## Can hypercomputation compute the impossible?

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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 – 2019-04-13T06:02:44.950

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 – 2019-04-13T06:05:57.837

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 – 2019-04-13T06:08:28.023

part 4 -- A random search brought up this article: https://www.researchgate.net/publication/228702268_ON_THE_PHYSICAL_POSSIBILITY_OF_ORDINAL_COMPUTATION where they speculate on which ordinal levels are attainable based on various structures of spacetime. So this field is developing!

– user4894 – 2019-04-13T06:14:08.053

@user4894 Someone changed the original title. Have you read the question's body? I did not mean whether hypercomputation was physically impossible but whether hypercomputational models could compute a trivialist model and, even more, whether some hypercomputational model could even compute all logically impossible/illogical things that would arise in a trivialist system that we are not able to conceive since they are impossible to describe/imagine/conceive. There are more details in the question's body – Sue K Dccia – 2019-04-13T10:32:26.540

Any computation is based on the framework you set it in. If you have a computer with a framework different than classical logic and mathematics, it works until the system itself is limited. It's actually a good way to see if a logical/mathematical framework is consistent/good, by using computers to make these calculations. If we can't make the framework itself, the question itself is meaningless. If we come to the point where we manage to get a ASI (Articifial Super Intelligence), and ask it the right questions, then if it IS possible, that's probably how we will do it. – Lexipaichnidi – 2019-04-13T10:42:32.193

This question is based on a confusion. It is easy to have computers output 2+2=5, or implement whatever paraconsistent or trivial logic one wishes, no oracles are needed, just as there is no problem with comprehending Russell's set or the Liar sentence. Computation (and language, generally) is indifferent to possibility or impossibility of what it represents. Hypercomputation and oracles are needed to produce answers that Turing machines can not compute, but whether they are "possible" or not is irrelevant. – Conifold – 2019-04-13T11:53:19.407

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– Conifold – 2019-04-13T11:54:48.190

@Conifold the problem here is that although we can create computers that output 2+2=5 and work with paraconsistent or trivialist systems (just as we do with our brains) there are things in those systems that would be uncomputable (we would need a hypercomputer to compute them). But then, there are logically impossible/illogical things that are impossible to describe/compute (thing that, if we hypothetically saw them, we would se nothing since there would not be any mental state that could represent it in our brains). – Sue K Dccia – 2019-04-13T18:31:07.297

@Conifold 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? – Sue K Dccia – 2019-04-13T18:31:13.287

There are uncomputable things already in the classical logic and mathematics, it has nothing to do with trivialism or paraconsistency. Indescribable and uncomputable are also two different things, to even ask about computability one has to describe what it is to be computed. Hypercomputer is not an "algorithm that can not exist", and it is certainly not "forbidden by logic itself", it is a computer with capabilities surpassing Turing machines. In short, the analogy between logical possibility and computability you are working from just does not exist. – Conifold – 2019-04-14T00:08:23.180

@Conifold I read that only some models (particularly some those which would have a black box and an oracle) could have a logically impossible algorithm hidden by that black box. In any case, what I am saying is that there are logically impossible/illogical things that we cannot conceive (like, imaging a circle cutting a straight line in 3 point in Euclidean geometry or factorizing the number 181 in more numbers than 181 and 1, without changing anything in the axioms of mathematics and using "classical" maths) – Sue K Dccia – 2019-04-14T02:47:24.337

@Conifold and we cannot even describe these things because they are impossible to describe (how would we describe such a circle or such a solution for the factorization?). So what I'm asking is: Despite all of this, could we have a hypercomputational machine that somehow (for example using a logically impossible/illogical algorithm) would be capable of "conceiving" those illogical/logically impossible things I mentioned before? – Sue K Dccia – 2019-04-14T02:49:26.317

@Conifold Or maybe, could our brains somehow evolve enough in the future to allow us to conceive these things (even if they could not have any mental state representing them since they would be impossible to describe)? – Sue K Dccia – 2019-04-14T02:49:38.300

What we can conceive has little to do with the axioms, we hardly remember them when we do the conceiving most of the time. The best I can make of what you are trying to say is that perhaps if our brains are computational then maybe there can be hypercomputational brains which are capable of what we are not. But you'll have to think it through some more because currently it is hard to make sense of your way of describing the difference. It might help to give a reference to what you read about the "impossible algorithms". – Conifold – 2019-04-14T02:53:32.307

@Conifold the reference was basically I conversation I had with a computer scientist, I could send you the rest of the conversation if you want... Now, if axioms are not related with what we can conceive, would it be there a way (even if incredibly hypothetical) to get hypercomputational machines (or very-evolved hypercomputational brains) that could compute/conceive all these illogical/logically impossible things that cannot exist? – Sue K Dccia – 2019-04-14T10:11:01.850

Do not take off the cuff remarks in conversations as a basis for reasoning, who knows what they really meant. Ask the person for a published reference. – Conifold – 2019-04-14T10:14:44.830