There are things which are illogical/logically impossible (like 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 inconsistent and illogical/logically impossible.

There are other types of logic systems apart from classical logic, like paraconsistent logics or even trivialism, that allow these contradictions to occur, prove them as right and work with them.

We can make a paraconsistent or trivialist system and work with it. For example, with trivialism, in theory, we would be able to derive and state everything we would want (since literally everything, even including illogical/logically impossible inconsistencies and contradictions), but we as humans (or as brains), are limited and can't conceive everything we want (at least to what I know). Therefore, no matter how many trivialist models we create and how much time we would work with them, we would never find or conceive many illogical/logically things because they are just that: impossible. There are impossible things to describe and conceive. For example, Russel's set is the set of all sets that do not contain themselves. If Russel's set contains itself, then it cannot contain itself, since it only contains sets that don't contain themselves. But if Russel's set does not contain itself, then it must contain itself, since it contains all sets that don't contain themselves. There are quite a few logic bombs like this. You cannot ever compute the contents of Russel's set, and there are more formal, mathematical ways to present it. All of them have in common that you can't actually compute what the set is, whether you do it by hand, in your head, or on a computer. It's just a statement that cannot be fully logically processed. If you take every possible state the human brain can be in, none of them include the computation of Russel's Set's contents. That is, not only can the contents not be computed, they cannot even be represented. No stimulus can cause us to comprehend Russel's Set, since such comprehension is not possible to begin with. It does not have a solution. Even if we try to solve it using trivialism, we would just be able to write a solution that does not make sense and prove it has sense and it is the real solution, but we could not be able to have a solution that would make sense "outside" the realm of trivialism (for example in classical logic), even though, using trivialism, we could prove that such solution would have sense in whatever context and logic system.

But what about hypercomputational machines (for example oracle-like machines)? I've read about some models of hypercomputation which are compatible with paraconsistent or trivialism logics. I've also read there are some models of hypercomputation (particularly those oracle-like models which use a black box) where, essentially, the hypercomputer is an algorithm that cannot exist. It might be because such an algorithm is fundamentally forbidden by logic itself (which is hided in a black box). Would any of these be capable of computing/"conceiving" these impossible things I wrote before? Do you know of anything that would help?

  • I was going to write up a long answer but only have time for a short one. If you start with the class of Turing machines (TM's) and add an oracle for the Halting problem, you get a new system TM' that still has a problem it can't solve, call it H'. So you invent an oracle for HT', add it to TM' to get TM'', but then you have impossible problem H''. In short, you end up reinventing the ordinal numbers. Now as it happens, Turing wrote his doctoral thesis on Ordinal models of computation. He already figured all this out in the 1930's. The more you learn about him the smarter he gets. – user4894 Apr 13 at 6:02
  • part 2. So we DO have an entire hierarchy of models of computation. The problem is that the very first transfinite one, TM', is already unrealizable in the physical world. Now I am going to throw out a speculation. The next great revolution is solving this problem. There will someday be new physics that allows some kind of actual infinity; and we will then be able to work our way up Turing's ordinal hierarchy. I want to throw in some other visualizations. Say you have all the computable real numbers. An oracle H is like adding a single noncomputable number. An oracle for H' is – user4894 Apr 13 at 6:05
  • part 3 -- like adding two noncomputable numbers. And so forth. Yet another parallel is that if you start with an axiomatic system like ZF, then ‎Gödel says it's incomplete. So we add another axiom, like the axiom of choice. But that's still incomplete, so we add yet another axiom. Same hierarchy. Essentially the same thing, though I should state for the record that I'm technically ignorant past this point. So take it all with a grain of salt. Bottom line is that we need a breakthrough in physics that allows us to represent a few small transfinite ordinals in a physical substrate. – user4894 Apr 13 at 6:08
  • part 4 -- A random search brought up this article: researchgate.net/publication/… where they speculate on which ordinal levels are attainable based on various structures of spacetime. So this field is developing! – user4894 Apr 13 at 6:14
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