Why Gödel Failed to Prove His First Incompleteness Theorem

ABSTRACT:

This paper shows how Gödel failed to prove his first incompleteness theorem and shows a standard of proof that completes a system containing genuine, not contrived, undecidable statements.

Gödel's First Incompleteness Theorem is often stated as follows:

"Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F."

On its face, this theorem appears to be asserting that any consistent formal system (F) cannot consistently prove its statements or their negations, i.e., it is incomplete. The statements are of the language of F, which may be constructed from Peano arithmetic axioms and Gödel's numbering system where letters, words, symbols, etc. are encoded into natural numbers. Whether or not you agree Gödel proved his theorem, consider the following proof of the transitive property:

After stepping through it, are you convinced that A = C? Well, you shouldn't be, because the main characters in this proof are not what they seem--they are each code for a statement in a similar but different language. Let's label the original characters' language "X." and the different language "Y." Below is the Rosetta Stone or table that translates X into Y and vice versa:

Now, examine the proof in the Y-language:

As you can plainly see, A doesn't equal C in the Y-language. Let's assume languages X and Y are subsets of a greater language. Let's call it language F. In system F, which contains the language of F, we have the following:

Let's assume system F is consistent. That being the case, we can't assert that we proved A equals C, and A doesn't equal C, so we declare that neither is proved, i.e., the statement A = C is unprovable. I know this seems absurd, but it should have a familiar ring if you are familiar with the process Gödel used to prove his theorem. It is the process of encoding one language into another and labeling the result the "language of F," making sure the conclusion in language Y negates the conclusion in language X. Such a process violates the law of non-contradiction, and, if it is applied throughout all mathematics and logic, then nothing can ever be proved or disproved!

The corollary, of course, is all statements that can be proved are provable again, including perhaps the infamous Gödel sentence (G) if we insist on conforming to the law of non-contradiction. At line 6 below is the diagnal lemma used to prove that G is unprovable. Translated into English it says, "System F proves G if and only if G has the property of unprovability."

Not only does this lemma violate the law of non-contradiction, but its statement was arbitrarily chosen (not proved). If one examines the proof of the original lemma, the proved lemma reads, "System F proves G if and only if G has the the property of A." Any property can be assigned to the placeholder A. Gödel chose "unprovability." It is fortunate he didn't choose "vampire powers," otherwise his theorem might claim that some statements in F have vampire powers.

Anyway, claiming that F proves G only if F can't prove G is contradictory and should not be validated. The contradiction is done away with if we stick with the original language of F (language X) where natural numbers are not code for statements (in language Y). In language Y the statement might read, "Statement G can't be proved in system F." In language X it is just #G', the Gödel number of the statement. At 7 below we make that adjustment:

After making a substitution, the new statement reads, "F proves G if and only if there is a natural number #G'." No contradiction here.

To prove there is a natural number #G' in system F, use Peano's fifth axiom: "If a set contains zero and the successor of every number is in the set, then the set contains the natural numbers." That brings us to line 9 which forces the conclusion at 10 which is system F proves G.

"System F proves G," of course, is not the result Gödel wanted, so he made certain that the natural number #G' is code for a different statement, a statement that violates the law of non-contradiction and negates "F proves G."

Given the preceding arguments and evidence, it appears Gödel failed to prove his first incompleteness theorem. Notwithstanding this, his theorem may have some empirical support. For example, if you toss a coin in the air, while it is spinning, the statement "It is heads," can't be proved, nor can the statement's negation "It is tails," can be proved. The coin is in a fuzzy state of superposition. The usual standard of proof, where a statement is either true (heads) or false (tails) is inadequate, i.e., the system is incomplete. There is, however, a way to remedy this.

The negation of "true" is "not true." We can decide that a statement is only "true" if it is proved true, and, a statement is "not true" if it is proved undecidable or false. Doing this completes the system if a complete system is where any statement or its negation can be proved. Circling back to the spinning coin, the statement "It's heads," is "not true." If and when the coin lands and shows tails, the statement is still "not true." It's only true if heads is face up.

In conclusion, Gödel failed to prove his first incompleteness theorem because no statement in logic or mathematics can be proved or disproved if that statement is code for a different statement that negates the proof. By contrast, all that is decidable remains decidable if the law of non-contradiction is not violated. In circumstances where there is genuine undecidability, we can impose a standard of proof where a statement is not either "true" or "false," but either "true" or "not true." Such a standard of proof completes the system.

References:

1. Wolchover, Natalie. 07/19/2020. How Godel's Proof Works. wired.com.

2. Godel's Incompleteness Theorems. 11/11/2013. Stanford Encyclopedia of Philosophy.

3. Craig, Robin. 1992. Holes in the Heart of Reason. TableAus.

4. Godel's Incompleteness Theorems. Wikipedia.

5. Raatikainen, Panu. 2020. The Diagonal Lemma. Stanford Encyclopedia.

6. Hosch, William L. Peano Axioms. Britannica Encyclopedia.

7. Raatikainen, Panu. 2020. Godel Numbering.

The Incompleteness of Gödel's Theorem

ABSTRACT:

This paper shows what happens when Gödel's theorem is turned on itself, and the logical inconsistency that occurs when self-referencing statements are converted to Gödel numbers.

Gödel's First Incompleteness Theorem (GFIT) is often stated as follows:

"Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F."

Here's another way GFIT is stated: "In any reasonable mathematical system there's always true statements that can't be proved."

What happens if we turn the mirror of GFIT towards itself? Is the theorem still valid? Let's assume GFIT is a theorem in system F. We want to prove that a Gödel statement G is true yet unprovable. We can express G in the form of an equation: G = "This statement cannot be proved or disproved in system F."

Let's assume that every statement in F is proved or disproved except for G. If GFIT is true, then G is also true, since there must be at least one statement in F that can't be proved or disproved. But if G is true, then GFIT is false because G was proved by a theorem in F, namely GFIT. Now, if GFIT is false, then its negation ~GFIT is true: there are no statements in F that can't be proved or disproved. Since ~GFIT is true, G is false. If G is false ~GFIT is still true and GFIT is still false.

Of course one might argue that what I have presented so far is unfair. GFIT should exist outside of F as some sort of elite theorem exempt from its own rules and with zero self-awareness. If it could talk, it might say, "Rules for thee but not for me." Perhaps the best argument in favor of placing GFIT outside of F is it clearly fails to work inside F. So let's try removing it from F to see what happens.

So G is still inside F and there are no axioms or theorems that prove or disprove it (because we also removed ~GFIT), thus G is true and so is GFIT. Albeit, G is now its own axiom and it proves itself, so now GFIT is false. But wait ... G also turns false, since its message is consistent with GFIT, so nothing proved it--and it along with GFIT is true again, then false again ... ad nauseum.

Placing GFIT outside of F creates a paradox. Further, it makes sense to place GFIT (better yet, ~GFIT) inside F because GFIT is an F-system theorem. Its language makes that very clear. But let's ignore that and see what happens if we place both GFIT and G outside of system F.

Being outside of system F, G is not proved or disproved by any axioms or theorems inside F, so it's true and so is GFIT. So far, so good. Now, here's the beautiful part: Since G is true, it's its own axiom again, but it is not an F-axiom, so it remains true and so does GFIT! Look ma, no paradox! Unfortunately, none of this proves the validity of GFIT, since according to GFIT, "statements that cannot be proved or disproved" are inside system F.

So far we've tested GFIT with words which have shown GFIT to be paradoxical at best and false at worst. Let's try using Gödel numbers and express our statements mathematically and see what happens. Let's focus on the Gödel sentence G. G = "This statement can't be proved." Function N converts the statement to a Gödel number: N(G) = N("This statement can't be proved."). Now let's convert the statement into variables. "This statement" = G; " can't be proved" = P. Making some substitutions we get N(G) = N(G,P), but we erred. G has two different values: G = "This statement" and G = "This statement can't be proved." However, if we choose just one of these values to make our equation consistent, then it is no longer an equation: N(G) != N(G,P). The Gödel number on the left side no longer equals the Gödel number on the right side. This invalidates the G statement and shows that self-referential statements like G are logically inconsistent. Thus using G in a proof of GFIT would only make that proof logically inconsistent and the truth of GFIT logically inconsistent.

In conclusion, it seems reasonable to assume that our systems are not perfect. If a theorem is required to make the point, let's not assume the theorem was created by magical beings from Galaxy M64. Let's turn the critical eye of that theorem on itself and dare to see what happens.

References:

1. Wolchover, Natalie. 07/19/2020. How Godel's Proof Works. wired.com.

2. Godel's Incompleteness Theorems. 11/11/2013. Stanford Encyclopedia of Philosophy.

3. Craig, Robin. 1992. Holes in the Heart of Reason. TableAus.

4. Godel's Incompleteness Theorems. Wikipedia.