[FOM] 622: Adventures in Formalization 6
Freek Wiedijk
freek at cs.ru.nl
Thu Oct 8 08:05:26 EDT 2015
Dear Larry,
>And then what is the status of the “real” natural
>numbers, which are surely present somewhere unless they are
>actually constructed to be a subset of the complex numbers,
>which would be even stranger?
It's different. When defining, say, the integers from the
natural numbers nat, then one first defines the integers as
a set int' (for instance as a quotient of nat x nat), with
an embedding i: nat -> int'. And then the integers int that
the rest of the library actually uses everywhere are then
defined as int := nat U (int' - i[nat]). And similarly for
the other number types. So, yes, the "real" natural numbers
(the finite Von Neumann ordinals, of course!) are a subset
of the complex numbers. And a subset of the ordinal numbers.
And a subset of the extended reals.
Henk Barendregt once told me that Van der Waerden describes
exactly this same construction in one of his books.
Freek
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