Re: Uncountable sets

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Date: Tue, 1 Jun 93 07:12:22 EDT
From: sowa <sowa@turing.pacss.binghamton.edu>
Message-id: <9306011112.AA21514@turing.pacss.binghamton.edu>
To: phayes@cs.uiuc.edu, sowa@turing.pacss.binghamton.edu
Subject: Re: Uncountable sets
Cc: cg@cs.umn.edu, interlingua@ISI.EDU, jw_nageley@pnlg.pnl.gov

Pat,

There's almost nothing in your last note that I disagree with.
In particular, I would like to endorse your conclusions:

> So, using KIF presupposes its semantics, but it does not presuppose the 
> framework in which those semantics are formalised: and anyway, it could 
> get by with a kind of schoolboy's set theory. That the KIF designers have 
> chosen a Maserati rather than a Ferrari doesn't mean that we have to ever 
> drive at unsafe speeds.

> Both of these essentially mathematical activities are relevant to, but not 
> identical with, the philosphical problem of finding a secure consistent 
> foundation for mathematics. Most of the controversy surrounding set theory 
> (including all the high-falutin' worries that Nageley refers to, without 
> pointing out that this stuff is all at least 30 years old) has been 
> concerned with this last goal. I don't think this has ANYTHING AT ALL to 
> do with Krep in AI, and can be safely ignored in our discussions. That 
> the semantics of KIF is stated in set-theoretical terms, or even that KIF
> has chosen to use the VNBG off-the-shelf formalisation, is of no particular 
> importance to anyone not interested in the foundations of mathematics.

The only point I would quibble with is the presupposition that anything
that is 30 years old is something that the general scientific community
has understood, assimilated, and come to terms with.  Science moves very
fast in the details, but more fundamental paradigm shifts take a long
time.  The ultimate foundations of mathematics are similar to the ultimate
foundations of physics:  the overwhelming majority of working mathematicians
and physicists can go about their daily work without worrying about the
nature of quarks or the foundations of set theory.

To give you the last word for a change, I would like to repeat one of
your sentences that I completely agree with:

> ... I don't think this has ANYTHING AT ALL to
> do with Krep in AI, and can be safely ignored in our discussions.

John