Explosion use in math
pratt at cs.stanford.edu
Wed May 25 17:30:32 EDT 2022
Harvey Friedman offered the following as simple examples of propositions
supposedly most easily proved using "explosion" (reductio ad absurdum).
The emptyset is included in every set.
There is one and only one set that is included in every set.
While contemplating his claim (for the first I would have avoided the use
of explosion by arguing that "for all x in emptyset, x is in Y" holds
vacuously) it occurred to me that the second is only true of the category
Set up to isomorphism, but that in that sense of "only one" there is also
only one singleton in Set, one doubleton, and so on. In set theory however
there really is only one empty set, but many singletons etc.
So Harvey's example serves not only his original purpose but also the
purpose of illustrating that category theorists view sets differently from
how set theorists view them.
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