Do logical limits point to the limits of what is possible or to limits of our thinking? — Fooloso4

Logic is a formal language.

There is a mismatch in power between what we can say (formal language) and what we can solve (system). The solution in the development of set theory was to add constraints to restrict formal language as such to prevents particular questions from being asked. ZFC was extended with three "hacked" axioms for that. They have no other function than to restrict what can be asked. Still, the strategy ultimately failed because you can still do it (Gödel's incompleteness).

You're just expressing beliefs about the content of logic. It's beside the point. I am asking whether being omnipotent involves having control over the content of logic. You can't provide any insight into the answer by just telling me more and more about the content of logic.

No, it is to do with the concept of power. To have maximum power one needs to be able to do anything at all, for if there is something one cannot do, one lacks the power to do it. And lacking a power to do something is to lack a power.

You're just expressing beliefs about the content of logic. It's beside the point. I am asking whether being omnipotent involves having control over the content of logic. You can't provide any insight into the answer by just telling me more and more about the content of logic. — Bartricks

Well, I am only pointing to the history of "smartass" questions in mathematics.

After adding a rule to fix Bertrand Russell's "smartass" question, simply by making it impossible to ask (by introducing the axiom of restricted comprehension), another genius, Dmitry Mirimanoff, discovered a new way of asking "smartass" questions, because hey, "look, the system is full of bugs", and "look at this", and "look at that", and "Is this normal?", "Didn't I tell you so!?" ...

So, between 1917 and 1920, Mirimanoff published his endless rant, in a long series of articles, showing that there are "non-well-founded" sets that can cause all kinds of mischief in set theory. It is actually simple. The expression:

A = { A }

will easily cause havoc, because it means that you can replace A by { A }. Therefore:

A = {{ A }}
A = {{{ A }}}
A = {{{{ A }}}}
...
... ad nauseam ...
...

So, what did they do to get rid of Mirimanoff "smartass" questions?

Well, they (=Zermelo and Fränckel) introduced a new rule, titled "the axiom of regularity", which explicitly forbids asking this kind of "smartass" questions by making it impossible to do so. Problem solved.

Well, not really.

In 1928, Hilbert asked: If we forbid asking "smartass" questions, simply by outlawing them, can all yes/no questions be answered with a yes or a no?

Das Entscheidungsproblem. As late as 1930, Hilbert believed that there would be no such thing as an unsolvable problem.

In 1931, the first blow came with Kurt Gödel's incompleteness theorems. It is the language of logic itself that is the problem and that allows for asking "smartass" questions. So, fixing the system, by adding new rules, will not help. In 1936, Alan Turing and Alonzo Church then independently proved that a yes/no question is answerable only if there exists a purely mechanical procedure that can answer it.

Given the history of "smartass" questions in mathematics, and the 1936 conclusion, my position is:

I do not know of a purely mechanical procedure that can answer your yes/no question. Unless you can point out the existence of such procedure, your question must be deemed unanswerable.

Note: Computability, computation, and computer systems propel epistemology to the forefront. Knowledge-justification methods are truly key now.

There you go again - more content. You're not addressing the question.

And I answered the question and gave my reasons for the answer. Those reasons justify that answer. You seem to be thinking they don't - why?

And I answered the question and gave my reasons for the answer. Those reasons justify that answer. You seem to be thinking they don't - why? — Bartricks

Where is the purely mechanical procedure that would answer your question?

You see, a machine must be able to come to the same conclusion as you do. So, the very first part of the answer must consist in the description of an algorithm. I am just applying Alan Turing's and Alonzo Church's conclusions here, with regards to David Hilbert's question concerning the decidability of yes/no questions.

I appealed to reason.

Whatever it is that you're saying - and it isn't clear to me what you're saying - you're either attempting to appeal to reason, or you're not. If you are not, then what you're saying counts for nothing. If you are, then show me how what I've said fails to report what reason says.

Reason says that to be maximally powerful is to be able to do anything. Reason says that if you can't do something - anything - then that's a restriction on your power. Therefore, a maximally powerful agent is not in any way restricted in what they can do. Thus, a maximally powerful agent is not bound by logic. They are the author of logic.

I appealed to reason. — Bartricks

So, do you believe that Alan Turing's and Alonzo Church's answer to David Hilbert's question is not reasonable? On what grounds would you then believe that?

No, why do you think I think that?

We're talking at different levels. my question is about whether or not an omnipotent agent would have control over logic. What you're doing is talking about the content of logic. What you're talking about it is irrelevant. Whatever you say about the content of logic, my point is that an omnipotent being isn't bound by it.

If you say no question can be given a yes/no answer, the omnipotent being can give you a definitive answer to any question you ask. And so on.

A programmer can create a level he cannot complete, and then a cheat that enables him to complete it. In relation to the game-world, the programmer is omnipotent. Any constraints he may have are 'otherworldly'. — unenlightened

If an omnipotent being created a stone they could not lift, and then created a means to lift it, it would no longer be a stone they could not lift, and therefore not impossible. An omnipotent being has the capacity to make anything possible - even what is considered impossible from a certain perspective.

Wouldn’t any constraints on this being then be irrelevant to what is created?

We're talking at different levels. my question is about whether or not an omnipotent agent would have control over logic. What you're doing is talking about the content of logic. What you're talking about it is irrelevant. Whatever you say about the content of logic, my point is that an omnipotent being isn't bound by it.

If you say no question can be given a yes/no answer, the omnipotent being can give you a definitive answer to any question you ask. And so on. — Bartricks

Well, everything you are saying, is simply not decidable, unless you manage to point out a purely mechanical procedure that calculates your conclusion as its conclusion.

This requirement is simply part of the limitations of formal knowledge. You have never demonstrated that your yes/no question would fall within the boundaries of Turing-complete decidability/computability. If you do not demonstrate this convincingly, the only conclusion left, is that it falls outside these boundaries, and that formal knowledge-justification methods cannot reach it.

I refer you to my earlier answer. You're not addressing the question, or realizing that you're not addressing the question.

I refer you to my earlier answer. You're not addressing the question, or realizing that you're not addressing the question. — Bartricks

I am pointing out the issue of the addressability of the question. You seem to have pre-1936 views on this matter. Unlike what you seem to believe, disregarding the entire issue of decidability/computability will not help solving your question.

By the way, I am not answering Russell's and Mirimanoff's questions either. These problems were not solved by answering the question but by declaring them unanswerable.

Wouldn’t any constraints on this being then be irrelevant to what is created? — Possibility

Yes indeed, that is my point. From the POV of the game-world character, the fact that the programmer needs a comfort break. or occasionally cannot work out what will happen if he inserts this code, does not limit him, because he can always wind things back and do things over in another way until it goes just as he wants He sees that it is good. his potency over the game world is unconstrained even by the limits of the computer, because that will affect the speed the game runs globally but will be unnoticable to the game characters. Even the logic of the computer does not constrain the physics of the game.

Logic is a formal language. — alcontali

That is one concept of logic, but certainly not the only one.

No, it is to do with the concept of power. — Bartricks

Yes, omnipotence has, as the term indicates, to do with power. But the question is whether logical constraints are a limit on power, whether a logical contradiction limits what is possible or merely limits our understanding of what is possible.

my question is about whether or not an omnipotent agent would have control over logic. — Bartricks

And so, the paradox has to do with logic and not just power.

That is one concept of logic, but certainly not the only one. — Fooloso4

Yes, agreed.

It is a formal language along with transformation/rewrite/inference rules. The EBNF grammar of a particular language of logic does not capture its rewrite rules, but can/must be used to express them. So, a logic is indeed a complete formal system. There are, of course, numerous variants of different expressive power.

Still, the problem of computability/decidability is not caused by these inference/rewrite rules. These rules are just axioms, while it gradually (historically) became clear it is not the axioms causing (or solving) the problems. They are caused by the excessive, expressive power of universal quantifiers; which is a language problem.

It is a formal language along with transformation/rewrite/inference rules. — alcontali

The case is analogous to mathematics. There are fundamentally different views of their ontology. And, of course, there are different views of the ontology of language and thought. Are thinking and being the same? Is the logos a human activity or fundamental to being? Is it merely a human invention that attempts to give an account of what is or is it that which shapes and determines what is? That is, is logic merely descriptive or causal in the Aristotelian sense?

[Added: That is, the logos is regarded by some as an active, organizing principle that is prior to and and make possible formal languages and make possible the connection between thinking and being.]

I will not attempt to answer these questions, for any answer is based on certain assumptions that are not held in common by those who offer a contrary view.

I will not attempt to answer these questions, for any answer is based on certain assumptions that are not held in common by those who offer a contrary view. — Fooloso4

Well, yes, the ontology of logic and of mathematics are up in the air, and will probably remain so for the foreseeable future.

I was only referring to what causes the problem of undecidable questions in logic. According to Gödel incompleteness theorems, they are not caused by the axioms, or lack of axioms, and cannot be fixed by the axioms (including the axiomatic inference rules).

It is the language of logic itself that causes the issue. It occurs when propositional logic gets extended to first-order logic, by adding existential quantifiers. The decidability damage is caused by introducing just two symbols: ∀ and ∃.

It is the language of logic itself that causes the issue. — alcontali

I regard the paradox as a pseudo-problem since an omnipotent being is a hypothetical, but I do not think the problem of logical contradiction here is a language problem. I do, however, think the problem is compounded when one attempts to solve it on the basis of an abstract symbolic system.

A 2 dimensional person can be trapped inside a closed square but, according to how I understand it, a 3 dimensional being can just jump over the sides of the square. In other words impossibility is relative. — TheMadFool

This is good, as a circle perpendicular to the 2D square, like a rainbow.

Here is my 4-sided circle: O.

Sides one and two are the inside and the outside; sides three and four are the perpendicular top side and bottom side.

I also made an infinitely sided polygon circle: O.

I regard the paradox as a pseudo-problem since an omnipotent being is a hypothetical, but I do not think the problem of logical contradiction here is a language problem. — Fooloso4

The Creator of the real, physical world cannot be existentially contained in it. So, it is not a question of about the real, physical world. The question then becomes: Can human knowledge even reach outside the universe in order to answer questions about what we would observe there?

Furthermore, how are we supposed to ascertain that any purported answer really is the answer? We cannot try to inspect anything outside the universe. This objection would indeed involve the semantics of the question. But then again, if there are already syntactic issues, why even involve semantics?

I do, however, think the problem is compounded when one attempts to solve it on the basis of an abstract symbolic system. — Fooloso4

Yes, the abstract symbolic system will already fail on the bureaucracy of formalisms involved. They have limitations which prevent us from answering a whole range of questions about abstract, Platonic worlds, i.e. mathematical ones. Inasmuch as theories about the real, physical world -- in this case, even outside its boundaries -- make use of this bureaucracy of formalisms in order to maintain their own consistency, they will also start failing.

That is why the question, "Is it uberhaupt possible to address that kind of questions?", will automatically get propelled to the forefront. I wonder how anybody could take decidability/computability for granted, knowing that there is an entire field of investigation about just that issue?

Computability is the ability to solve a problem in an effective manner. It is a key topic of the field of computability theory within mathematical logic and the theory of computation within computer science. The computability of a problem is closely linked to the existence of an algorithm to solve the problem.

This entire field would not even exist, if all possible questions were solvable ...

The Creator of the real, physical world cannot be existentially contained in it. — alcontali

Some hold that there is only God and God's manifestations.

So, it is not a question of about the real, physical world. — alcontali

It is a question of whether an omnipotent being exists. The question takes it as a given, even if only for the sake of the argument, that such a being exists. That is not a question that is reducible to the physical world, but it is also not a question for which we have a common agreed upon answer. Hence, it is a pseudo-problem about a hypothetical.

Can human knowledge even reach outside the universe in order to answer questions about what we would observe there? — alcontali

I would frame it differently: can a being that is not omnipotent comprehend a being that is? This leaves open the question of whether an omnipotent being exists as well as the question of the limits of the "real" world.

I have answered the question. If you insist that the question is unanswerable, demonstrate that by showing my answer to be false.

Yes, but the way in which it involves logic is that it tells us something about what the nature of logic would need to be for there to be an omnipotent being. All Alcontali is doing is talking - irrelevantly - about the content of logic.
The point, though, is that an omnipotent being would have to be the author of logic. That is incompatible with some concepts of logic. Well, either those concepts are the ones that have something answering to them -in which case we can conclude that no omnipotent being exists - or we have good evidence that an omnipotent being exists, in which case we can conclude that the alternative concepts do not have anything answering to them.
So we can learn something about the nature of logic from this kind of inquiry.

Yes, but the way in which it involves logic is that it tells us something about what the nature of logic would need to be for there to be an omnipotent being. — Bartricks

One could base a logic on such an assumption, but to require logic to conform to such an assumption is not something many of us would support.

The point, though, is that an omnipotent being would have to be the author of logic. — Bartricks

Okay, but we do not know what such a logic would look like. Do we? How would this logic resolve the problem of contradiction?

Well, either those concepts are the ones that have something answering to them -in which case we can conclude that no omnipotent being exists - or we have good evidence that an omnipotent being exists, in which case we can conclude that the alternative concepts do not have anything answering to them. — Bartricks

The logical problem exists whether such a being exists or is simply posited.

So we can learn something about the nature of logic from this kind of inquiry. — Bartricks

I do not think that what you come away with from this inquiry is the same as what regard the problem to be. I do not think the problem is with logic, but rather what one expects from it.

It might be demonstrable that logic requires a god. The god it requires would be omnipotent because the god in question would have control over both its existence and content.

It might be demonstrable that logic requires a god. — Bartricks

Do you have such a demonstration?

The god it requires would be omnipotent because the god in question would have control over both its existence and content. — Bartricks

Well, I think we can agree that logic exists in some form or other. If we accept a logic that forbids contradiction then we must then address the contradiction or apparent contradiction present in the paradox.

You posit an imaged God-given logic without saying how it resolves the paradox.

I have answered the question. If you insist that the question is unanswerable, demonstrate that by showing my answer to be false. — Bartricks

I have never said that I think that your answer is false. I am saying that I am absolutely sure that your answer is not even false.