News for hackers. Stuff that matters.

A comment on Category Theory Illustrated – Sets.

Re:Category Theory Illustrated – Sets by hackandthink on Wednesday September 13, 2023 @01:47PM (#37499450)
>In particular, a set can contain itself.

There are many kinds of mathematics, this is an unusual one.

"In Zermelo–Fraenkel set theory, the axiom of regularity and axiom of pairing prevent any set from containing itself."

https://en.wikipedia.org/wiki/Universal_set

[ Parent | on HN ]
Re:Category Theory Illustrated – Sets by axblount on Wednesday September 13, 2023 @02:06PM (#37499699)
This gives rise to Russell's Paradox:

Does the set of "all sets that do not contain themselves" contain itself?

[ Parent | on HN ]
Re:Category Theory Illustrated – Sets by tikhonj on Wednesday September 13, 2023 @02:17PM (#37499811)
You can have set theories that allows sets to contain themselves without allowing Russell's Paradox. You can read about non-well-founded set theory[1] if you're curious.

[1]: https://en.wikipedia.org/wiki/Non-well-founded_set_theory

[ Parent | on HN ]
Re:Category Theory Illustrated – Sets by hackandthink on Wednesday September 13, 2023 @03:27PM (#37500633)
I like the diagram of "the set containing itself". It illustrates non-well-foundedness niceley.
[ Parent | on HN ]
Re:Category Theory Illustrated – Sets by l33t7332273 on Wednesday September 13, 2023 @10:06PM (#37504086)
Which diagram are you referring to?
[ Parent | on HN ]
Re:Category Theory Illustrated – Sets by hackandthink on Thursday September 14, 2023 @04:43AM (#37506438)
This one:

https://abuseofnotation.github.io/category-theory-illustrate...

[ Parent | on HN ]
Re:Category Theory Illustrated – Sets by seanhunter on Wednesday September 13, 2023 @02:22PM (#37499847)
Resolving this paradox is discussed in TFA as being the founding rationale for the Zermelo–Fraenkel set theory in fact.
[ Parent | on HN ]
Re:Category Theory Illustrated – Sets by righttoolforjob on Wednesday September 13, 2023 @02:31PM (#37499944)
simple: there is no such set.
[ Parent | on HN ]
Re:Category Theory Illustrated – Sets by bmacho on Wednesday September 13, 2023 @02:55PM (#37500259)
It is mentioned 2 sentences earlier that it was the case in naive set theory.
[ Parent | on HN ]