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.