[FOM] Laureano Luna on Cantor

Gabriel Stolzenberg gstolzen at math.bu.edu
Tue Jul 3 16:06:24 EDT 2007

   In volume 55, issue 3, Laureano Luna asked:

> Cantor seemingly believed the set of possible definitions of reals was
> not countable: how was this possible?

   Viewing definitions of reals as a subset of all formal words from some
alphabet, there is a classically definable counting of them.  So, it is
classically a countable set.

   On the other hand, because the counting is classically definable,
we can diagonalize and produce another classical definition of a real
that is not on the alleged list of all such definitions.

   So, classically, the set of definitions of real numbers is both
countable and uncountable, which refutes the anyway dubious assumption
that there is such a thing as the set of definitions of real numbers.

   I have no idea whether this bears on what Cantor had in mind but I
can see that it might.

   Gabriel Stolzenberg

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