Re: Uncountable sets
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Date: Mon, 31 May 1993 17:12:10 +0000
To: sowa <>
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Subject: Re: Uncountable sets
Cc:, interlingua@ISI.EDU,

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
>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

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