/sci/ - Science & Math

>>9645910
If you model the ordered pair of two things x and y, i.e. (x,y), via
(x,y) := {{x}, {x,y}}
(this is Hausdorff's definition, the standard one)
and if you model the natural numbers via
0 := {}
1 := {0}
2 := {0,1}
3 := {0,1,2}
etc.
(Neumann ordinals, the standard one)
then, evidently, the number 1 is in the set (0,7).

If you only got sets, then you use models to talk about stuff and then you introduce those "auxilliary theorems", stuff you weren't able to prove in raw category theory, for example.

>>9645814
Sounds interesting, be a bit more concrete with your integral and periodic function definitions.

I'm a fan of creatures such as
https://en.wikipedia.org/wiki/Generalized_mean
https://en.wikipedia.org/wiki/Product_integral
or q-analogs such as
https://en.wikipedia.org/wiki/Jackson_integral

but those games can lead to endless seas. Like the whole branch of fuzzy so and so's. Or parametrized so and so's. I suppose sheaves are a good example for where it worked.

Similar to "fuzzifications", I'm a bit sad when I see people lose themselves in some generalizations, like e.g. the manifold generalizations, or breaking up topology axioms in this and that subject to play with a bit more general objects. Good if it works and you can tackle some tanglibe or at least popular problem, but often it doesn't work.