement claimed were being formally proved within Q were 1. and 2. Neither of those statements require induction. 1. allows us to treat Na as, at most, N. 2. allows us to treat Na as, at least, N. Both together allow us to treat Na as N (within limits). Only 1. and 2. are internal, there are no other formal statements. That 1. and 2. are sufficient for demonstrating that N and Na are semantically the... (read more)
The goal of this post is to outline a loophole of sorts in Gödel's Second Incompleteness (GSI) theorem (and of Löb's theorem by corollary) and give a somewhat sketchy description for how it might be exploited.
I'll begin by describing the loophole. Essentially, it's merely the observation that Second Incompleteness is a syntactic property rather than a semantic one. Assume that is a provability predicate which is subject to GSI. GSI does not prohibit the existence of a predicate not subject to GSI such that , where is the set corresponding to the predicate in the standard semantics. That is to say, it's possible for two predicates to be the same, semantically, while one is subject to GSI... (read 1902 more words →)
Considering how much time is spent here on this subject, I'm surprised at how little reference to distributional semantics is made. It's already a half-century long tradition of analyzing word meanings via statistics and vector spaces. It may be worthwhile to reach into that field to bolster and clarify some of these things that come up over and over.
I do worry that there's some technical subtlety not obvious until the end game that makes this particular construction impossible. I've been trying to spot some places where something might go wrong, and there are a few parts I don't know how to fill in. I think there might be a problem relativizing a proof of the totality of ordinal multiplication which may prevent either the proof predicate relativization or the transfinite induction derivation from working. Or there may be some more subtle reason the steps I outlined won't work. I was hoping that someone in the community might be able to say "I've seen this before, and it didn't work because... (read more)