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From: gjk@talcott.UUCP (Greg Kuperberg)
Newsgroups: net.math
Subject: Re: Re: Re: Nova's Mathematical Mystery Tour (truth of CH)
Message-ID: <360@talcott.UUCP>
Date: Fri, 15-Mar-85 22:22:39 EST
Article-I.D.: talcott.360
Posted: Fri Mar 15 22:22:39 1985
Date-Received: Sun, 17-Mar-85 21:53:14 EST
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>> = Lambert Meertens
> = Andries Brouwer

>>I do not see an essential difference between axioms and inference rules; an
>>axiom is simply an inference rule with an empty set of antecedents.
> 
>But I *do* see a difference between an axiom and a rule of inference.
>The former is a requirement on one particular situation; the latter
>(in the context of our discussion) formalizes a way of reasoning
>that is supposed to be universally valid.

In the particular case of CH, I think that Lambert is right in saying that
one could conceivably add new rules of inference that would make CH
decidable without causing a contradiction.

But about axioms versus inference rules in general I'm not so sure.  My
only objection is that inference rules are much more fundamental than
axioms;  if you changed De Morgan's law then the new rules might well be so
radical as to not be useful.  Certainly in the case of an unproved result,
such as the Riemann Conjecture, the mathematical community can, and does,
treat is as an axiom of sorts. (There are papers which start off with
"Assuming the Riemann Conjecture..." just as there are papers which start
off with "Assuming the Contiuum Hypothesis....")  The reason is that if RC
turns out to be true, then all these theorems will be very useful (of
course, if RC is false, then they will be garbage, but that's another
story).  On the other hand, one does not have the privilege at the current
time to change the rules of inference so that RC is decidable in an easy
way.  And if one did this, all of the "theorems" that would result would
not necessarily be useful if RC were to be decided by conventional means.
In short, at the current time most open problems turn out to be true or
false, rather than unsolvable, so I don't see the utility in changing
the rules of inference just yet.

Of course, in the year 2150 say, there might be a "Goedelian crises",
whereby there will be lots of new axioms and few new theorems.  At that
point we might see the light and switch to a different set of inference
rules.
---
			Greg Kuperberg
		     harvard!talcott!gjk

"No Marxist can deny that the interests of socialism are higher than the
interests of the right of nations to self-determination." -Lenin, 1918