Explosion use in math
Patrik Eklund
peklund at cs.umu.se
Thu May 26 01:56:42 EDT 2022
And it's maybe trivial to point this out, but we always need to be
careful about not identifying ø with {ø}, with {{ø}}.
Somebody recently asked about monads. Studying the powerset and term
monads are useful. For the powerset functor P there is the
multiplication of the powerset monad that brings PP back P, but if we
study the double powerset PP we have to be care as PP can be composed by
the covariant P with itself or by the contravariant P with itself. Both
PPs are covariant but with different properties. For instance, the
filter monad is a submonad of the contra-contra PP but not of the coco
PP. Then the Eilenberg category of the filter monad is interesting also
as it it becomes to be isomorphic with the category of compact
topologies. Understanding these underlying phenomena is also useful.
As for the term monad we often say it is idempotent TT=T since a "term
of terms is a term", but when we really examine that T as a monad we
realize it's not an equality TT=T but it is equal "up to isomorphisms".
Studying compositions like PT (sets of terms) is interesting as this
brings in the "swapper" transformation t : TP -> PT and Beck's
distributive laws. Again, studying such things will show how PT is
doable, but TP as a monad not, since a swapper s : PT -> TP does not
exists. A similar doable and not doable is in the case of P(X x X) and
PX x PX.
Finally, for the purpose of this e-mail, if we bring in many-sortedness
into T, the extension from one-sortedness to many-sortedness is not
trivial. Doing it with sets enables to bend the curves, but when you
construct that many-sorted term monad categorically, you will see how
subtle it is.
---
So indeed, be careful with the emptyset and the brackets around it, and
Vaughan is right on target on "the purpose of illustrating that category
theorists view sets differently from how set theorists view them".
Somebody also pointed out earlier how some categorists prefer to go
morphism before going object, where "singleton" is a property of a
morphism, not a feature of the object. And, as Vaughan says, there is
one empty set, but many singletons. And clearly, depending on the
category, this situation is different. In Set, the empty set is the
unique initial object and singletons are terminal objects, but in many
other categories it looks different. This connection with terminal and
initial again shows how categorists like morphisms to precede objects.
---
Best,
Patrik
On 2022-05-26 00:30, Vaughan Pratt wrote:
> 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.
>
> Vaughan Pratt
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