Could we develop a notion of conceivability that would allow us to imagine impossible/inconsistent/illogical things? There are things that we cannot imagine because they are impossible (like a solution to Russell's set theory, since it is impossible to reach that solution because it is illogical.)

Some months ago, discussing with a user in this site, he said that although no one has reached that notion, it could be the case that someone in the future could achieve it, but he did not know the topic well enough to make any meaningful conclusion. (https://chat.stackexchange.com/transcript/73718/2018/2/28)

Maybe someone here could help me

So could we develop a notion of conceivability that would allow us to imagine all of these things? Even all things that do not exist or cannot exist or are impossible like a solution to Russell's set problem or a thing that is illogical but at the same time is logical or inventing a new logic (or illogic) system...etc? What would be needed to reach such notion of conceivability? A change in the laws of physics? Or just biological evolution in our brains??
There's a serious science of the theoretical limits of representation / information processing in CS, in complexity theory &amp; computability theory. You can pull some of that stuff across to cognitive science, the question 'what level of the complexity hierarchy does language sit at' has a decent track record of productivity, it's not totally crazy to ask the same thing of cognition in general.

For me, there's a satisfying answer in https://mitpress.mit.edu/sites/default/files/sicp/index.html, although if you start at the beginning, it takes a little while to get there. In the lecture videos, it's the bit where he puts on the wizard hat. Because if you're going to answer a question as profound as this one, it's best to wear a cool hat when you do.

This is not the only answer, some folks believe in literal quantum cognition, so you could go chat to them too. They're kinda fringe at the moment, and I'm not a subscriber, but they are very smart people and the only game in town if you want think about going beyond Church-Turing.