Your statement is pure opinion. The only fact here is that you have not pointed to even a single specific flaw in the proof that I did in fact offer. Claiming I didn't give a proof is rhetoric, I'm the only one here doing mathematics. That statement flows from the definition of mathematics, not opinion.
That is just precisely what I proved.

But lest we get into unfruitful territory, it suffices to observe that it is impossible to prove that arithmetic is consistent, based on the statement you just made. So tell me again, do we have a theorem (something proven) that arithmetic is incomplete, or don't we? Yes or no.

Thinking some more about the literature.

First there are various popular expositions - Hofstadter being the best known I suppose. Then there are more technical works - I'll mention "Computability and Unsolvability" by Martin Davis. I'm sure that people can add many more at every level.

But I'll draw attention to what I think is a significant oddity. The Oxford Reading in Philosophy series had a title "The Philosophy of Mathematics", edited by Hintikka. His aim was to present, in his words, "... the kind of reading which I judge to be by far the the useful for a student of the philosophy of mathematics ... the reader soon discovers that a .. a book or article is not based on anything like an adequate acquaintance with the ... body of materials ...".

But this book in the series was replaced, in 1996, by an identical title, edited by Hart, that favours (apparently) far less technical articles. (There is no reason given for the replacement of the original title.)

I'd recommend both - and from Hintikka the paper by Smullyan ("Languages in which Self-reference is Possible"), from Hart the paper by Isaacson ("Arithmetical Truth and Hidden Higher-Order Concepts").

I would naturally welcome any recommendations from other readers of this thread (including any feedback on "An Introduction to Godel's Theorems" by Peter Smith).

There is a poopularization, Godel's Proof, by Nagel and Newman that you might look at.

I don't generally like popularizations of mathematics. For the real thing I recommend Godel's Incompleteness Theorems by Raymond M. Smullyan and
Recursion Theory for Metamathamtics also by Smullyan.

There are also Godel's original papers, available over the internet, but be aware that Godel's original proofs have since been improved by removal of the hypothesis of omega-consistency and replacement with ordinary consistency. http://www.research.ibm.com/people/h...n00-goedel.pdf

You have offered nothing that even vaguely resembles a proof, and one cannot find a flaw in the empty set.

You are not and have not done any mathematics. Nothing at all. You seem to be harboring a delusion in this point.

If you think you have a proof of something please present it in recognizable form. That would be a clear statement of the hypothesis followed by a clear argument labeled "proof".

We have Godel's theorem which implies that if the axioms of arithmetic are consistent then they are also incomplete.

That means that it is impossible to prove that arithemetic is consistent if it is in fact consistent. If however, it is inconsistent, then it is possible to prove that it is consistent (yes, you read that correctly). Since there are transfinite proofs that arithmetic is consistent, it is probably fair to conclude that a proof of consistency is not possible.

It is difficult to exhibit ignorance and lack of understanding by merely formulating a question, but you have managed to do just that. By your insistence, as a sophomoric debating tactic, on "yes" or "no" you fail to recognize the role played by consistency in the Godel theorems.

The important aspect of the theorems is the implication of incompleteness given consistency and the inability of a consistent set of axiom to support a proof of its own consistency.

Although you fail to understand this point, inconsistent sets of axioms are totally worthless, and hence are not the subject of consideration by real mathematicians. That is because
1) Given an inconsistent set of axioms, ALL sentences are both true and false
2) Given an inconsistent set of axioms, one can prove that the set is consistent -- because you can prove any sentence that can be formulated in that situation.

So, given a consistent set of axioms, of sufficient richness to admit the natural numbers, Godel has shown us that using only those axioms and second order logic
1) The axioms are either incomplete or inconsistent
2) If the axioms are consistent it is impossible to prove that they are consistent.

The points that you have made that are different from these two items are trivial and semantic in nature and exhibit not only no insight but in truth negative insight -- you are missing the forest for the trees.

Specifically, there is no value whatever in considering the notion of completeness or incompleteness in the face of inconsistency.

If one were to adopt the definition that a set of axioms is incomplete if either it is inconsistent or if there are true sentences that cannot be proved, then there would be no substantive difference from case in which a set of axioms is complete if is is consitent and if there are true statements that cannot be proved. Inconsistency is simply not relevant to the real issue that is addressed via the notion of completeness.

Here is the proof once again:
That is a mathematical proof, not a "delusion". It's just a short one, elegant and simple, but it's still mathematical, and I'm sure I'm not first to have seen it. Indeed, it was precisely the argument I've been making all this time, and you have now described as "absurd", "ludicrous", "ATM", "wrong", and now you've added "the null set." But what you have not done is find a flaw. Done, and just by quoting.

In case I need to repeat that too, the hypothesis is "certain knowledge that an axiomatic system that supports a Godel statement is incomplete allows you to prove it is inconsistent." In short, it is impossible to have certain knowledge an axiomatic system of any richness is incomplete, in stark contradiction to your claim above:
I do not believe this claim, because I believe "nearly all" mathematicians:
1) know what it means to be a theorem
2) know that a theorem can be used to prove other things
3) know that if the incompleteness theorem was really a theorem that said Peano arithmetic was incomplete, then one could use that theorem to prove it is consistent (in one line), and
4) if they could prove it is consistent, they could also prove it is inconsistent.

Now, there's a second mathematical proof for you, one that I have also already offered in this thread. It is a kind of corollary to the fact that Peano arithmetic cannot prove its own consistency-- it cannot prove its own incompleteness either. Ergo, the Godel proof is not a proof of incompleteness. Q.E.D., no ifs and or buts. You are wrong.

Yup, we certainly agree there.
Yup.

Obviously.

Ah, the sands are shifting: "probably fair to conclude"? We are not talking about an argument in rhetoric here, involving "probability" and "fairness". The issue here is not if it is "probably fair to conclude" that arithmetic is consistent, everyone knows that, it's why we teach arithmetic. The issue at hand in this thread is whether or not its incompleteness is a theorem (your word). All I have ever said, which initiated all your accusations of not understanding math and logic and all that hooey about the null set, it is not a logical truth that arithmetic is incomplete. You are welcome to find the post where I said exactly that, and repeated it over and over, and never said anything other than that, except to claim that I proved that this is true, which in fact I did.
What is difficult is heaving around insults without ever actually finding a single specific flaw in someone else's argument. But that's all you have done here, repeatedly. Remarkable stance for a mathematician-- dropping all mathematical argument and resulting to "appeals to what is interesting" as you did here:
or "appeals to semantics" as you did here:
or "appeals to delusion" as you did here:
but not a single appeal to an actual flaw in my proof, other than "labeling it the null set." Nope, it's a proof, it starts with a hypothesis, it follows logical steps, and it reaches a conclusion. You know what the "null set" is, and that ain't it.

So you will not even give a "yes or no" answer to this simple question:
You can't even give a yes or no answer to that most basic question? Why not? Because you'd have to contradict your own words.
Ah, the plot thickens. Before "almost all mathematicians" thought it was an incompleteness theorem, but now we merely have an "implication" of incompleteness. Do you believe that an implication is the same thing as a theorem? Do you believe that I ever claimed there was not an "implication" here? Reread the thread, all I ever claimed, so "absurdly" and "ludicrously", demonstrating how little mathematics I know, is that it has never been proven to be incomplete, so it is not a theorem that it is incomplete. Yup, that's what I've said, it's the difference between a logical truth and what we believe to be true, which is just what this thread is about (reread the OP). I was right then, and I'm right now, and you are just starting to admit it now as the sands shift to being an "implication of incompleteness." But I knew that all along.


This has to do with what statements of mine? I see zero relevance, it's just you talking to yourself. You keep overlooking that knowledge of consistency is a totally different issue than consistency itself. Everything I've said has been about the former, as the ramifications of the latter are obvious and uninteresting, and certainly of no relevance to this thread. What the thread does is establish, beyond all doubt, that it is not a logical truth that arithmetic is incomplete, and the more important corollary that is independent of your semantic objections, it is not a logical truth that arithmetic is consistent. That is quite obvious. But you are mistaken that the issue of interest here has anything to do with the semantics of our definitions. The issue here is, and would continue to be with your definition, whether or not it is a logical truth that Peano arithmetic is consistent. All I have ever said, you can scan the whole thread, is that this is not a logical truth, so we do not know it with complete certainty. This is also what my sig has been trying to suggest to anyone who is open to hearing it, and is the core purpose of the thread to establish. And it has been established, to anyone who cares to read the logic of the thread carefully enough.

__________________
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.

"But as soon as you know it [the axiomatic system] 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."

A Godel sentence asserts its own unprovability within a specific formal system; it does not assert its own falsehood.

Sorry I missed this before, and yes I have seen that Hume piece and find it quite compelling. The differences here are semantic-- what you are calling the "logic of experience", I am calling "actual truth", and I am distinguishing that from formal logic. Formal logic is not the logic of experience, experience is not always logical. Formal logic is symbolic logic, it carries no meaning, but it renders complete certainty because it is part of the definition of knowing, it is the way our intelligence defines as the sole path to certainty. About all else, we must harbor some doubt, for the rest requires sensory inputs that could be mistaken.
But there is no such axiom in formal logic. I accept your alternate terminology as valid, it's just not the terminology of this thread-- here "logic" is defined the way it will be defined in a mathematics text (and I'm glossing over its various alternate forms for simplicity). In no mathematics text will Kant's axiom be included as part of logic, but it is part of what we are willing to call "actual truth", or experiential truth.

Yes, that "inner truth" is what I'm calling "actual truth" in this thread, to distinguish it from logical truth. The former carries meaning but no certainty, the latter carries certainty but no meaning.

Actually, one requires more than just that the universe exists, one also needs that our senses and our ability to assimilate the input from the senses are completely reliable testifiers to that existence. This is where the axiom falters into the realm of "actual truth," leaving the realm of "logical truth", because it is also an actual truth that our senses are not completely reliable, both in what they sense and how we assimilate them.
Yes, here I believe he is echoing my claim that logic conveys certainty but without meaning, whereas only experiential truth has access to meaning. Nevertheless, this makes experiential truth something different from logical truth, and the Godel proof is much better understood in light of such differences.
Yes, again I see the tension between logic and meaning in these statements. You are bringing up points that are completely relevant to the purpose of the thread.
That's not a direct quote, but to make it something I'd agree with, it would depend on the interpretation of the word "obey." If this word is taken literally, then that is indeed a reflection of something I'd say, but if it is taken loosely, then I would not say that (being, after all, a physicist, I invoke the loose meaning all the time).
That simply isn't true in general. You are right that measurement has limitations, but no physicist thinks that this is the sole fundamental limitation on the laws we make. More often, the laws break down before the measurements do, because the laws make assumptions that are never realized in practice (Newton's laws make assumptions about low speeds, quantum mechanics makes assumptions about the independence of the particles being treated, thermodynamics makes assumptions about the ergodicity of the system and about what is important and not important to track, relativity makes assumptions about the precision of the concept of spacetime, and all physics theories require idealizations applied to the boundary conditions in space and time such as closed systems. These are all fundamental idealizations-- no system can actually satisfy them, the problem is not just with the system it is with the impossibility of idealizations).

But that's just it, that is not a universal law, we already know it's not a universal law because it doesn't obey general relativity except under idealized conditions that would be impossible to achieve to arbitrary accuracy. So we simply make the idealization and accept the error-- the universe does not, it hasn't got that luxury.
The point of this thread is, in part, to establish why that statement is not true. I claim there is an essential difference between logical truth and actual truth, that we must track or we confuse ourselves. Perhaps the difference should not be labeled an inconsistency per se, for one should never assert the opposite of the other, but there are times when one asserts something but the other is moot on the point (as in whether or not arithmetic is consistent). That is the kind of difference I'm talking about-- we can have certainty only where we do not have meaning, and we can have meaning only where we do not have certainty.

Yes, that's what I would call part of the definition of actual truth. That's the contention that this thread is aimed at defeating.

__________________
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.

Not only is it not a mathematical proof, it is not any sort of proof. It is not elegant. It might be simple, but only in a trivial sense. It is just plain wrong.

The Gödel sentence does not assert its own falsehood, but rather its own unprovability within the axiom system. Those are two very different things.

What Gödel showed was that, under the assumption that the axiom system is consistent, the Gödel sentence is in fact true, and hence it is not provable. Thus if the axiom system is consistent it is incomplete. That most certainly does not make it inconsistent. The theorem makes no conclusions regarding consistency. In fact consistency is part of the hypothesis.

That is precisely why the theorem is called, quite correctly, an incompleteness theorem, and why you are all wet.

So, as we also see you have not offered a valid proof of anything, In fact your assertion is simply an example of badly flawed logic, and you reached a conclusion that is simply erroneous.

For the benefit of those trying to assimilate the material in the chapter regarding Gödel’s two incompleteness theorems I offer this informal summary.
For a more rigorous treatment one is invited to read the original or translations of Gödel’s original papers or texts such as Smulkyan's Gödel’s Incompleteness Theorems.

Definition A set of axioms is consistent if no sentence is both provable and refutable. It is inconsistent otherwise.

It is a fundamental fact of logic that a false premise implies any conclusion whatever. Thus if one has an inconsistent set of axioms there is a provable sentence (therefore “true” within that system) that is also refutable (hence “false”). That sentence therefore serves as a valid hypothesis, which is nevertheless false and hence serves to prove the truth of any sentence that can be formulated within the system. Thus all sentences are true. Since one can also form the negation any sentence, all such negations are also true and therefore all sentences are both true and false. Thus one sees that inconsistent sets of axioms are pathological and utterly worthless. Once you know that all possible sentences are both true and false, there is nothing left to talk about. Moreover, since you have a valid false premise all sentences are also provable, since that false premise implies any sentence whatever as a conclusion. That is why inconsistent systems receive no study and no consideration by professional mathematicians.

Note that given an inconsistent system one can prove that it is consistent (and inconsistent). In the case of an inconsistent system “truth” and “falsehood” become meaningless.

Henceforth we will assume that any set of axioms under consideration is consistent, though we may state this explicitly for emphasis.

Definition: A set of axioms is called complete if all true sentences are provable. It is called incomplete otherwise.

Gödel’s First Theorem Given as set of consistent axioms that admits Peano arithmetic, that set of axioms is incomplete.

The proof of the theorem is rather involved and uses a technique for numbering sentences, and showing that the number of true sentence exceeds the number of provable sentences. It goes further and uses that numbering system to exhibit a sentence that is true but not provable
.
While the precise formulation of the specific “Gödel sentence” requires a rather pedantic application of symbolic logic the basic idea is that Gödel formulated the English sentence A=”This sentence is not provable” in a formal manner. Now that sentence must be either true or false. Suppose that it is true. Then A is not provable, but since we suppose that it is also true we have an example of a true but unprovable sentence and the theorem follows. Suppose that it is false. Then the sentence is not provable, but that makes the sentence true, which is a contradiction. Hence the Gödel sentence is a true but unprovable sentence and the system of axioms is incomplete. Note the implicit use of the assumption of consistency of the axioms.

Mathematicians usually state this as saying that the axioms of arithmetic are incomplete, making the implicit assumption that the axioms are consistent. There are proofs of the consistency of the axioms of arithmetic, but those proofs use transfinite techniques that lie outside the bounds of the rules of symbolic logic imposed by Gödel. Mathematics does not restrict itself to first or even second-order logic, so this situation is not unusual. First-order logic, in fact does not admit the necessary quantifiers to permit construction of the integers.

Gödel’s Second Theorem Given any consistent set of axioms admitting Peano Arithmetic, it is impossible to prove the consistency of those axioms on the basis of the axioms alone.

Gödel’s Second Theorem is really nothing more than the statement that not only is arithmetic incomplete, but also (still assuming consistency) one of the true but unprovable sentences is its consistency.

A corollary of Gödel’s Second Theorem is that an axiom system (of sufficient richness to admit the natural numbers) that admits proof of its own consistency is in fact inconsistent. This shows that a proof of consistency (using only second order logic and the axiom system itself) is of little value, except as a demonstration of inconsistency.

You can pretty much disregard KenG’s assertions regarding the Gödel theorems They tend to be either a) gibberish, b) trivial or c) wrong.

I think this can conclude the discussion of the Gödel Incompleteness Theorems. They most certainly are incompleteness theorems, are aptly names, and they are universally accepted as such by the professional mathematics community. If there are clarification questions they might be appropriate but I see no benefit to any further argument with KenG, especially given the mistakes and misunderstanding shown in his "proof".

You are mistaken, but at least this time you gave an argument, so I can point out your mistake. That's the wonderful thing about mathematics, it is not a matter of opinion, even if mathematicians sometimes wished it were.
I'm quite aware of that, indeed that's why I started this thread. If you look at my proof again, you will see quite clearly that the proof does indeed involve proving the Godel statement, just as I said it did. I repeat, if we know (not just "probably fair to conclude") that the axiomatic system is consistent, we can use that in our proofs. That's also why, to make the point crystal clear in the main version of the proof I gave and repeated, I referred to axiom A1 being affixed to the system, which is the axiom that the system is consistent. For example, this appeared a few posts back:
Your attempt to find a flaw in this proof has failed, as the proof does involve proving the Godel statement, which does falsify it. This is also the logic used by Godel himself.
Yes, I know that, I said it many times in this thread. I guess it makes more sense when you say exactly the same thing.
The reason it is called an incompleteness theorem, understood by careful mathematicians, is that it is a theorem that in some way relates to incompleteness. Uncareful mathematicians interpret this as a theorem of incompleteness. I disproved that incorrect interpretation by proving that if the Godel proof is really a theorem of incompleteness, then it can be used to prove the Godel statement, thereby falsifying it. Again, you are the one who is wet here, sorry, it's all right here in black and white.

__________________
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.

Where did I say that the Godel sentence asserts its own falsehood? What I did was to take a theorem about consistency and use it to prove the Godel sentence. That's what falsifies it, proving it. I have no doubt that this is the standard way to show that arithmetic cannot prove its own consistency, I'm just using to show that arithmetic cannot prove its own incompleteness either, so admits no theorem of incompleteness, contrary to the claims in this thread.

Yep mathematics is objective and your conclusion, hence "proof" is completely incorrect.

The truth of the Godel sentence is proof of the incompleteness of the system of axioms, which is why it is an incompleteness theorem.

However, your assertion is that that the axioms are inconsistent. That is just plain wrong.

I give up. Will somebody else try to straighten out this guy ?

These appallingly false claims are totally unsupported by any logic from the thread. I can easily summarize every claim I have made in this thread, and anyone is welcome to review the thread and see that these are precisely the claims I have made:

1) logical truth (what is proven) is different from actual truth (what is established by experience) in important ways, including the fact that logical truth conveys certainty but not meaning, and actual truth conveys meaning but not certainty. What's more, the Godel proof, along with many other things involving logic and truth, are all about this difference, and make it forever impossible to claim they are the same things.

2) no axiomatic system rich enough to include a Godel statement (I generally use "arithmetic" or "Peano arithmetic" as my examples, as that's what we all care about) can prove itself to be consistent. I offered a proof of same, though the conclusion has not been contested.

3) since arithmetic cannot prove itself to be consistent, we can never know with formal logical certainty that arithmetic is consistent, because to know it would be to be able to use this knowledge in proofs, including a proof of its own consistency, which would render it inconsistent (and I gave the proof of that).

4) arithmetic also cannot prove itself to be incomplete, so we can never obtain certainty that arithmetic is incomplete as a formal axiomatic system. As soon as we could know it is incomplete, like if we had a theorem that it was incomplete, we could use that to prove things, including the Godel sentence. That would also render the axiomatic system inconsistent.

5) given that we cannot know that the axiomatic system that is arithmetic is consistent without rendering it inconsistent (or going to other axiomatic systems that can prove things about arithmetic but only by opening themselves up to the exact same issues we are talking about here), we will simply never get to assert its consistency as a logical truth. We can, however, act on the assumption that it is an actual truth (which is why we teach it and use it). What's more, we can continue to do this for a million years if we like, and at no time in that process will it ever be a logical truth that it is consistent, so at no time will we ever be certain that it is consistent. What's more, it might actually be inconsistent, but as long as we never discover how or even know it, that undiscovered inconsistency would never in any way limit our ability to use the axiomatic system exactly as we do today-- as a guide to actual truths involving arithmetic.

These are my assertions in the thread, and they are all true. Throughout the thread, Dr Rocket has claimed these very same assertions are 1) wrong, 2) ATM, 3) unproven, 4) many other unpleasant and incorrect things, but at no time has he ever been able to cite any actual errors in the arguments that were not actually errors on his own part that I pointed out immediately. That's the fact of the matter here. We certainly seem to have rather a difference of opinion, but fortunately, it isn't a matter of opinion-- everything I have said here is right there in the thread, in black and white, and all based purely in logic, so if it had any flaws, they'd be locatable. Instead, any time an actual claim was made against the logic, instead of making empty insults, I addressed the falseness of the claim directly. That's all I can do.

__________________
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.

What you keep missing is that the issue here is not the truth of the Godel sentence, it is our knowing that the Godel sentence is true! That's a totally different animal, because our knowing it within the axiomatic system that is arithmetic lets us use that knowledge in proofs in that axiomatic system, including a proof of the Godel sentence. That's what renders it false, and the axiomatic system inconsistent, and that's why it is not a theorem of incompleteness. I'm just repeating myself, but maybe at some point it will click.

No, what you keep missing is that you have not produced a proof.

Firstly, we do know that the Godel sentence is true. That is part of the proof of the first Godel Incompleteness Theorem. It is an example of a sentence that is true but unprovable using symbolic logic. That is its purpose in the proof. Godel proved that the Godel sentence is both true and unprovable using symbolic logic.

Note that part of what Godel showed is that it is possible to prove statements that are not provable using only symbolic logic by using methods that are more powerful than symbolic logic.

Secondly, the truth of the Godel sentence does not show that the system of axioms is inconsistent. It shows that the system is incomplete. Incomplete is not inconsistent.

Consider this. If your "proof" were correct, then one need only consider two cases. Case 1. The axioms of arithemetic are inconsistent, hence arithmetic is inconsistent by assumption. Case 2. The axioms are consistent. Then according to your "proof", the Godel sentence is false and the axioms are inconsistent. A contradiction. In either case the axioms of arithmetic are inconsistent. Now THAT would be Earth-shaking.

But your "proof" is simply invalid.

You keep saying this, but then I show why you are mistaken. I will do so again here.
Within an axiomatic system, we know it is true? That's what I'm calling logically true, and it means we can use it in proofs. It suffices for me to assert, correctly, that:

we do not know that the Godel statement is true well enough to use that knowledge in any proofs.

That's the kind of knowing that I'm saying we do not have about the Godel sentence, the kind of knowing that is called a logical truth using whatever brand of logic we have adopted and mean by "provable" whenever we are translating the Godel sentence, the kind of knowing I have described over and over as being certain within that logical scheme, the kind of knowing that does not derive its truth from any meaning but rather follows from pure logic. And of course I'm right, for that kind of knowing would render arithmetic inconsistent (as I've proved over and over, and you haven't even disputed, because it's a darn simple proof).No, the Godel sentence is only true if the axioms are consistent, so you cannot claim to know the Godel sentence is true unless you already know the axioms are consistent. The possibility that the axioms are not consistent is not treated anywhere in your logic, which is a fallacy-- you are doing an argument by cases in which you have not treated all the cases.

Now I know what you are going to say, because you keep repeating it, but it isn't correct-- you are going to say that you don't care if the axioms are consistent or not because you have to assume they are consistent to make them worth anything. But I don't care what you think they are worth, that's not the issue here. You can assume they are consistent if you choose, but if you do, anything you prove will have to be predicated with "if the axioms are consistent." That leaves out a case, so you end up with "if the axioms are consistent, arithmetic is incomplete." I'm quite well aware of that, I've never said anything else-- if the axioms are consistent, arithmetic is incomplete, and if they are inconsistent, arithmetic is complete. We do not know which as a logical truth. That's what I've said, over and over, and proved, over and over. Never have I said anything else on that score.

The Godel sentence works by playing off the tension between provability and actual truth. It asserts a sentence that is only actually true if it is unprovable, and if it is provable, it is actually false. That's it, it does no more than that. The rest is done by you-- you claim that it is unthinkable that it could be provable and actually false, so it has to be unprovable and actually true. It's pure appeal to incredulity, and it is a logical fallacy.

Look at your own words-- Godel proved it was true! You just said that. Please not the Godel sentence is not "this statement cannot be proved using symbolic language", it is, "this statement cannot be proved". Period. And you just contradicted that, yet claimed the sentence is known to be true. The Godel proof is not a proof of the Godel sentence, it is a proof about the ramifications of the existence of the Godel sentence.
Note nothing of the kind, for it is not true. Again, the Godel sentence does not assert it is unprovable using symbolic logic, it asserts it is unprovable within the axiomatic system. You cannot use one meaning of "prove" when it is referred to in the Godel sentence, and another when you talk about what Godel did. That's inconsistent.
I already addressed that. You just said Godel proved the Godel sentence. Then I can use that sentence for other theorems too, obviously. Like this one: if Godel proved that the Godel sentence is true, then the Godel sentence is provable. No, not by assumption. If the axioms are inconsistent, then they are inconsistent by fact. That we do not know if this fact is a fact or not is another fact.
I'm way ahead of you, I offered that argument pages ago, and repeated it many times since. But you are mistaken about the implications of that contradiction-- it doesn't make my proof wrong, it makes it the proof that the system cannot prove its own consistency!

Those sound quite interesting, especially that second title. It sounds like it will go quite deeply into these issues of the difference between logical truth and meaningful truth. The big questions would seem to be, what is the meaning of self-reference, and what are lower-order versus higher-order concepts. It sounds like he is saying that each brand of logic, each "order", spawns its own set of logical truths. I guess there is only so far we can get without really knowing how Godel's translation of the Godel sentence actually occurs-- what happens to meaning under that translation, and is meaning translatable? These seem to be linguistic questions are much as they are logical ones.

A Godel sentence in a system is either true or false. If it is true then it cannot be proved in the system, which is, therefore, incomplete; if it is false , then the system is inconsistent. A Godel sentence can be constructed in arithmetic.

I recently ran across a Taxonomy of Logical Fallacies, and was going to keep it all to myself, but...

I'm a sucker for the open source approach, so there it is.

Each fallacy his hotlinked to both definitions and examples, althoug I don't think either are the best available. What I like about it is the ability to see how they all relate to one another, visually, rather than trying to decipher a left-most list of all fallacies and figure it out for one's self.

The surpemely logical aren't usually noted for their creative and meaning-laden layouts.

Just for reference, the anthology "The Undecidable", edited by Martin Davis, with original papers by Godel, Church, Turing, Rosser, Kleene, and Post, is available from Amazon - a Dover edition republished in 2004.

A very precise. concise, and correct, statement of the situation.

Right-- and note that it is also completely consistent with absolutely everything I said on the topic.

Only you could reach that conclusion.

I wonder where you get this delusion that you speak for anyone but yourself, have you polled everyone? Do you fancy yourself an expert on human cognition, and that's how you know that no one else could reach that conclusion? Either way, your claim is just poor logic. What's more, anyone with basic logical reasoning ability could come to my conclusion quite easily, as it is in fact consistent with everything I said, as it is also a statement that I already knew.

Certainly they could. I have seen freshmen and sophomores, mediocre students at best, do such things. But if they make errors of that nature with regularity, they then have to repeat the class, or find another area of study.

My post, approvingly quoted by Dr Rocket:

"A Godel sentence in a system is either true or false. If it is true then it cannot be proved in the system, which is, therefore, incomplete; if it is false , then the system is inconsistent. A Godel sentence can be constructed in arithmetic."

was intended as a rebuttal of earlier claims by Ken G.

I will also say that I can make no sense of, and find no evidence for, his other claim: that mathematicians in general do not accept the concept of Incompleteness.

I believe that the foundations of mathematics do have philosophical implications, about which there is room for considerable and significant debate, but that the debate is impoverished by the persistent misunderstandings about those foundations - and about Godel's work in particular.

But that is simply not the case, because I knew every part of that statement, and it refutes nothing I've said, indeed it is the starting point of what I've said. To recap, what I said was:
Choose your axioms and your logic. Now you have a system for making proofs (like Peano arithmetic, or one with real numbers, it doesn't matter). That system can never include a theorem that it is incomplete. If it does (and I proved this), that system is inconsistent. Ergo, there is no "incompleteness theorem" within an arithmetic system. Within said system, either the system is incomplete, or it is inconsistent, and this is just precisely the statement quoted above. No rebuttal there.
That's because I never made any such claim. Mathematicians act on the suspicion that arithmetic, as a system for proving things, is incomplete. I have never said anything else. What I have said is that they know well that they cannot prove that within that system for proving things, therefore, they know well that they do not know it is a logical truth (and this whole thread has been about the difference between logical truths and the things we suspect from experience are true!). The same can be said for the reason that the quoted statement above includes inconsistency as one of the enumerated possibilities. That's why the quoted statement is in fact correct, as I have said it is correct. Not a single thing I said has been refuted anywhere on this thread, but there are sure a lot of misconceptions.

Agreed. And a very concise rebuttal it was, if one understands basic logic, making the same point in a single sentence that several of us have been trying to explain, somewhat more verbosely, throughout this thread.

KenG apparently cannot distinguish between that which rebuts his position and that which supports it. No wonder his "logic" is so hard to follow, and no wonder it is so hard to get a point across to him.

I would suggest substituting "fact" for truth.

Physics is the analysis of "facts" (Observations) to construct models and theories. If the analysis of observations to construct models is defined as science, then mathematics is not science. Mathematics is science like in that it is analytical and logic is applied to reach valid or invalid conclusions.

A fact is an observation which is not interpreted. Practically most observations require some theoretical assumptions, so what is a fact can from time to time change.

Logic is analytical linking of facts. CSI for example. There is a maximum amount of blood in a person. Blood at a crime scene that exceeds that amount indicates .... More analysis and data is required to reach a justifiable conclusion.

Logic can be correct or incorrect. Logic can prove something or it may not prove something.

Sorry folks, but I am not prepared, nor would the moderators permit me, to continually assert that Ken G is mistaken. I shall, therefore, flag this post at some time, and convey to the administrators why, for this and other reasons, I am withdrawing from BAUT.