Don't get me wrong, it's a very important point (the core of the thread) that the Godel theorem is not actually an incompleteness theorem. Obviously the tendency is to imagine it is, ergo its name, so I'm glad you pointed out its name. If you find any fault in the proof I gave that it is not an incompleteness theorem, or my argument that the possibility that arithmetic might be inconsistent is the only thing that saves it from being inconsistent, then I'd like to hear it. Surely it must be admitted this is a fascinating and surprising aspect of arithmetic.

Indeed. I will try to clarify:

As stated in an earlier post, things experienced thru our senses are a priori (conceived beforehand, self evident) - my understanding is they need no proof - however by way of comparison, I liken "proof" of a priori to the mere fact it is experienced by the senses. That is what I meant in last post by handing you the chair. We both came to know the chair in the same time and place thru our senses. It is true to us both a priori. But the synthetic a priori, while based upon truths derived a priori, can not be concluded true in the same physical manner, but rather by intellect alone, and therefore, can not be "proved" with our physical senses, yet are true none the less.

Campred to Godel statement:

If I have it straight, not all true statements in a theory can be proved true by the axioms in the theory, thus conclusion is the theory is incomplete. My analogy to the above is that - the staments in a theory that are proved true by the theory itself, are similar (in theory) to real a priori - i.e. their truth was "self" evident ("self" being the theory).

However, true statements that can not be proved by the axioms, I am likening to synthetic a priori.

Hope that clears it up.

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Where did everyone go?

I think that is what most people would call "a posteriori", that which you don't know until after you have experienced/perceived it.

OK, then you are making the same distinction I am but with different terms. What you are calling "a priori" is what I called "actual truth", and what you are calling "synthetic a priori" is what I called that fraction of actual truth that is not logical truth.

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Logic is the grammar of truth.

Meaning and absolute certainty are incompatible, and profound meaning and absolute certainty are profoundly incompatible.

The only thing intelligence is capable of is recognizing itself.

This sentence was in error: "By that it sounds like you are saying that truth is what you get at the end of a logical process." That was not what I was saying nor was it implied. Was that error in logic a misunderstanding or a misrepresentation? The point may not longer be at issue, as the discussion has passed it by, but I consider it important to decide if further discussion would be fruitful, depending on whether you can or cannot admit to being wrong.

You seem to be placing more constraints on the basic definitions than am I. Moreover, I think you have it backwards. You write: "A key aspect of precision involves objectivity and repeatability", but I would reverse that and say that a key aspect of objectivity and repeatability is resolution. Resolution, as a description of a datum, is more fundamental than the operations you may or may not perform upon it.

Have you not elsewhere stated that meaning is what we call truth? Mathematics performs operations on symbols. It is we who substitute data for the operands and meanings for the results.

The issue in my statement is not about the applicability of the "Godel proof", but of your behavior. I dislike what appears to be your use of the Socratic method as it appears to be a bait and switch tactic.

I'm not a mathematician but I know a little about communications. The issue seems to be about the differences between the abstract and the concrete. I don't want to get into a discussion about archtypes. It might be simpler to note that perception of data and meaning is individual and that is called Dialogism. Meanwhile, mathematics, other forms of logic and many forms of communications assume that data and meaning are all experientially the same and that is called Monologism. The latter can be more precise, by manner of it's assumptions about meaning, however, the former is more accurate because it takes into consideration a range of observations that has a higher fidelity with perceptions of reality.

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"What you think you thought you saw you did not see." Agent J, MiB - Manhatten Bureau

Here the issue is in regard to whether or not resolution by itself characterizes logical truth, or whether there are also important issues of objectivity and repeatibility. You are claiming the additional elements are redundant because they are included in your concept of resolution. If that is how you define resolution, then we are in full agreement-- you can certainly interpret my adding repeatibility and objectivity to resolution as a means of ensuring that the concept of "resolution" as you are using it includes these important elements.
Absolutely, that's what I mean by logic connecting the meanings. Once we make those substitutions, the combined result exemplifies logic connecting the meanings that we have substituted. Logic is never responsible for those substitutions, so logical truth is assessed independently of meaning. I'm raising two points in regard to the result of these substitutions: if the meaning we are substituting is not true, then perfectly correct logic will not necessarily connect them to other true meanings, so logic can take no responsibility for the truth of the meanings. What's more, if the meanings we introduce are close to true, there is no guarantee that correct logic will not connect to meanings that are farther from true.
Everything I'm doing is intended to get at the "truth" about logic and truth. Nothing I'm doing is devious or hidden, all my statements are to be taken completely at face value. The Socratic method is merely a means of elliciting discussion. Don't shoot the messenger, the question is, do you want to understand the connections between truth and logic? I do.
This is an interesting issue, we might tend to associate logical truth with abstract truth and actual truth with something more concrete, but I actually think that it would be more correct to classify both abstract and concrete truths as actual truths. This is because logical truth is pure syntax, it cannot have any meaning whatsoever (as you point out, the meaning is all from our substitutions, whether we are substituting an abstract concept or a concrete one). This is important, because my core point here is that mathematics, as a means at arriving at both concrete and quite abstract truths, still cannot naively associate logical truths with actual truths. They are still different animals, even when abstract. That is what the Godel proof is telling us (and that proof manipulates actual truths that are about as abstract as they come).

Then you can interpret this thread as establishing, with Godel's help, that monologism does not work categorically for mathematics. One must notice the difference between symbolic data (I take "data" to mean the symbols and their defining syntax, devoid of their meanings to us) and actual meaning, or else one cannot understand why mathematics might not be inconsistent. I did indeed prove above that if everything that is true about the symbolic data must be the same as what is true about its meaning, then mathematics has to be inconsistent (because then the Godel statement can be proved and it asserts that it cannot be proved).

ETA: added the omitted if.

For what its worth, thats about what I get from it.

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Where did everyone go?

An inconsistent set of axioms does not simply admit untrue things that can be proved. It does do that, but in an inconsistent system all statements that can be formulated are both true and false.

Despite what you have said elsewhere, the inconsistency of a set of axioms renders it completely useless. There is no recovery.

There are proofs of the consistency of ordinary arithmetic. What Godel proved is than any set of axioms that is sufficiently rich to admit the natural numbers is either inconsistent or does not admit a proof withiin that set of axioms of its own consistency. That does not rule out proofs of the consistency of the original set of axioms in some other, possible larger and stronger, set of axioms.

So, for instance Peano arithmetic is provable within Zermelo Fraenkel set theory, or using transfinite methods as in Gentzen's proof.

If the axioms of arithmetic were inconsistent the result would be catastrophic for mathematics and for those disciplines that rely upon it. For instance, the statement 1+1=2 would be false (and true ).

In fact Godel's proof procedes by showing that if the axioms of arithmetic are consistent then the Godel sentence is true. That is the crux of the proof.

On the other hand, if the axioms of arithmetic are inconsistent then the Godel sentence is both true and false, as is every other sentence that can be formulated.

So in either case the Godel sentence is true.

Well, the proof proceeds by translating the Godel statement into an arithmetical expression, and therefore it takes on the same status as all arithmetical expressions, as either actually true, or actually untrue, as well as either provable from the axioms, or not provable from the axioms. That is as far as Godel needs to go, he does not need to prove the Godel statement (he cannot), he need only say that if it is provable, then it is a false arithmetic statement, and that renders arithmetic inconsistent, since a provable falsehood = inconsistency, or if it is not provable, then it is a true arithmetic statement, and an unprovable truth = incompleteness. It must be one or the other, but making the assumption that the axioms of arithmetic are consistent would be disastrous. I showed that above. Here is the flaw in your claim:

If you are right, that Godel can add the axiom (call it A1) "the Peano axioms are consistent" to the Peano axioms, and then use the new set of axioms to prove the Godel statement, then the Godel statement is a false arithmetic expression in the new system, and the new system Peano + A1 is inconsistent. But all we did was axiomatize the assumption that the Peano system is consistent, and we render the resulting system inconsistent! Clearly, it is essential that we not know that the Peano system is consistent if we are to have any hope that it actually is.

This is the subtlety that this whole thread has been about. We can only say we know the Godel sentence is true if we draw no distinction between actual truth and logical (provable) truth, but the point of this thread has been to show how essential that distinction actually is. What we actually know is that the Godel statement, translated arithmetically, is an expression that is either actually true or logically (provably) true, but we also know it cannot be both, and we do not know which one it is. If it is provable, then it is actually untrue, and arithmetic is inconsistent. If it is actually true, then it is unprovable, and arithmetic is incomplete. We still don't know which, and it would be a Really Bad Thing if we ever do-- because as I said, the only one we could ever be certain of is that it is inconsistent.

Now you see why it is so important to maintain a difference between actual truth and provable (logical) truth! If we do not make that distinction, then any assumption that arithmetic is consistent (adding axiom A1) will render it inconsistent. It is only by maintaining the distinction I refer to above that we can use the Peano axioms without A1, and be saved from inconsistency by our own sincere uncertainty about whether A1 is a valid axiom. If we do not recognize that distinction, then your argument goes through, with the disastrous result that we know the Godel statement is true, so we have proven the Godel statement, so arithmetic is inconsistent.

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Logic is the grammar of truth.

Meaning and absolute certainty are incompatible, and profound meaning and absolute certainty are profoundly incompatible.

The only thing intelligence is capable of is recognizing itself.

If you add your A1 ("arithmetic is consistent") to the system that Godel showed to be incomplete, then would we not have to address that new and different system? In fact such a system will have its own Godel sentence.

As before, all I can say is that there is an extensive literature.

I'd concur that no ideas are ever dismissed on the basis of the originator. Merely on the basis of the ideas themselves.

And yes, sometimes it's an uphill battle! Hang it there...

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If I set the budget, we'd have Ares and more. Unfortunately, I don't set the budget, and Ares is just too expensive and too far out for us to accomplish our goals within the budget we were given.

If we halt the ISS, all versions of Ares, and transport Orion and Altair aboard DIRECTv3's Jupiter family of Shuttle-Derived Launch Vehicles, we just might make it back to the Moon by 2020.

I presume that is known by all.Not so, I was right then and I'm still right. Indeed, it is obvious that an inconsistent set of axioms can be quite useful if the inconsistency is never encountered. To wit, it is clearly possible that a set of axioms might admit a proof of both A and ~A, yet a million years of mathematics might never encounter such a proof. Such a problem is then perfectly harmless. Indeed, this could be exactly the state of affairs of the arithmetic axioms. How can you prove otherwise? (Any proof you offer, if correct, would also prove that arithmetic is inconsistent-- yes, you heard me, any proof that arithmetic is consistent proves that it is inconsistent.)
Correct.

Certainly, for example I can add the axiom "The Peano axioms are consistent" and prove it in one step in that larger system. Of course, in that case, the larger system would be inconsistent, as I showed above.

Quite so, but we still don't know if those larger sets are incomplete or inconsistent, so such proofs do not assert any truth that is independent of those axiom sets. They are logical proofs only-- they are semantic proofs, tautologies. The moment we know the Peano axioms are consistent by their meaning, that is the moment we know they are inconsistent by their logic-- because if they are consistent by their meaning, then the Godel statement cannot be untrue or it would be provable and untrue, which proves the Godel statement, which makes its meaning untrue, which makes the system inconsistent by logic.
But that would be harmless if we didn't know it.

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Logic is the grammar of truth.

Meaning and absolute certainty are incompatible, and profound meaning and absolute certainty are profoundly incompatible.

The only thing intelligence is capable of is recognizing itself.

Why would we jump from "unprovable within a particular formal system" to "untrue"?

First of all, Godel did not show the system to be incomplete, I proved that already. If you don't see it yet, just add "the Peano axioms are incomplete" to the Peano axioms-- that should be no problem if Godel proved it. But, it will be a problem, because that new set will be inconsistent, because if we assume the new axioms are consistent, we can use them to prove the Godel statement for that set, which contradicts the Godel statement. What does it mean when if you include an axiom that asserts "A" to a set of axioms, and the result is inconsistent? It means that A is not consistent with those axioms. So you are claiming that Godel proved something from the axioms that is not consistent with the axioms, you are claiming the Godel proof is wrong.

Secondly, if you attach A1 to the Peano axioms, you will indeed have a new Godel statement, but the relationship of the axioms to that Godel statement will be quite different-- those axioms are inconsistent (as I showed), so the Godel statement is now provable. That axiom set is thus irrelevant to the Peano axioms.If it doesn't include my argument, then the "extensive literature" is missing something rather important. Maybe no one here has read enough of it.

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Logic is the grammar of truth.

Meaning and absolute certainty are incompatible, and profound meaning and absolute certainty are profoundly incompatible.

The only thing intelligence is capable of is recognizing itself.

Yes we get an infinite regress of incomplete formal systems. Is that a problem?

It would not be a regress of incomplete systems-- the very first time we put in the axiom "the Godel statement is true in this system" (or "this system is consistent", or "this system is incomplete", it all amounts to the same end), we no longer have incomplete systems. We have inconsistent systems. This is also why the Godel proof is not an incompleteness theorem, despite the proclivity for hopeful (but naive) mathematicians to call it that. There is no problem with hoping that the system is incomplete, the problem is with claiming you know it is incomplete, which if true, renders it inconsistent and complete.

Your assertion is simply absurd, and it conclusively demonstrates a total lack of understanding of the nature of axiomatic mathematics and the content of the Godel theorems.

As you have recognized elsewhere, any set of axioms that is inconsistent has the property that every sentence that can be formulated is both true and false. That is simply an untenable state of affairs. No such set set of axioms could possibly be "quite useful", or in fact of any use whatever. In that situation 1+1=2 and also 1+1=7. Perhaps that would serve your purposes, but it would not serve the purpose of most others.

You were not "right then". You are not "right now". You are simply oblivious.

Whether or not one is aware of the inconsistency is completely irrelevant. Lack of awareness, simply sticking your head in the sand, does not cure the problem. It is no more a cure than ignorance of one's cancer is an assurance of prolonged life.

This point is too ridiculous for any further debate. We've been through this before and there is simply no educating you.

This is just ridiculous. If you start with a system in which the Peano axioms are inconsistent, then nothing that can be added will make them consistent, most certainly not a simple statement that they are consistent. In fact, all that you have done in that case is add one more inconsistent sentence.

On the other hand if the axioms are consistent, then adding a statement that they are consistent adds nothing.

So in either case you have demonstrated nothing except a lack of understanding.

If you think that this statement makes sense, then I suggest that you go get some help.

I am not going to debate this further. There is no point, But it is important that there be a counter point to your assertions, which are , most assuredly, ATM.

Not only is my assertion not "absurd", it is obviously true. Do you actually expect me to believe that it is not possible that mathematics could go a million years without encountering such a proof, for some set of axioms? How on Earth are you going to establish that impossibility? I'm all ears. Oh, you're not, your "proof" is simply going to be a claim that anything else is "absurd." That's what you call a logical argument? Yes, that's perfectly obvious, but only insofar as one takes "true" = provable, not in terms of its meaning. I told you that distinction is important.
That's not only not obvious, it's not true. It is perfectly tenable if no such proof is every encountered. That is just a fact, I cannot imagine how you could think otherwise, but you certainly have not lent any support to your claim. Again: the existence of a proof that 1+1=7 is not any kind of problem if no such proof is ever encountered by any mathematician, perhaps because it requires an intellect to produce that no human has ever achieved.
Not if there exists proofs that 1+1=2, while no proof for 1+1=7 is ever encountered.

You are missing the point. Read what I said again.

Irrelevant reasoning by analogy. This is your argument, it's like cancer?
I'm sad that you cannot make your case and must resort to such silly claims. I expect more from an expert.
Obviously. I think if you actually read what I said, you'll find I proved that adding that axiom would make it inconsistent, not consistent. Yes, I proved that, nor have you found any flaw in that proof. It's not very helpful if you do not even know what I'm saying.Nevertheless, adding that axiom to the Peano axioms would indeed render the new system inconsistent. Let me submit the proof again. Here is the proof that if you take the Peano axioms, and add another axiom (A1) that says "the Peano axioms plus this axiom results in a consistent set of axioms", you end up with a system that is inconsistent:

Form the Godel statement G in the system Peano + A1. Prove (from its axioms) that PA1 is consistent, so if G does not have a true meaning, then G cannot be proven. But G asserts that it cannot be proven, so if G does not have a true meaning, then it has a true meaning. The system PA1 is therefore inconsistent.

Now, instead of complaining about how "absurd" this is, find a flaw in the proof, or admit that you cannot.

Then you should have no trouble finding a flaw in my proof. And if you cannot, then which of us has demonstrated a lack of understanding?
I wish! No, I'm sure my proof is quite well known by mainstream mathematics, it is the proof that no consistent system that is rich enough to form a Godel statement can ever prove its own consistency. Which is precisely why if the Godel theorem was really an incompleteness theorem, it could use Peano arithmetic to prove the consistency of Peano arithmetic, which would render Peano arithmetic inconsistent, and that would be pretty unpleasant. So quite fortunately for Peano arithmetic, and unfortunately for my fame, my proof is not ATM.

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Logic is the grammar of truth.

Meaning and absolute certainty are incompatible, and profound meaning and absolute certainty are profoundly incompatible.

The only thing intelligence is capable of is recognizing itself.

There is clearly no form of words that I can supply that will persuade Ken G that Godel's theorem is an Incompleteness Theorem

I could continue to say that it is called that because we assume that arithmetic is consistent, not because it proves arithmetic consistent, but it is now clear that this would never be persuasive.

Well you could always try a proof, like the one I supplied that it is not an incompleteness theorem, because I know what those words mean and how to manipulate them mathematically. Seriously, what it this place, an opinion sounding board? If someone supplies a proof, either you can find a fault in it, or you must accept it. This is mathematics, not rhetoric.
Well, you certainly could say that, and you'd be agreeing with what I've said all along. Yes really, that is just precisely what I've said all along. You see, my sin here was only one thing: the absolutely true statement that we cannot say the theorem is an incompleteness theorem because the axioms might be inconsistent. That's it, period. Obviously you can hope or assume or imagine anything you like, it's a free country! All I did above was to add something that I think is quite interesting-- to be certain that Peano arithmetic is consistent is to be able to prove that it is inconsistent. Whether or not that tells you anything is up to you, but it is still true, even though Dr Rocket has proclaimed it "absurd" and "ATM", without finding any flaw, mere "argument by labeling". But yes, I'm the hardhead here, for being right. Please make up your mind, am I:
1) wrong
2) ATM
3) saying something totally different from anything I've actually ever said, or
4) consistently saying exactly the same thing and proving its truth, only to have you begrudgingly accept that truth and somehow claim that it wasn't what I've been saying and wouldn't even "persuade" me if it came from you instead of me.
Sheesh.

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Logic is the grammar of truth.

Meaning and absolute certainty are incompatible, and profound meaning and absolute certainty are profoundly incompatible.

The only thing intelligence is capable of is recognizing itself.

Ken G, in my opinion you are wrong.

1 and 2

As to 3, you have consistently misunderstood the Godel theorem and basic logic.

As to 4 you, have not proved anything of import.

To be precise, the Gödel sentence asserts that it is not provable, with second order logic within the set of axioms themselves.

Note that the proof of the first Gödel theorem involves a proof of the truth of the Gödel sentence, but not within the limitations imposed. That is in fact precisely what the first Gödel theorem is.

The second Gödel incompleteness theorem states that it is impossible to prove completeness of a set of axioms

Gödel's first and second theorems are accepted by nearly all mathematicians as an incompleteness theorem, and here is why:

1) The case in which the axioms are inconsistent is completely uninteresting. If any set of axioms is inconsistent, then within that set of axioms all possible sentences are both true and false. That is a totally useless situation, and no one is willing to work with inconsistent axioms. Your protestations to the contrary are just plain ludicrous.

2) It is an accident of semantics that "inconsistent" is commonly taken to include "complete". In point of fact that it a trivial thing. Once it is recognized that a set of axioms is inconsistent, it does not matter one whit whether you also call it consistent or inconsistent. They are utterly useless nevertheless and any and all theorems related to those axioms are worthless -- since all possible theorems are both true and false. Hence there is nothing of importance lost if one simply states that "the fundamental axioms of arithmetic are incomplete", except of course the trivial semantic point that you keep bringing up.


Note also that that if one could prove the consistency of a set of axioms using only those axioms it would not be a useful theorem. That is because is is not only possible, but a requirement, that an inconsistent set of axioms be able to prove its own consistency (and also its own inconsistency since all sentences are both true and false).

But you can find no fault in my proof, apparently. I wonder how your mind can hold a mathematical opinion that it cannot support with a mathematical argument.

Actually, what I proved was Godel's second theorem. So that is neither 1 nor 2. How can you be a mathematician and not see that?
Well, I proved it, and you called it wrong and ATM. And I misunderstand it?
Just Godel's second theorem, that's all.

A fine discussion of the relation between truth and logic is at http://plato.stanford.edu/entries/ka...ity/#KanAnsHum From this examination of the views of Kant and Hume on the nature of causality and time, it can be suggested that Godel lacked necessary axioms to explain the relation between truth and logic, by neglecting the function of reason as seeing the necessary conditions of experience. In this post I respond to comments from Ken G at Truth and Logic

Hume held that we have no perceptual access to the necessary connection between cause and effect (hence skepticism), but we are naturally compelled to believe in its objective existence (hence realism). As a necessary explanation for the regularity of the universe, the law of cause and effect in motion through time and space provides the logic of experience. The truth of causality as universal law is not derived from experience, but from mathematical reason applied to experience.For Kant, causality is a central synthetic a priori necessary truth of reason, an axiom or highest value. We apply the axiom of the universality of causality to give form and meaning to experience. Taking as axiomatic that the universe exists prevents the Godel cul de sac of incompleteness. This opens the relation between logic and experience in science. The ‘inner truth’ of causality is discovered in the facts of nature, eg that DNA makes proteins or that stars make elements. These observations indicate a necessary connection between cause and effect, ie that DNA necessarily makes protein and stars by their nature necessarily make elements. Their existence necessarily implies the existence of causation as a logical feature of matter. It depends on your method of proof. If we take as axiomatic that the universe exists, then the necessary conditions for experience, including causality, space and time, establish a logical connection between proof and observation. The truth of observation is accessible by logic, in that we know there is a law-governed physical process which enables us to predict the future regarding celestial mechanics, as an objective correlate to the accuracy of syntax. This separation between logic and observation is too simplistic. As Kant said, perception (observation) is blind without concepts (logic), just as conception (logic) is empty without perception (observation). The quoted article on Kant and Hume discusses the distinction between “logical grounds” and “real grounds,” both of which indicate a relationship between a “ground” (cause or reason) and a “consequent” (following from this ground).
You say the universe generally does not obey ‘our laws’, presumably meaning the laws of physics. This is a strange claim, given that science is founded on the axiomatic assumption that the laws of physics are universal. Difficulty of prediction is a result of complexity of experiment, not of a failure of law-like behaviour by matter.A closed system is not typical of physical law. The contrary point can be made by taking as an example the inverse square law of gravity which is universal in the actual universe. By any real standards, your sentence is not an actual truth, but just a semantic construction. It does not refer to anything real, which should be the criterion of actuality. It is almost like a Mobius strip of words, disregarding the normal meaning of actual in order to build a fanciful system of logic. Any so-called ‘logical truth’ that is inconsistent with actual truth is eo ipso not real. Everything real is consistent with actual truth. Truth should be logically confined to reality. If logic is not actual then it is not true, and does not detract from the completeness of actual truth.The problem is that Godel’s set of axioms is incomplete. If mathematics takes as axiomatic that the universe exists, then inconsistency is not possible. The claim of necessary incompleteness is only possible by giving real status to ideas that are not actually true.

That is completely irrelevant to the issue of whether it is an incompleteness theorem. If it is an incompleteness theorem, surely you must agree that it is proven, yes? It is proven that the axioms in question are incomplete, this is your claim here. Not that it is likely, not that we prefer imagining it is true, not that there's no point in assuming it isn't true, but that it is proven. You're a mathematician, you are supposed to know what proven means. Now, how does the "interestingness" of the axioms matter to such a proof? What kind of logic is that, proof by interest?
It is logically obvious that no one can be "unwilling" to work with axioms if they do not know they are inconsistent. Argue otherwise, with logic, stop just claiming the converse is "ludicrous."
Then why do you keep bringing up this "accident" every five sentences? At some points, your whole argument seems to hang on that accident, at others, you dismiss it is unimportant. Get your story straight.
And I care little about that, nothing I'm saying hinges on this. What I'm saying does hinge on the fact that as soon as you know with certainty that an axiomatic system that supports a Godel statement is incomplete, you know with certainty it is consistent. That is just a fact, and it is something you care about. But as soon as you know it is consistent, you know the Godel statement is true, you have then proven the Godel statement by this very argument. That renders the Godel statement false, and your system is inconsistent. I keep repeating this, you keep failing to find the slightest flaw in the argument, relying instead entirely on rhetoric instead of mathematical logic.

Actually, what I keep bringing up is a proof, and you keep failing to find any flaw in it that is not "rejection by labeling". Keep saying it's "ATM", "absurd", "ludicrous", and now "semantic"-- but never point to any logical flaws. This is a fact, anyone can see this fact, it's right here. And I'm the one who doesn't understand how mathematics works?

It would be a lot worse than not useful-- it would prove the system is inconsistent. That's exactly the proof I keep giving, and you keep ignoring, apparently because you don't approve of the outcome.

Yep, I am a mathematician. The real thing.

No, you did not prove Gödel's second theorem. Neither have you proved Gödel's first theorem.

In fact you have not offered anything in this thread that would begin to be accepted as a mathematical proof of anything.

BTW Gödel's second theorem states that any consistent set of axioms sufficiently rich as to admit the natural numbers does not admit a proof of its own consistency. You most certainly have not produced a proof of that statement. The fact that you think that you have is merely one more in a long line of demonstrations that you don't understand the problem.

Does Godel's question ask if the natural numbers are real?

Surely inconsistency with reality is ruled out by logic? Observation of the universe shows that the structure of elements is determined by the natural number of protons and neutrons in the nucleus of the atom. Counting the particles gives the periodic table. Hence for a mathematics that describes the universe, the natural numbers are embedded in the structure of matter.

How could a true logic possibly be inconsistent with the natural counting of the elements?

I don't know what you mean by "Gödel's question". There is nothing that goes by that name in mathematics.

In mathematics "real" has a specific meaning, and it is quite different from that of common usage. In that mathematical context, the natural numbers are a subset of the real numbers and therefore are real.

However, "real" in this context is not "real" in the philosophical context or the "real" of general usage (which to me agree with the philosophical meaning).

I think you are asking a question with regard to the Platonic view of mathematical objects. In that case I think it is fair to say that most mathematicians view mathematical "truths", numbers, and constructions as being "real" (in the sense of everyday language) and therefore as being things that mathematicians discover as opposed to things that they invent. But this is really a personal philosophic stance, and has no real bearing on the formal structure of mathematics. It is rather akin to the philosophical stance taken by many of the better theoretical physicists, which is that nature is described by some (as yet unknown) set of mathematical laws and it is up to them to discover those laws. That may or may not be "correct" but it does serve the researcher well as a motivational factor in his research.

Mathematics makes no claim to describe the universe, or anything other than mathematics itself. The connection between mathematics and physical laws is made by physicists, not mathematicians (at least speaking narrowly).

Hence inconsistency with "reality" is not a consideration within the formal structure of mathematics. "Reality" in the sense of the physical world is not a part of formal mathematics.

However, as a matter of day-to-day pursuit of mathematics as a research subject, it is fair to say that a mathematician would take as an indication that it is not true, the fact that some conjecture would be contradictory to a mathematical relation that is known to be physically applicable and accurate. This is a purely heuristic and motivational statement, however, and again not any part of formal mathematics.

In that vein is perhaps valuable to note that correct mathematical theorems are usually "guessed" and once guessed, then proved using the rigorous methods of proof governed by mathematical logic. Mathematics as a research topic is not the deductive monolithic structure that is often perceived by non-mathematicians. Unfortunately what is published and shown to the non-specialist is the polished, and streamlined, final form of the proof a theorem and not the inductive steps, false starts, and illuminating examples that led to the theorem in the first place.

I don't think you really mean "true logic" in this context.

The logic used by working mathematicians is basically the same logic used by others, though perhaps not so formal and restricted as that considered by logicians (working mathematicians do not restrict themselves to first order logic for instance).

The natural numbers, the counting numbers, are a result of the basic axioms of mathematics. That could be the Peano Axioms or the more formal Zermelo Fraenkel axioms (usually with the axiom of choice added). In any case, with the exception of logicians who explore the ramifications of formal axiomatic systems in a rather abstract way, all mathematicians work with a set of axioms that admits the natural numbers and ordinary arithmetic. So essentially all of mathematics is based on the natural numbers in a rather direct manner.

I am aware of no mathematics that does not admit the natural numbers, and cannot imagine any such set of axioms that would be of interest to the professional mathematical community or be taken seriously. In mathematics, as in all disciplines, there is a factor of "taste" and without the natural numbers mathematics would be at best tasteless, and frankly completely uninteresting (and useless to those who apply mathematics).