/sci/ - Science & Math

>>12517081
Very much fun.

>>12517111
So, toposes are basically generalisations of category of sets (and functions) if you want to approach it from the direction of logic, and then you get to do all sorts of logic stuff in them with their internal logics (which are usually non-classical). This allows one to formulate pre-existing theories in a new way using the language of topos theory, and then generalise those by replacing the underlying sets with objects in other toposes (like one can replace the underlying set of a group with a underlying space, smooth manifold etc.) and so on. According to my intuition, this would allow logicians to break weaker theories with a suitable choice of topos, but I don't know if they actually do that.

Alternatively, one can approach from the direction of sheaves. Given any topological space X, we can take its topology, order it by inclusion and consider that poset as a category. This turns any presheaf over X into a contravariant functor from the (category of the) topology poset. This, as well as the sheafification of a presheaf, generalises from topologies to small categories when one equips the category with a Grothendieck topology (thus making it into a site), and then takes the contravariant functors from that category to the category of sets. The Grothendieck topology will now allow us to sheafify those functors and we will obtain the category of sheaves over the given site, called a Grothendieck topos. These are a special case of toposes and generalise sheaves over a space.

If those approaches seem even remotely interesting, then please:
(1) google the jargon to find out if you are actually interested;
(2) ask additional questions from someone who actually knows the stuff instead of me;
(3) start studying that stuff.

>>12517202
To me, it looks like the reader does not know that any real numbers exist, so the problem is the existence of any such x. Assume there are no real numbers, then a real number x will exist by 1.1.1.