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Message-id: <199305312208.AA19315@dante.cs.uiuc.edu> Date: Mon, 31 May 1993 17:12:10 +0000 To: sowa <sowa@turing.pacss.binghamton.edu> From: phayes@cs.uiuc.edu X-Sender: phayes@dante.cs.uiuc.edu (Unverified) Subject: Re: Uncountable sets Cc: cg@cs.umn.edu, interlingua@ISI.EDU, jw_nageley@pnlg.pnl.gov

John, I agree that this discussion, while fascinating, is not of deep relevance to KR work and especially the KIF/CG discussion. And I agree with most of what you say on it, but with one important difference: ........ > 4. This raises a question about the semantics of KIF, which is being > modelled using a version of VNBG (von Neumann, Bernays, Goedel set > theory), which does support uncountable sets. How can one use KIF > without presupposing its underlying semantics, which does seem to > lead to the dreaded "swamp of confusions" or "Cantor's paradise"? > Easily. The semantics of KIF is defined using set-theoretical terminology, but it does not need the full resources of VNBG or any other fully formalised set theory. It is very SIMPLE. It uses set-theoretical language for the reasons Nageley, citing Fraenkel, so lucidly gives: >The overwhelming majority of mathematicians, even those who are theoretically >impressed by the critical arguments [against set theory], continue to apply >the >methods of set theory. Very few, if any, have followed the example of Hermann >Weyl, who confessed that he avoided certain analytical research subjects >because they were immersed too deeply in set theory. > >In short, as Cantor himself observed, mathematics has within it its own >corrective: if a theory is not useful--"fruitful"--it will soon be discarded. > >And the history of his theory has shown that it can be fruitful. That's all: its just using modern mathematical methods, in that way that an engineer might use partial differential equations without getting all hot under the collar about the exact foundational status of a concept like an infinitesimal. Its worth emphasising that the aspects of axiomatic set theory which are most worrying - the axiom of choice, notably - are not relevant to the kind of simple construction utilised in defining first-order models. One has to keep clear the distinction between this pedestrian use of set-theoretical ideas in defining models (just standard mathematical practice) from the actual formalisation of set theory in the logical language whose semantics is being discussed. Even this might be regarded as pretty normal these days, and the various formal versions are pretty well understood, but it is still an area not entirely free of controversy, viz. Nageley's article, and in which new work is still being done, eg consider Aczel's 'hypersets' used so neatly by Barwise and Etchemendy in 'The Liar'. (These were inspired by CS ideas, by the way, c.f. Nageley's closing remarks.) 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. Pat Hayes ---------------------------------------------------------------------------- Beckman Institute (217)244 1616 office 405 North Mathews Avenue (217)328 3947 or (415)855 9043 home Urbana, IL. 61801 (217)244 8371 fax hayes@cs.stanford.edu or Phayes@cs.uiuc.edu