About basic approach

[Jano]

TOE must be a mathematical original set theory, because set is the most universal structure and only mathematics has sufficiently abstract method. Physics has no chance to discover TOE, because it studies only a part of natural phenomenons.
TOE must be built as an axiomatic theory, because the axiomatic principle is the only one scientific approach in building scientific theories. TOE's axiomatic system must be without contradiction and it must be independent ( it means that this system needs not another information beside this system to "work" ).

[arivero]

"Topos" is more general than "set".

[Jano]

"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".

[arivero]

OK, topos (a special cathegory having some intersection, union and limit constructions) is less general that your intuitive concept of set, but still it is more general that the axiomatic definitions of set. And it is capable of containing the same logical structures. So Goldblatt and other people headed in the 60's a movement to use topoi as a fundamental notion instead of set.

[kennap]

well this set would be the set containing all other sets. This is the set of exisistence, we could call it E-set. I agree that physics alone will not solve it.But physcis is so intertwined with math. You can have a generalized mathematical model, but this will not be acceptable unless physics agrees.

[Marketa]

any axiomatic theory can't be without contradictions. Consider: It was proved that: All consistent axiomatic formulations of number theory (but as well of any other mathematical theory) include undecidable (=contradictional) propositions.

[michellemfry]

I like your definition of set and it's use to begin the TOE. I used universe instead of set and universe(s) instead of sets in Russel's Paradox and got some clarity. If your suggestion pans out, and I think your logic is ironclad, that would or could lead to an end to Russel's Paradox. I answered that the universe that doesn't contain a universe is what existed before the Big Bang. So, what existed before the Big Bang was an equivalent to a paradox, which is what nothing can be likened to, the paradox of paradox's.

If a paradox is...Hmmm....

[Guille]

Neither topos or set or group... Or whatever else you defend. The most fundamental structure in the universe is Unity. And then, later on, the rest derive.

[subversion]

any axiomatic theory can't be without contradictions. Consider: It was proved that: All consistent axiomatic formulations of number theory (but as well of any other mathematical theory) include undecidable (=contradictional) propositions.

and thus the TOE will be consistently inconsistent, which simultaneously makes it both consistent and inconsistent

which means scarily that it is both true and false