[FOM] Some historical questions

Richard Heck rgheck at brown.edu
Sun Nov 7 19:10:11 EST 2010

On 11/07/2010 01:15 PM, G. Aldo Antonelli wrote:
> On 11/7/10 9:00 AM, fom-request at cs.nyu.edu wrote:
>> Paul Henrard has a clever way of defining equinumerosity between sets
>> that doesn't use ordered pairs, by using overlapping unordered pairs.
>> Some time in the 1970s. I don't *think* he ever published it.  Allen Hazen
>> independently discovered (essentially) the same trick. Others may have
>> done the same.  Does anyone know anywhere where such a gadget is
>> published?  It would be good to get the history right.
> It's known that, in general, unordered pairs (= symmetric irreflexive
> binary relations) can be used to encode arbitrary binary relations. The
> reference I have is the following:
> Anil Nerode and Richard Shore [1980], Second order logic and theories of
> reducibility orderings, The Kleene symposium (Jon Barwise, H. Jerome
> Keisler, and Kenneth Kunen, editors), North-Holland, Amsterdam, pp. 181-200.
I saw a philosophical paper some years ago, by Cian Dorr, in which he 
argued, on
the basis of this kind of result---he may have rediscovered it 
himself---that, in some
fundamental sense, there are ONLY symmetric relations. (I'm not sure he said
anything about reflexivity.) Of course, the philosophical claim goes 
well beyond the
technical result, but it struck me at the time as (a) interesting and 
(b) insane. ;-)


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