Oh no ! Yet Another Flawed Incompleteness Proof

From the collection of obviously flawed incompleteness proofs, here is yet another:

A Flawed Incompleteness Proof by Jason Steinmetz

Page last updated 12 May 2025

A Jason W. Steinmetz has written what he calls a “detailed and rigorous analysis of Gödel’s proof of his first incompleteness theorem” and which has been published on Arxiv as An Intuitively Complete Analysis of Godel’s Incompleteness. However, a quick examination demonstrates that he makes untenable assumptions in order to get his result.

When Steinmetz refers to Gödel’s Proposition V, although ostensibly he is presenting an analysis of Gödel’s proof, he does not include Gödel’s Z function in his presentation of the proposition, and which is an essential part of Gödel’s Proposition V.

Instead, Steinmetz’s version is:

for any primitive recursive relation R(x₁, …, x_n ), where n ≥ 1, there exists a primitive recursive numeric formula φ with n free variables such that:

1.   R(x₁, …, x_n ) → Prov(φ(x₁, …, x_n ))

2.   ¬R(x₁, …, x_n ) → Prov(φ(x₁, …, x_n ))

But, just like Gödel, he declines to give anything even approaching a rigorous proof of this proposition. But, leaving that aside for the moment, he then continues (note that in the following φ_n is the formal formula whose Gödel number is n):

1:   Q(x, y) ≡_df ¬(φ_x ProofOf φ_y(y))

Due to Gödel’s theorem 5 there is a numeric formula φ_q(x, y) that implements or computes the relation Q. Therefore, due to theorem 5 and the definition of the relation Q:

2:   ¬(φ_x ProofOf φ_y(y)) → Prov(φ_q(x, y))

3:   φ_x ProofOf φ_y(y) → Prov(¬φ_q(x, y))

The formulas φ_p and φ_r are now defined:

4:   φ_p(x) ≡_df ∀n φ_q(n, x)

And then where p = [φ_p(x)]:

5:   φ_r(x) ≡_df φ_q(x, p)

Hence, it is immediate due to 5:

6:   φ_q(x, p) = φ_r(x)

But when Steinmetz states p = [φ_p(x)], he is simply assuming that formulas of the formal system can express the Gödel numbering function, since he has previously defined the square brackets to indicate the Gödel numbering function, see in his Section 3: Gödel Number of a Formula: [φ_n ] = n.

But it is patently obvious that the formal system cannot express the Gödel numbering function, since its free variable has the domain of sequences of symbols of the formal system, and there is no such variable in the formal system. This renders Steinmetz’s result completely invalid (see also my paper PDF Paper: The Impossibility of Representation of a Gödel Numbering Function by a Formula of the Formal System).

Besides the above obvious flaw, Steinmetz manages to completely confuse formulas of the formal system and Gödel numbers that correspond to such formulas. As noted above, having defined that φ_n is the formal formula whose Gödel number is n, he proceeds to claim in his Section 3: Gödel’s Provability Predicate, that Prov(φ_x ) is Gödel’s relation 46, Bew(x), which of course is complete nonsense, since the free variable x of Bew(x) has the domain of natural numbers, whereas the domain of the free variable φ_x of Prov(φ_x ) is formulas of the formal system.

I could go on but I can’t take any more of this. Steinmetz has managed to fill 36 pages with what he claims is a rigorously detailed analysis of Gödel’s proof, yet all it demonstrates is his compete misunderstanding and confusion. How can people be so blind to this nonsense?

Also see Errors in incompleteness proofs and Analysis of incompleteness proofs.

 

Other obviously flawed incompleteness proofs can be seen at:

• An Incompleteness Proof by Francesco Berto
• An Incompleteness Proof by Bernd Buldt
• An Incompleteness Proof by Dan Gusfield
• An Incompleteness Proof by Byunghan Kim
• An Incompleteness Proof by Dennis Müller
• An Incompleteness Proof by Sebastian Oberhoff
• An Incompleteness Proof by Arindama Singh
• An Incompleteness Proof by Antti Valmari