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Originally Posted by arivero "Topos" is more general than "set". |
Sorry, bur I do not know imagine more general structure than set. I believe, that you too. By intuitive mathematical definition "Set is an arbitrary collection of elements, that have an arbitrary quality". The only condition to be a set is to have a quality. More general property of structures does not exist.
This principle really does not enable for any structure to escape from competence to sets. Is set a structure, which has not any quality? Yes, because "not to have a quality" is quality. Is set a structure, which is not a set? Yes, because "not to be a set" is a quality. Do not let yourself surprise by the contradiction "a set, which is not a set", because generaly sets can be contradictive. But it is another problem related with declaration and insertion of logic to the TOE.
Finally, I must explain one big problem. The ZF ( Zermelo-Fraenkel ) and NGB ( von Neumann-Godel-Bernays ) set theories are not suitable for being a foundation of TOE. The most important cause is bad definition of set in mentioned theories ( !!! ). But about this possibly sometimes in the future. It depends on your interest.
Ad topos:
I do not know, what kind of structure topos is, but, if this structure has more specific qualities besides the quality "to have a quality", it is less general than set. The proof, that topos is a set, is very trivial: Topos has a quality "to have a name topos".