FOM: Precursors and Cantor's originality

Richard E. Grandy rgrandy at
Thu Jun 13 20:18:30 EDT 2002

It seems to me that a number of Cantor's precursors noticed that some 
infinite sets are "bigger" than others in the proper subset sense, 
and a number noticed that sometimes a proper subset could be put in 
one-one correspondence with the whole set.  Those who  noticed both 
concluded that no sense could be made of sizes of infinity.  Cantor's 
originality was, after recognizing both,  to reject the proper subset 
criterion and make the existence of a 1-1 correspondence definitive 
of equinumerosoity even when the proper set criterion indicated 
otherwise.  And of course to show that an enormously  rich 
mathemtical theory results!!

There are a number of instances in the history of science where two 
notions were conflated for a long time and it was a major 
breakthrough to separate them.  In this case it is proper subset vs. 
equinumerosity, others are  velocity vs acceleration, heat vs. 
temperature, and perhaps the fact that the earth is locally flat but 
globally spherical.

Richard Grandy
Philosophy and Cognitive Sciences
Rice University
Houston TX USA

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