[FOM] Dedekind's theorem

meskew at math.uci.edu meskew at math.uci.edu
Fri Jun 17 03:28:01 EDT 2011

In "Was sind und was sollen die Zahlen?" Dedekind gives a noncircular
proof of the statement that a set is finite if and only if it cannot be
put in bijective correspondence with a proper subset.  By "circular" I
mean in this context that you should not prove it by simply saying that a
proper subset of a finite set will have a smaller cardinality; this
theorem should be taken as the ground for the well-definedness of the
finite cardinals.

Regarding the "only if" direction, which establishes that finite ordinals
are cardinals, was Dedekind the first to publish a proof of this?  Did
Frege give a proof independently?  Galileo?  Leibniz?  Some medieval monk
perhaps?  It would seem strange if this basic aspect of the concept of
number was not reflected upon for so many centuries.


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