**I am sorry, have I disproved Gödel’s incompleteness theorem?**

In a book called ”Introduction to Metamathematics” by Stephen Cole Kleene, a standard work about Gödel’s theorem (claims to contain the complete proof for Gödel’s theorem) with over 500 pages. On page 205 (following a theoretical background of about 200 pages) Kleene gives a heuristic ”proof” for the theorem, which I will present here:

By the construction of A [a proposition],

(1) A means that A is unprovable

Let us assume, as we hope is the case, that formulas which express false propositions are unprovable in the system, i.e.

(2) false formulas are unprovable.

Now the formula A cannot be false, because by (1) that would mean that it is not unprovable, contradicting (2). But A can be true, provided it is unprovable. Indeed this must be the case. For assuming that A is provable, by (1), A is false, and hence by (2) unprovable. By (intuitive) reductio ad absurdum, this gives that A is unprovable, whereupon by (1) also A is true. Thus the system is incomplete in the sense that it fails to afford a proof of every formula which is true under the interpretation (if (2) is so, or if at least the particular formula A is unprovable if false).

The negation of A (not-A) is also unprovable. For A is true; hence not-A is false; and by (2), not-A is unprovable. So the system is incomplete also in the simple sense defined metamathematically in the last section (if (2) is so, or if at least the particular formulas A and not-A are each unprovable if false).

The above is of course only a preliminary heuristic account of Gödel’s reasoning. Because of the nature of this intuitive argument, which skirts so close to and yet misses a paradox, it is important that the strictly finitary metamathematical proof of Gödel’s theorem should be appreciated. When this metamathematical proof is examined in full detail, it is seen to be of the nature of ordinary mathematics. In fact, if we choose to make our metamathematics a part of number theory (now informal rather than formal number theory) by talking about the indices in the enumeration [the Gödel numbering], and if we ignore the interpretations of the object system (now a system of numbers), the theorem becomes a proposition of ordinary elementary number theory. Its proof, while exceedingly long and tedious in these terms, is not open to any objection which would not equally involve parts of traditional mathematics which have been held most secure.

End quotation.

So we have two statements:

(1) A means that A is unprovable

(2) False formulas are unprovable

One can easily replace (1) with either “False A is unprovable” or “True A is unprovable”. (See below)

“A means that A is unprovable” can only devolve upon that A is unprovable, because to say “A means that” is just an added appendage to saying “(this claim) A is unprovable”. So the full sentence “A means that A is unprovable” is a predication in which A is either true or false. Unprovable means that something cannot be proved true. So we come to the question of not-A, i.e. false A.

(3) A means that A is unprovable (if false A or if true A)

(4) False formulas are unprovable

We cannot initially put an equal sign between the premise “A means that A is unprovable” and “False formulas are unprovable”, because we do not yet know if A is false or true. The following are all four heuristic possibilities for a theorem which I am going to exam very shortly:

A = false and provable

Since A cannot be false and provable I will leave this sentence aside.

A = true and provable

If A is true and provable it does not contradict “False formulas are unprovable” (4) and hence (true and provable) is still valid and thus also is independent from (4) which is rendered superfluous.

A = false and unprovable

“False A means that false A is unprovable” is a true proposition. It does not contradict with (4). (See the asterisk in parentheses below (*))

A = true and unprovable

And of course, if A is true and unprovable it does not contradict (4), because true A is supposedly just unprovable (at present date) and not false.

(*) Remember that “is unprovable” means that something cannot be proved true. “Unprovable” doe’s not mean that A is both not true and true at the same time, or even undecided, because that is impossible anywhere but in quantum mechanics. A true proposition cannot be unprovable, and a false proposition can never be proved true. A false proposition can perhaps be proven false, but it would still not contradict (4).

Someone may suggest that we have to transform the formulations above into basic math-rules like this, and strip it of digits:

(- +) = (-) (imaginary)

(+ +) = (+) (true)

(- -) = (+) (true)

(+ -) = (-) (imaginary)

The following is an explanation of what I am claiming here:

a) We would get (- +) = (-) (imaginary) if A could be false and provable, which it cannot. False propositions cannot be proved true.

b) We get the formula (+ +) = (+) (true) if it is true and provable, which certainly wouldn’t conflict with (4).

c) We get (- -) = (+) (true) if it is false and unprovable.

d) Thus we get the formula (+ -) = (-) (imaginary) for the true and unprovable.

I realize that labelling “unprovable” as a negative equaling with “false”, by assigning it too the negative (-) when “true” represents the solid plus (+), can open up for an interpretation of the above four a), b), c) and d) as erroneous thinking all-in-all. Because “false A is unprovable” means that false A cannot be proven true, but false A can still be proven false which seem to correspond with the negative (-) much better. And that would have been correct if it hadn’t been impossible to prove false A true, as we have accounted for in and above the deterministic expressions. So what we are left with, is that false A can never be proven true, that is, false A (-) must always be followed by (-) for “unprovable” and that means that this proposition (- -) is true. A true proposition cannot be unprovable, and a false proposition can never be proved true.

In the original theorem it is claimed:

• A means that A is unprovable. That means that A cannot be positive (+) if unprovable is (-) since true A cannot be unprovable. Because everything true is provable, and (+ -) = Imaginary = (not true). Therefor A = not-A = -A. And the formula must read (- -) = (+) or true.

• False formulas are unprovable. Wherein the false formula equals (-) and unprovable equals (-). Therfor (- -) = (+) = true.

Even though “unprovable” is a factor in the proposition, there is no contradiction.

The important thing is that the plus (+) is indicating existence, and the minus (-) is indicating non-existence, so that the result equals one of two things – true or imaginary. For the fun of it one can maintain, that this is the explanation of why the universe exists and that it is a God proof as well. Let us assume that (- -) represents the two unexplained fundamental entities; the universe and God. Since two non-existing of anything (- -) equals plus, i.e. a positive number = (+), the universe and God are destined to exist however unlikely they seem to be. In fact the improbability of their existence separately, could be a precondition for their very co-existence, (-) God (-) universe = (+) existence. And if it (math) is a precondition for their very co-existence, then the existence of the universe and God suddenly seems very plausible. And if either the universe or God fails to exist the result is that neither of them exist (+ -) = (-). But we exist, and therefore God exist. But is this God proof conclusive? Of course not, no God proof is conclusive. I am just having fun.

We have to revise the semantics in certain suggested variants of formulas for Gödel’s incompleteness theorem and Plato’s theorem, but Edmund L. Gettier’s theorem stands as a shining example still.

A suggested variant of formula for Gödel’s incompleteness theorem:

In any logical system for mathematics, there are statements of speech that are true, but that cannot be proved.

This statement cannot be true

Must be either true or false.

If the claim is false, it can be proved. Then it must be true. Which is a contradiction, therefore, the claim is true.

This is therefore a mathematical claim that is true, but cannot be proven.

The mathematical implication is: What if the Riemann hypothesis would prove to be true, but is impossible to prove?

____________________________________________________________________

It seems to me, this suggested variant of formula for Gödel’s incompleteness theorem gets entangled in it’s own semantics. It is certainly a logical argument based on the theorem, but you cannot use the order in which the words follow, mathematically. What do I mean? Well, the sentence: ”This statement cannot be true” must indeed be either false or true, but if it is true then you should – if it is possible to translate it into a mathematical formula that says something about something other than semantics – replace the words ”cannot be true” with the words ”is not true”, which makes it correct without the inconsistencies. Y can stand for ”is not true”, and X can represent ”must be true.” A can stand for the opening words ”this statement”.

——————————— X ”must be true” (+)

A X or Y, but it cannot be both!

——————————— Y ”is not true” (-) (cannot be true)

The theorem as it appears above the unbroken line, perhaps proves that it semantically can be either false or true, though it cannot be proven. But does it prove anything, with mathematics, of the nature of the world beyond it? No, it rather seems to disprove the theorem itself! The theorem doesn’t help us understand the world. Perhaps one cannot conclude a solution from the first (“This statement”) or A with both (“must be true”) and (“cannot be true”) for it to be a correct formula? Either “This statement” or A is true or it’s not true, so Y should read “is not true” if it should be adjacent with “This statement” or A since “Can” is a statement that says that something either is, or is not, but not both at the same time. When you put “not” after “can” (cannot), you are either saying (can; as in must[+] not[-]) = (-) = (“is not true”) which translates into a mathematical language + (-) = – or with other words it is a negative. Or you are saying (can; as in not[-] not[-]) = [+] = (“must be true”) which mathematically translates into – (-) = + or with other words it is a positive and henceforth must be true. Conclusion; “is not true” or “can not be true” is a correct wording, but not “cannot be true”! And what is the statement A? We don’t know! What we are doing is to apply the label of an unknown statement to a formula. But we cannot say anything about any actual statement. Is that logical? Surely “this statement A” is not a statement!? So what we have got left in “Y” is “A is not true” or “A is false” + – = – or just plain -.

Maybe we need to accept the fact that the answer to the Riemann hypothesis involves no more pattern in any sequence of prime numbers, than do the number sequence in Pi, and still the enigma could be solvable – if we look outside the box.

The above picture with the text “the next sentence is true” and “the previous sentence is false” is an anomaly if you translate it into a mathematic language. Think about how wrong it is semantically to not say anything about the sentence we read for the moment being, but instead say something about the second sentence which we do not read for the moment and haven’t had the opportunity to infer anything from at the moment. The sentence we are reading does not in any way entail the other sentence but are merely referring to it. These two combined sentences in the above picture with the dinosaur are related to the first suggested formula on Gödel’s incompleteness theorem This statement cannot be true, but only separated into two individual sentences and without the inconsistencies that comes with the word/words “cannot” (can; as in must [+] alt can; as in not [-] + not [-]) from the bipolar word “can” and “not” which the originator didn’t split up like I did here. The above statements in the picture is like saying “x+1=something in another formula (the next sentence) here not specified or even correlating (with the next sentence)”. It translates into (the next sentence[x] is true[1]) and then goes on to saying (the previous sentence[y] is false[-1]). The two sentences are simply not translatable into any logical algorithm one can solve, it only states that x=1 and y=-1. Or maybe you should say that x=-1 and y=1, but it still does not translate into any logical algorithm with a plausible answer. There is no mathematic connection between the two sentences, not even an equal sign. It is like saying; (the next bun [x] is tasty [1]) and (the previous bun [y] is disgusting [-1]). You could also shift the meaning in the two statements “the next sentence” and “the previous sentence” and get (this sentence [y] is true [1]) and (this sentence [x] is false [-1]). “This sentence is true”, is always a true sentence. “This sentence is false” if it is a true statement it must be false. If it is a false statement it must be true. It’s a pun that is transferable into a solvable mathematical formula (x=-1). Thus [x] is false and when one read it in its mathematical formula one can see no further implications because x=-1. It shows that there can be something illogical and subjective with the semantics we humans use.

I have other philosophical examples as well, of how semantics can mess it up, when trying to convey it into logical theorems (read below). The presentation of the criteria (for how we could be considered to have knowledge of anything) is construed by Plato, and problematized by the renowned philosopher E. Gettier. It has been considered an unsolvable problem for many years. The problem is related to Gödel’s incompleteness theorem, because of their semantic nature. I consider myself to have solved the enigma of Gettier’s problematisation of Plato’s theorem:

AN EPISTEMELOGICAL AND RATIONAL CONCLUSION FROM PLATO’S THEOREM AND E. GETTIER’S EXAMPLE WITH THE WOLF.

1:st example: A train is running on the railway tracks past a meadow. In the meadow there is a wolf. The passengers can see the wolf from the train.

According to Plato we require three criteria for enabling us to have knowledge of it.

(1) It should be a conviction.

(2) It must be consistent with reality.

(3) We must have rational reasons to accept it.

All three criteria are met.

2:nd example: Now suppose that, as in E. Gettier’s example, the wolf is actually a dog dressed up as a wolf. But a little further beyond the dog in the meadow there is a real wolf. The three criteria are still met, and this is E. Gettier’s problematization of Plato’s theorem, for the wolf we see is not a wolf at all, and hence the theorem is faulty even if it is true, according to Gettier.

Can we have knowledge that there is a wolf in the meadow (that the theorem is satisfied) by observing the dog, and applying the three criteria? The answer is that we cannot! The theorem’s correctness is completely independent of our observation of the dog (we do not know that the “wolf” is our costumed dog or that there is a real wolf just behind the dog in the meadow).

Or should we perhaps say that the theorem, on the contrary, is totally dependent on our observation, because our observation results in our belief (1), and our rational reasons to accept it (3). But thereby follows that our observation leads to a faulty conclusion, for the visible wolf is false. The theorem is still true, but Plato’s theorem requires an alternation applied to the unique situation.

(1) It should be a conviction.

(2) It must be consistent with reality.

An omniscient archangel must be the judge of whether the theorem is consistent with the real situation. Or in other words:

(3) ONE must have rational reasons to accept it.

Thus premise (3)’s rationality (as above) is not based on observations from our side. By changing premise (3) to; one must have rational reasons to accept the belief, we move the decision for what is rational from the group, to an omniscient archangel. One obvious objection you might come up with is that one can say that premise (3) is not needed then, because to claim premise (3), is the same as to claim premise (2).

The ideal type theorem itself is not critical to getting an epistemological answer to an investigation of the rational conclusion of the theorem. The key is to know when a rational answer emanates from the premises, not when a premise is rational. “A rational answer” is comprehensive of the whole situation with the wolf and takes into account both the wolf and the dog as distinctive entities (even in mathematics!). The original premises (1),(2) and (3) have not led to a rational answer to Plato’s theorems inconcistencies in this unique situation from Gettier’s example with the wolf and the dog simultaneously located in the meadow but where we only see the dog, because that is what the whole point with Gettier’s fictional example is, that Platos theorem is inconsistent. Here the archangel in my modified third premise that equals the second premise, come into the picture. Or should I say – it eliminates the third premise and leave us with only premise two and premise one.

In one possible Gettier reality applied on Plato’s original theorem, all of Plato’s original premises are not fulfilled: Say that in one occasion there is a dog dressed up as a wolf in the meadow (premises 1 and 3 are satisfied), while there is not a wolf behind the dog (premise 2 is not fulfilled), then the conclusion we make about the so called “wolf” is not a correct conclusion, because the “wolf” is actually a dressed up dog. If we had been able to make a correct conclusion, it would not have been our belief that there is a wolf in the meadow.

In our second example from above (read 2:nd example in bold black letters above) from Plato’s original theorem, there is a real wolf standing behind the dog, and all three premises are met. Let me first say that a correct conclusion would be as seen from a correct supervision of an omniscient archangel’s judgment about what constitutes a proper conclusion. On this occasion, we cannot make a correct conclusion based on our position on the train, that there is a wolf in the meadow, because we do not see it, we only see the dressed up dog. We believe however that the conditions are in order, (which they actually are, but not as we think, because we believe that the dog is the wolf in the meadow), and from it derives a conclusion that happens to be true, based on our false beliefs and Plato’s original premise, (from which I say that we have achieved an “Accidental Conclusion”, which we may call it). It also requires that the dressed up dog really looks like a wolf for us to be able to make a true (but not overall correct) conclusion. If there had been a water fountain or a Dachshund dressed up in front of the wolf rather than a German shepherd dog dressed up, we had never come to the conclusion that there is a wolf in the meadow, by looking at the fountain or Dachshund. The conclusion is true in this our other example, where all three original Platonic premises complied with the conclusion, but it is not a correct one. For this to be a correct conclusion requires that the premises implicitly take into account all the underlying facts. (Read and compare with my deconstruction of suggested formulas posing as Gödel’s incompleteness theorem!) Again; the theorem itself is not of crucial importance. The key is to determine when the premises amount to a rational conclusion. And here is where the archangel and my modified premise comes to use, for here it is the archangel’s insight that is the standard, and not my insight, and from that follows a rational answer to the theorem. The fact that the original theorem is true in this unique situation where we see the dog but not the wolf, is a pure coincidence (read blot on Plato’s behalf) and not relevant to how we should set up the premises properly. To make a true conclusion based on faulty underlying facts is something that has happened before in history. For example, there was an ancient Greek (Plato) who said that the earth was round long before anyone else had thought of it, and he founded this conclusion from that the shadow the earth cast on the moon could not be a likeness of the Earth’s shape, if the earth was flat. He believed that the earth cast its shadow on the moon, when in fact the moon (usually) is shaded by itself and its position relative to the sun as seen from our perspective. In light of this, Plato’s original theorem appears quite absurd, and Gettier’s situation with the wolf, in the context of Plato’s theorem is revealing deeper thoughts about the nature of epistemology, how we humans are limited and how we can be wrong without realizing it. I don’t know if Gettier was conscious about it, but that is what Gettier’s example implicates. The theorem “proves” more than it can prove, just as the moon’s shadow can do for those who have certain beliefs.

There is another way of going about Plato’s inconsistent theorem. The belief ((1) we believe there is a wolf in the meadow) and the rational reasons ((3) we have rational reasons to accept that there is a wolf in the meadow) with ((2) there is a wolf in the meadow) may seem to be waterproof as a logical framework. But the premise (2) should be read/understood and set up like this: The wolf is false, but there is a real wolf in the meadow that we do not see = it must be consistent with reality, and the whole complete reality with every underlying fact taken account for, if the belief is to conform with truth. This is how we must see the adapting of the situation with the wolf and the dressed up dog, I think. Had we just said; It must be consistent with reality, yes, it would have been correct. But should we allow the reality of our second premise to be so simplified as to say “there is a wolf in the meadow”? If so, the premise would not be completely true, or at least not entirely complete. Look at the example with the costumed dog which had a wolf behind it. We have rational reasons to accept the belief that there is a wolf in the meadow when we run by in the train, according to the original theorem. We have the illusion of the dog as a wolf. But coincidently there was a wolf in the meadow. Leaving aside premise (2), here in the form: “it must be consistent with reality, and the whole complete reality with every underlying fact taken account for”; is premise (1) and premise (3) merely cosmetic? They are at least “ideal types” constructed from our own shortsighted perspective, but still inconclusively constructed since they in Plato’s original theorem are not based on any actual situation in an all in all complete situation with at least as in this case, the dog and the wolf in E. Gettier’s example. Premise (1) and premise (3) are merely convictions, which by chance happens to mess it up in at least one of the cases written above, where the wolf and the dog coexisted in the meadow simultaneously, in Gettier’s example – hence “Accidental Conclusions”.

In conclusion, we have to revise Plato’s theorem, or abolish it as a whole. And E. Gettier’s example reveals more about the world or epistemology than he perhaps thought it would. I’m sorry I in previous versions 1-9 did not recognize Gettier’s genius potential!

Conclusion 1: One has to have rational grounds for accepting the belief.

Conclusion 2: Convictions leads to “Accidental Conclusions.”

Conclusion 3: The costumed dog must look like a wolf, and not a Dachshund or a water fountain, for the theorem to work.

Conclusion 4: The theorem proves more than it can prove, by the principle “the earth casts its distinctive shadow on the moon, and therefore the Earth is round”, which is false for some part.

In a textbook used at Lund University in the B course, called ”Philosophy of Language a contemporary introduction” by William G. Lycan from University of North Carolina, chapter 13 on ”Implicative relations”, page 198 it says to read in the first lines: ”Sentences entail other sentences, and in that strong sense imply them. But there are several ways in which sentences or utterances also linguistically imply things they do not strictly entail.”

It describes the chapter’s content very briefly. Anyway, in this chapter you can read an interesting thing that you can directly connect to and make of use to Gettier’s problem without that Lycan, or rather Grice, seems to have had any intentions in that direction.

There you can read: ”-Here as in many cases, a good way to investigate the nature of these different kinds of implications, is to ask about the penalty or sanction that ensues when an implicatum is false. When S:1 entails S:2 and S:2 is false, the penalty is that S:1 is false. When S:1 semantically presupposes S:2 and S:2 is false, then S:1 is sent ignominiously to zip. When someone utters S:1, thereby conversationally implicating S:2, and the conveyed meaning or invited inference S:2 is false, then the penalty is that, even if S:1 is true, the speaker’s utterance is misleading. If S:1 conventionally implicates S:2 and S:2 is false, then S:1 is misworded even if not false.”

One can implicate and translate this into Gettier’s example with the wolf directly like this:

”-Here as in many cases, a good way to investigate the nature of these different kinds of implications, is to ask about the penalty or sanction that ensues when an implicatum is false. When a ”wolf” in the meadow (S:1) entails a belief (S:2) and the belief (S:2) is false, the penalty is that the wolf (S:1) is false. When the wolf (S:1) (semantically) (I here chose to put this word within parentheses) presupposes a belief (S:2) and the belief (S:2) is false, then the wolf (S:1) is sent ignominiously to zip. When someone utters wolf (S:1), thereby conversationally implicating a belief (S:2), and the conveyed meaning or invited inference of the belief (S:2) is false, then the penalty is that, even if the wolf (S:1) is true, the speaker’s utterance is misleading. If the wolf (S:1) conventionally implicates a belief (S:2) and the belief (S:2) is false, then the wolf (S:1) is misworded even if not false.”

To translate this, one must resort to some drastic interpretations. Among other things, one must interpret the following sentence – “When someone utters Wolf (S: 1), thereby conversationally implicating a belief (S: 2), and the conveyed meaning or invited inference of the belief (S: 2) is false, then the penalty is that, even if the wolf (S: 1) is true, the speakers utterance is misleading.” – as utterances never are trustworthy regardless of whether they are true! But thereafter a complementary interesting thing is mentioned, namely: – “If the wolf (S: 1) conventionally implicates a belief (S: 2) and the belief (S: 2) is false, then the wolf (S: 1) is misworded even if not false.”

Also the philosopher Bertrand Russell addressed the self-contradictory logical problems one can construct with semantics and set up in an equally contradictory theorem, in Russel’s paradox or ”Performative Contradiction”. The paradox is as follows: When people say; ”all truths are relative” they make an absolute claim, and thus it becomes a contradiction in terms. I can answer with saying that; if all truths are relative, they are not truths, they are but a misch-masch or a composite of separate truths and non-truths and/or a misch-masch in the interpretation of the meaning of different non-hyphenated (usually) words, that need to be figured out separately, just like I did with the suggested variants of Gödel’s incompleteness theorem above. Either “the truth” (or in other words – the claim) is true, or it is false, but it cannot be half true in between!

Author: Roger Klang, civis Lundensis, Scania Sweden, updated version again (version 18 the 5th of september 2017). First translated into English 9/3/2011.