Path: utzoo!mnetor!uunet!seismo!sundc!pitstop!sun!decwrl!labrea!aurora!eos!ames!elroy!cit-vax!ucla-cs!hbe
From: hbe@math.ucla.edu
Newsgroups: sci.philosophy.tech
Subject: Re: Classifying the Axiom of Choice
Message-ID: <9759@shemp.CS.UCLA.EDU>
Date: 25 Feb 88 22:23:34 GMT
References: <8802251655.AA06216@duchamps.ads.arpa>
Sender: news@CS.UCLA.EDU
Reply-To: hbe@math.ucla.edu (H. Enderton)
Organization: UCLA Mathematics Department
Lines: 19

In article <8802251655.AA06216@duchamps.ads.arpa> rar@DUCHAMPS (Bob Riemenschneider) writes:
>=>   The a priori statements form merely a recursively enumerable set, ...

>I don't see why this must be the case.  Mightn't (some) people come
>equipped with knowledge of a more complicated set of truths, or with
>knowledge of a non-effective rule of inference?

For us to *know* a truth of arithmetic, we'll need to see some
effectively verifiable supporting argument--a proof.  If the set
of acceptable proofs is decidable (or even r.e.), then the set
of provable statements is r.e.  Here we are considering not simply
proofs according to some fixed rules, but arguments that could
*ever* be accepted (in theory) by human mathematicians.  Even so,
this still seems to lead to an r.e. set.

To "know" that Fermat's last theorem is true, it's not enough to
believe it and state it.  You need a supporting reason I can check.

--H. Enderton, hbe@ucla.math.edu