There are illogical / logically impossible things (for example, saying that 2 + 2 = 4 and 2 + 2 = 5. Without changing anything in the axioms of mathematics or logic, this would be a contradiction and would be incoherent and illogical / logically impossible.

Outside of classical logic, there are other types of logical systems, such as paraconsistent logic or even trivialism, that allow these contradictions to occur, prove them as being right and work with them.

We can create a paraconsistent or trivialistic system and work with it. For example, with trivialism, in theory, we could deduce and state everything we would like (since literally everything, even illogical / logically impossible inconsistencies and contradictions), but we, as humans (or in as brains), can not conceive everything we want (at least to what I know). Therefore, whatever the number of trivialist models we create and the time we will be working with them, we could never find or conceive many illogical / logical things because they are only that: impossible. There are things impossible to describe and to conceive. For example, the set of Russel is the set of all sets that do not contain themselves. If the set of Russel contains itself, it can not contain itself, because it contains only sets that do not contain themselves. But if the whole of Russel does not contain itself, then it must contain itself, because it contains all the sets which do not contain themselves. There are a lot of logic bombs like this one. You can never calculate the content of Russell's game and there are more formal and mathematical ways to present it. All have in common that you can not really calculate the whole, whether you do it by hand, in your head or on a computer. It's just a statement that can not be treated logically. If you take all the possible states in which the human brain can be found, none of them understands the calculation of Russel's content. In other words, not only can the content not be calculated, but it can not even be represented. No stimulus can lead us to understand Russel's set because such an understanding is not possible to begin with. There is no solution. Even if we try to solve it by using a trivialism, we would simply be able to write a solution that would make no sense and prove that it was good and that's the real solution, but we could not have solution that would make sense. "outside" the domain of trivialism (for example in classical logic), even if, using trivialism, we could prove that such a solution would make sense whatever the context or the logical system.

But what about hypercalculating machines (eg, oracle type machines)? I have read some examples of hypercalculators compatible with paraconsistent or trivial logic. I've also read that there are hypercalculus models (especially oracle-type models using a black box) where the hypercalculator is an algorithm that can not exist. This is perhaps because such an algorithm is basically forbidden by the logic itself (hidden in a black box). Would any of them be able to calculate / "conceive" those impossible things that I've written before? Do you know something that could help?

I also thought that maybe we could evolve enough to have brains capable of calculating all that … So, could our brains evolve to such an extent that we could conceive and calculate all those illogical / logically impossible things that would not can exist?