Arguments for the existence of God and Godel's Incompleteness theorem

[gnosis92]

Incompleteness_theorem


"Gödel's first incompleteness theorem shows that any such system that allows you to define the natural numbers is necessarily incomplete: it contains statements that are neither provably true nor provably false. Or one might say, no formal system which aims to define the natural numbers can actually do so, as there will be true number-theoretical statements which that system cannot prove. Some argue this refutes the logicism of Gottlob Frege and Bertrand Russell, which aims to define the natural numbers in terms of logic."

"The incompleteness results affect the philosophy of mathematics, particularly viewpoints like formalism, which uses formal logic to define its principles. One can paraphrase the first theorem as saying, "we can never find an all-encompassing axiomatic system which is able to prove all mathematical truths, but no falsehoods."

Perhaps the above discussion holds to any and all arguments for and against God's existence, that any arguments for or against God's existence requires axioms, that allow statements that are neither provably true nor provably false.

[RAFH]

Incompleteness_theorem


"Gödel's first incompleteness theorem shows that any such system that allows you to define the natural numbers is necessarily incomplete: it contains statements that are neither provably true nor provably false. Or one might say, no formal system which aims to define the natural numbers can actually do so, as there will be true number-theoretical statements which that system cannot prove. Some argue this refutes the logicism of Gottlob Frege and Bertrand Russell, which aims to define the natural numbers in terms of logic."

"The incompleteness results affect the philosophy of mathematics, particularly viewpoints like formalism, which uses formal logic to define its principles. One can paraphrase the first theorem as saying, "we can never find an all-encompassing axiomatic system which is able to prove all mathematical truths, but no falsehoods."

Perhaps the above discussion holds to any and all arguments for and against God's existence, that any arguments for or against God's existence requires axioms, that allow statements that are neither provably true nor provably false.

As noted elsewhere, proof is for mathematics (as above), courts of law (where anything can happen and usually does) and alcohol, which is the final resort for serious adherents of the previous.

[Antiplastic]

Incompleteness_theorem


"Gödel's first incompleteness theorem shows that any such system that allows you to define the natural numbers is necessarily incomplete: it contains statements that are neither provably true nor provably false. Or one might say, no formal system which aims to define the natural numbers can actually do so, as there will be true number-theoretical statements which that system cannot prove. Some argue this refutes the logicism of Gottlob Frege and Bertrand Russell, which aims to define the natural numbers in terms of logic."

"The incompleteness results affect the philosophy of mathematics, particularly viewpoints like formalism, which uses formal logic to define its principles. One can paraphrase the first theorem as saying, "we can never find an all-encompassing axiomatic system which is able to prove all mathematical truths, but no falsehoods."

Perhaps the above discussion holds to any and all arguments for and against God's existence, that any arguments for or against God's existence requires axioms, that allow statements that are neither provably true nor provably false.

Umm, what "axioms"?

Here is an argument. If living things show evidence of design, this is good reason to believe that God exists. Living things show evidence of design. Therefore there is good reason to believe God exists. Two premises and a conclusion. No axioms.

What on earth this has to do with number theory is anyone's guess. Certainly Godel's theorem says nothing whatsoever about any given argument requiring non-provable statements.

[Gawen]

Umm, what "axioms"?

Here is an argument. If living things show evidence of design, this is good reason to believe that God exists. Living things show evidence of design. Therefore there is good reason to believe God exists. Two premises and a conclusion. No axioms.I see your point. Creationists would consider your argument an axiom. But there is no self-evident truth or intrinsic merit in the argument at all. The only thing this could apply as an axiom is a statement accepted as true as the basis for argument or inference.

[Antiplastic]

I see your point. Creationists would consider your argument an axiom. But there is no self-evident truth or intrinsic merit in the argument at all. The only thing this could apply as an axiom is a statement accepted as true as the basis for argument or inference.

An argument cannot be an axiom, any more than an argument can be true or false. Arguments are valid or invalid. Only statements are true or false. "Axiom" has a precise definition in formal mathematical logic, and it is only with respect to this formal definition that talk about "axioms" has any relation to the completeness of formal systems.

[JohannGoodflag]

Perhaps the above discussion holds to any and all arguments for and against God's existence, that any arguments for or against God's existence requires axioms, that allow statements that are neither provably true nor provably false.

Well, yes --- however: (1) theists are rarely interested in valid formal logical deducations of the existence of gods within a formal frmework, and (2) the unprovable propositions may be about things that no-one, theists nor atheists, give a good goddamn about.

Godel's Incompleteness theorem can be quickly summarised as saying that no formal logical system is omnicompetent, regardless of how it's sentences are interpreted; whether it is considered to be mostly about number theory, theology, or cookie recipes doesn't matter. If it doesn't allow you to do number theory, that is a limitation of the system; and otherwise, it provably has some other limitation by Godel's Theorems. That's about it.

This doesn't mean no formal system can reasonably deal with gods: you can make any axioms about gods that you like and derive at least some things about them. The real question is whether any such model is in any way pertinent to the real world; and no theorem of logic or meta-mathematics can tell you this. That requires that you actually look at the world and see how useful your model is.

[Brian Bosse]

Hello Everyone,

Godel's Incompleteness theorem can be quickly summarised as saying that no formal logical system is omnicompetent, regardless of how it's sentences are interpreted; whether it is considered to be mostly about number theory, theology, or cookie recipes doesn't matter.

This is not correct. There are myriads of formal systems that Gödel’s Theorems does not apply to. For instance, sentential (propositional) calculus is a formal system that is complete. Gödel’s theorem only applies to those systems capable of expressing number theory. The reason for this is that any formal system that is less powerful is incapable of producing the special type of self-reference needed for the proof. Now, strictly speaking, his theorem only applies to formal axiomatic systems of sufficient strength. Arguments for God’s existence, although perhaps at times expressed formally, are part of informal systems. There is a subset of Christianity that speaks of an axiomatic system. But technically it is not a formal axiomatic system. However, even given their system, Gödel’s argument does not apply.

Sincerely,

Brian

[Witt]

Hello Everyone,



This is not correct. There are myriads of formal systems that Gödel’s Theorems does not apply to. For instance, sentential (propositional) calculus is a formal system that is complete. Gödel’s theorem only applies to those systems capable of expressing number theory. The reason for this is that any formal system that is less powerful is incapable of producing the special type of self-reference needed for the proof. Now, strictly speaking, his theorem only applies to formal axiomatic systems of sufficient strength. Arguments for God’s existence, although perhaps at times expressed formally, are part of informal systems. There is a subset of Christianity that speaks of an axiomatic system. But technically it is not a formal axiomatic system. However, even given their system, Gödel’s argument does not apply.

Sincerely,

Brian

Agreed. Not only propsitional logic is decidable and complete but modal propositional logic is also decidable and complete.

First order monadic predicate logic with and without identity is also decideable and complete, etc. etc.

But first order dyadic predicate logic, logic that includes relations R(x,y) etc.,
is not decidable. See: Alonzo Church (1936).

If we assume finite yet undetermined domains then all of predicate logic and higher order predicate logic with these finite domains, is decidable and complete.

[JohannGoodflag]

Witt, Brian: read what I wrote again, please.

I didn't say that all systems are incomplete; I said that no system is "omnicompetent", and proceeded to describe what I meant by this. But allow me to re-iterate more explicitly.

What was meant by "omnicompetent" was not that it is "totally competent" about what it is "supposed" to be about, but that it is "totally competent" about every subject about which one can discuss logically in which there is a fixed state of affairs which can be expressed with a finite axiom scheme; and I suggested that one such thing is number theory, so any theory which does not contain number theory is not "omnicompetent" for that very reason.

[Antiplastic]

Hello Everyone,



This is not correct. There are myriads of formal systems that Gödel’s Theorems does not apply to. For instance, sentential (propositional) calculus is a formal system that is complete. Gödel’s theorem only applies to those systems capable of expressing number theory. The reason for this is that any formal system that is less powerful is incapable of producing the special type of self-reference needed for the proof. Now, strictly speaking, his theorem only applies to formal axiomatic systems of sufficient strength. Arguments for God’s existence, although perhaps at times expressed formally, are part of informal systems. There is a subset of Christianity that speaks of an axiomatic system. But technically it is not a formal axiomatic system. However, even given their system, Gödel’s argument does not apply.


I always wished there could be a banner ad over internet discussion boards in the mode of Smokey the Bear, with the slogan "Only you can prevent Godel abuse." Seriously, it's just nails on a chalkboard to hear people spouting things like, what was it a few weeks ago, "the incompleteness theorem solves the paradox of omnipotence" or some such nonsense. If it brings theists and militant antitheists together in opposing it, you know it must be bad.

[Cheerful Charlie]

I see your point. Creationists would consider your argument an axiom. But there is no self-evident truth or intrinsic merit in the argument at all. The only thing this could apply as an axiom is a statement accepted as true as the basis for argument or inference.


Any number of people have worked to place religion on an axiomatic basis, Descartes and Samuel Clarkare good examples. This approach does not work very well. That is why it is not popular today. Axiomatic systems are supposed to pick the fewest basic, atomic axioms based on observed facts or ideas that cannot be further broken down and are not easily deniable. Then from these derive further ideas or facts. This means revelation cannot be a basis for an axiomatic religion as one has to make a choice of which revelation to use.

Axioms in such attempted systems are usually really only analogies as Hume noted. And axiomatic systems have trouble with things like the problem of evil and the problem of free will vs omniscience.

The most amazing abuse of the concept of axiomatic systems and religion was Scientology's L. Ron Hubbard who wrote a book with several hundred pompous "axioms".

Cheerful Charlie

[JohannGoodflag]

I always wished there could be a banner ad over internet discussion boards in the mode of Smokey the Bear, with the slogan "Only you can prevent Godel abuse." Seriously, it's just nails on a chalkboard to hear people spouting things like, what was it a few weeks ago, "the incompleteness theorem solves the paradox of omnipotence" or some such nonsense. If it brings theists and militant antitheists together in opposing it, you know it must be bad.

Yes, I agree with you. It's unfortunate, though, that the reaction to Godel abuse apparently makes it difficult for other people to bother reading what one writes, when one tries to present a new (or at least independently derived) interpretation for what it means.

[Brian Bosse]

Hello JohannGoodflag,

I didn't say that all systems are incomplete; I said that no system is "omnicompetent", and proceeded to describe what I meant by this...It's unfortunate, though, that the reaction to Godel abuse apparently makes it difficult for other people to bother reading what one writes, when one tries to present a new (or at least independently derived) interpretation for what it means.

I did not mean to be rude. I did read your post, and saw your use of the term 'omnicompetent.' I should have asked you to clarify this, but rather I assumed you meant 'incomplete.' My mistake. Nevertheless, Gödel’s Incompleteness Theorem can not be "quickly summarised as saying that no formal logical system is omnicompetent." This is not what it says.

Sincerely,

Brian

[JohannGoodflag]

I did not mean to be rude. I did read your post, and saw your use of the term 'omnicompetent.' I should have asked you to clarify this, but rather I assumed you meant 'incomplete.' My mistake. Nevertheless, Gödel’s Incompleteness Theorem can not be "quickly summarised as saying that no formal logical system is omnicompetent." This is not what it says.

It says that for any formal system which is powerful enough to express number theory, there is a well-formed sentence which cannot be proven, nor can its' negation be proven, if the system is consistent. Then, in a manner of speaking, either a formal system is competent about nothing (if it is inconsistent), it is not competent on the subject of number theory (if it cannot express it), or there exists a well-formed sentence about which it is not competent (that is, it is unable to either prove or disprove it).

In what sense would you say that "no formal logic system is omnicompetent" a poor summary of Gödel’s Incompleteness Theorem, given this idea of the competency of a logical system? What is your disagreement with this summary, or with the concept of "competency"?

[Brian Bosse]

Hello JohannGoodflag,

In what sense would you say that "no formal logic system is omnicompetent" a poor summary of Gödel’s Incompleteness Theorem, given this idea of the competency of a logical system?

The idea of competency that you referenced in your question was…

Then, in a manner of speaking, either a formal system is competent about nothing (if it is inconsistent), it is not competent on the subject of number theory (if it cannot express it), or there exists a well-formed sentence about which it is not competent (that is, it is unable to either prove or disprove it).

You are drawing conclusions based on Gödel’s Incompleteness Theorem. This is demonstrated by your use of ‘then’ to begin your explanation. These conclusions are not the theorem (they are further consequences you are drawing), and therefore are not a summary of the theorem.

What is your disagreement with this summary, or with the concept of "competency"?

Your definition of ‘omnicompetency’ is vague. I can create a formal axiomatic system that upon interpretation captures the concept of addition. It is consistent, it can express addition perfectly and there exists no true well-formed sentence in the theory that is unprovable. This seems to qualify as an omnicompetent system relative to addition, and as such contradicts your claim that there exists no such system. This omnicompetent system for addition is developed in my blog series Gödel’s Theorem (http://www.christianlogic.com/brianbosse/archives/2006/02/goedels_theorem.html). I did not invent this system, but reference the inventor in the blog.

Sincerely,

Brian

[JohannGoodflag]

You are drawing conclusions based on Gödel’s Incompleteness Theorem. This is demonstrated by your use of ‘then’ to begin your explanation. These conclusions are not the theorem (they are further consequences you are drawing), and therefore are not a summary of the theorem.

Well, I never claimed that the statement of Godel's Incompleteness Theorem is that no formal logical system is competent about all things. I said that it could be summarized as saying that, in the same sense that the first and second laws of thermodynamics can be summarized as saying that you can't win, and you can't break even. No-one pretends to use these statements to do physics; and these statements are also consequences of the laws of thermodynamics, contingent on the meaning of "win" and "break even" for that context, rather than clearly equivalent propositions.

The generally accepted meaning of "summary of a technical statement" is that it should be the image of a statement through a homomorphism, not an isomorphism.

Your definition of ‘omnicompetency’ is vague.

Natural language, even when used for the purposes of describing technical results, often is vague. But if you insist, I can do my best to formalize it.

A formal logical system is competent with respect to a well-formed sentence S of that system if either S or ~S is provable in the system, but not both.

This is just soundness and completeness restricted to a particular sentence. (Of course, competence about any sentence requires consistency of the formal system, which is a global property.) Of course, this doesn't capture what I want to assert as regards number theory, so I may extend it as follows:

A formal logical system F is competent with respect to a formal logical system G if there is an isomorphic image of the axioms of G in the theorems of F. F is said to be competent with respect to number theory if it is competent with respect to P (the axiom system used in Godel's article "On undecideable propositions in Principia Mathematica and related systems").

The idea of "omnicompetence" is more hand-wavy: it meant to be a sort of ideal which we would ideally like. I don't have a good idea of exactly how to formalize it, but the idea is that a formal system would be omnicompetent if you could use it essentially to study anything you want, and be confident that it is complete and sound. The idea would be that it could stand in for any other consistent theory. Two formal systems with different sets of axioms which are inconsistent with one another would have different, non-overlapping images in the theory. Basically, this is no more or less than the aim of Hilbert's program as I understand it.

In particular, we may require that an omnicompetent system would be competent with respect to its' own well-formed sentences, as well as number theory. Godel's Incompleteness Theorem states that no such system exists. Then, as a corollary if you like, no omnicompetent system exists.

I can create a formal axiomatic system that upon interpretation captures the concept of addition. It is consistent, it can express addition perfectly and there exists no true well-formed sentence in the theory that is unprovable. This seems to qualify as an omnicompetent system relative to addition, and as such contradicts your claim that there exists no such system.

Hopefully it is clear now that "omnicompetent" is not meant to be relative to a particular branch of mathematics, but was supposed to entail the ability to encapsulate all branches of mathematics.

I do know that I have not succeeded in completely formalizing the concept of "omnicompetence" (nor did I intend to give the impression that I was making any formal statements at all, earlier). If you feel that it is crucial to do so, in order for my summary to be quite appropriate, would you hazard a suggestion for how to progress towards a suitable definition of omnicompetence, in the spirit of Hilbert's programme?

[Brian Bosse]

Hello JohannGoodflag,

Before I begin, I would like to say that I appreciate the discussion. It has been fruitful for me.

I said that it (Gödel’s Incompleteness Theorem) could be summarized as saying that (no formal logical system is competent about all things), in the same sense that the first and second laws of thermodynamics can be summarized as saying that you can't win, and you can't break even.

I had to look this up, but it seems that the first law says that energy cannot be created or destroyed. I am not sure how this translates to “You can’t win” or “You can’t break even.” The second law deals with entropy. Again, I am not sure how this relates. But this is all beside the point, so please do not feel like you have to explain. The question is whether or not “Gödel’s Incompleteness theorem can be quickly summarized as saying that no formal logical system is omnicompetent.”

You have defined the omnicompetence of a system as being consistent and complete when interpreted as some type of “theory of everything.” Hilbert’s program was not this ambitious, but rather relegated itself to mathematics. Gödel’s Incompleteness Theorem only applies to axiomatic systems strong enough to express number theory. If there were such thing as an omnicompetent system, then Gödel’s Incompleteness Theorem would apply to that system. But to say "'no formal logical system is competent about all things' necessarily entails Gödel’s Incompleteness Theorem (i.e., summarizes it, or captures it)" is too much. Consider this abstract illustration:

P1 states ~X.
Let's say Z --> (X ^ Y)

Is it appropriate to say that P1 can be summarized by ~Z? P1 certainly leads to ~Z. However, ~Z does not necessarily lead to P1. If someone just has ~Z and does not know P1, then they can only conclude ~(X ^ Y). This leaves three possibilities:

(1) ~X (P1)
(2) ~Y
(3) (~X ^ ~Y) (this leads to P1 as well)

Out of these three possibilities, (2) has nothing to do with P1. Since a denial of Z does not necessarily mean a denial of P1, then Z does not capture the essence of P1. Let ~X stand for Gödel’s Incompleteness Theorem, and let Z stand for your omnicompetent system. Y could stand for a myriad of necessary things, one of which is that this system is a theory of everything. The point being that if ‘A’ captures ‘B’, then the assertion of ‘A’ must lead to ‘B’. If there is any possibility otherwise, then ‘A’ does not capture ‘B’. What do you think?

Sincerely,

Brian

[JohannGoodflag]

You have defined the omnicompetence of a system as being consistent and complete when interpreted as some type of “theory of everything.” Hilbert’s program was not this ambitious, but rather relegated itself to mathematics.

I hope I didn't say anything that would lead you to believe that an omnicompetent system would have anything to say, e.g., about physics or chemistry. I meant that it would be limited to "mathematics" --- which, however, is not a very strong limitation, as the consequences of any consistent axiom scheme in a formal logical system could defensibly be called "mathematics", or at least "thinly disguised arithmetic".

What tends to characterize mathematical study is the presentation of a set of axioms defining an object of interest (e.g., a group, a topological space, a graph) and structures of interest which can occur within or between such objects (a homomorphism, a connected set, a walk), followed by an examination of the properties of these objects. Since the turn of the 20th century, we have decided that we like the idea of being able to take these axioms and use them to identify objects in a larger theory, which has played the role of a widely encompassing foundation. When working with graphs, we don't usually care about the fact that the graph can be expressed as a set of vertices and edges, or that there exists an object called the set of real numbers, but by using ZF (or ZFC) as a foundation, we can make use of them when it is convenient.

By an omnicompetent system, I am trying to paint a portrait of an ideal such foundation; one which would allow us to perform number theory, as well as any concievable other mathematical subject (including onces not yet concieved). A system, in short, sufficient to any task we might want to put it to, in the context of doing "mathematics" --- that is, sufficient to any task to which we would ever wish to apply a formal system to; which we could be confident would not produce contradictions as theorems, but which can either prove or disprove any well-formed sentence of any consistent theory which it can simulate (which is all of them).

It seems to me that such a system was essentially Hilbert's programme. I believe it was Hilbert who said that anything which did not lead to a contradiction could be regarded as true (in defence of non-constructive proofs); if I have remembered the whole quotation, one presumes he means that it should not produce contradictions when adjoined to accepted mathematics such as number theory. It seems to me that this spirit is essentially one which embraces the idea of an omnicompetent formal system of the sort I have tried to describe. At least, thanks to the technique of Godel numbering, we now see that it was a consequence of Hilbvert's programme. Of course, we also know now that no such formal system can exist.

If there were such thing as an omnicompetent system, then Gödel’s Incompleteness Theorem would apply to that system. But to say "'no formal logical system is competent about all things' necessarily entails Gödel’s Incompleteness Theorem (i.e., summarizes it, or captures it)" is too much. Consider this abstract illustration [...]

Suffice it to say that you think that a summary of a technical result should nonetheless be equivalent to the statement, and not merely to try to communicate the impact or ramifications of the theory. I did gather that already, although you apparently did not gather that.

Do you have no suggestions as to how to sharpen my ideas for summarizing Godel's Incompleteness in terms of "the competence of a formal system" or "the things about which a formal system is competent"?

* * *

Incidentally, about:
I had to look this up, but it seems that the first law says that energy cannot be created or destroyed. I am not sure how this translates to “You can’t win” or “You can’t break even.” The second law deals with entropy. Again, I am not sure how this relates.

The first law says that energy cannot be created or destroyed. That is, there is no process by which the universe at large can make an energy profit; thus, "you can't win".

The second law can be stated as saying that there is no process which does nothing but consume heat energy to do work; e.g. some enewrgy will be lost as heat to a cooler system, from which you can only extract heat by losing heat to yet another cooler system, etc. That is, once you lose some energy as heat, you can never recover all of it; thus, "you can't break even".

[darkfox]

Hopefully it is clear now that "omnicompetent" is not meant to be relative to a particular branch of mathematics, but was supposed to entail the ability to encapsulate all branches of mathematics.


Erh! I don't profess to be an expert on the issue, but from what I have read you have pretty much encapsulated some of the consequences of the theorem. And I think you have more than elucidated your use of the term omnicompetent - that is to say, I get it!

[Brian Bosse]

Hello JohannGoodflag,

I hope I didn't say anything that would lead you to believe that an omnicompetent system would have anything to say, e.g., about physics or chemistry.

There were two things that lead me to think the way I did: (1) your use of the prefix ‘omni-’; (2) when you said, “The idea of ‘omnicompetence’ is…that a formal system would be omnicompetent if you could use it essentially to study anything you want, and be confident that it is complete and sound.” No biggy. I now understand you to mean any formal system that purports to capture the totality of mathematics would be considered ‘omnicompetent’.

In essence, there already is such a system. It contains the axioms of propositional calculus, along with predicate calculus with identity and the axioms of ZFC set theory. Practically all of mathematics can be derived from this foundation. This system is what I understand you to be calling omnicompetent – even though, as you say, it fails to be so.

Suffice it to say that you think that a summary of a technical result should nonetheless be equivalent to the statement, and not merely to try to communicate the impact or ramifications of the theory.

A summary of the statement should capture what the statement says. You claimed that “No formal logical system is omnicompetent” summarizes Gödel’s Incompleteness theorem. By this, you have clarified that you are referring to systems that represent all of mathematics. This does not work because Gödel’s Incompleteness theorem applies to much more than this. It applies to a whole host of formal systems that are not omnicompetent. Peano’s arithmetic is one example. Your summary would not apply to this formal system. Also, your summary would not apply to the system used in Russel and Whitehead's Principia Mathematica, and this is the very system Gödel used in his proof.

Do you have no suggestions as to how to sharpen my ideas for summarizing Godel's Incompleteness in terms of "the competence of a formal system" or "the things about which a formal system is competent"?

Here is how you defined ‘competence’: “A formal logical system is competent with respect to a well-formed sentence S of that system if either S or ~S is provable in the system, but not both.” It seems you are saying that a logical system is ‘competent’ if and only if it is consistent (does not prove both S and ~S) and it is complete (it can prove from all well formed sentences the sentence or its negation). This makes intuitive sense to me. In this case, my summary would be as follows…

Gödel’s Incompleteness Theorem (Competence Summary): No formal system strong enough to express number theory is competent.

What do you think? Am I understanding you now, or am I still missing it?

Brian

[JohannGoodflag]

In essence, there already is such a system. It contains the axioms of propositional calculus, along with predicate calculus with identity and the axioms of ZFC set theory. Practically all of mathematics can be derived from this foundation. This system is what I understand you to be calling omnicompetent – even though, as you say, it fails to be so.

It fails to be so because "omnicompetent" does not simply mean "able to act as a foundation for mathematics". You have forgotten the earlier intent that it should also be sound and complete.

Here is how you defined ‘competence’: “A formal logical system is competent with respect to a well-formed sentence S of that system if either S or ~S is provable in the system, but not both.” It seems you are saying that a logical system is ‘competent’ if and only if it is consistent (does not prove both S and ~S) and it is complete (it can prove from all well formed sentences the sentence or its negation). This makes intuitive sense to me. In this case, my summary would be as follows…

Gödel’s Incompleteness Theorem (Competence Summary): No formal system strong enough to express number theory is competent.

What do you think? Am I understanding you now, or am I still missing it?

Well, you have missed the fact that 'competent' as I defined it above is relative to a single sentence S; with the intent that, in a manner of speaking, 'omnicompetence' is meant to entail competence applied to all sentences not only in the formal system, and that the formal system include isomorphic images of all other formal systems of interest.

My comment was not supposed to entail that competence was a global property. I just commented that for a formal logical system, competence in anything requires soundness. So, the 'not both' requirement for competence with regards to a single sentence is uninteresting, for formal logical systems; one might imagine a more general sort of system, not necessarily a logical system, which is competent for broad classes of sentences while still being inconsistent for others. (Of course, in order for the label 'competence' to be meaningful, we'd have to have some other reason to consider the conclusions of this non-logical system worth considering. Short of an "intelligent being" of some sort --- which is itself a fairly ill-defined sort of beast --- I can't think of what such an occasionally consistent system would be; but never mind that.)

"Competence" is not meant to be a property of a formal logic system alone. By analogy, I am not competent to comment on psychology, but this doesn't mean I am not competent in anything; there are merely limits to the domain of my competence. Omnicompetence was meant to be the idea of "competence in everything", in a manner that makes sense given that we're talking about formal logical systems and not, e.g., intelligent agents acting in the universe.

I'm afraid I've gotten a bit tired of beating this dead and admittedly ill-defined horse. If you happen to have any ideas for how to refine my concepts of competence and omnicompetence for some future occasion, I'll be happy to listen; but if not, no worries.