David Pierce on Gödel’s incompleteness proof
Page last updated 16 Jun 2023
The mathematician David Pierce, on his web-page of 2018 On Gödel’s Incompleteness Theorem, states that he will review the mathematics of Gödel’s incompleteness paper in some detail. (Footnote: He also has some less detailed articles on the subject at Gödel, Grammar, and Mathematics and Mathematics and Logic.)
Pierce devotes several lengthy paragraphs detailing the first part of Gödel’s paper, to the point where a relation B(a, c) is defined such that the relation applies if and only if a and c are Gödel numbers of formal expressions A and C, and where A is a formal expression that, in the formal system, proves the formal expression C (Pierce writes this as a = ⌐A¬ and c = ⌐ C ¬ ). Then Pierce states:
There is then another formula, Q(x, y), also defining a recursive set, and for all counting numbers a and b, the sentence Q(a, b) is true if and only if:
- b = ⌐φ(y)¬ for some formula φ(y), and
- the negation ¬B(a, ⌐φ(b)¬ ) is true, so that a is not ⌐ Γ ¬ for any proof Γ - if it even exists - of φ(b).
Or, to put it rather more succinctly:
There is a relation Q(a, b), where a, c and y are natural numbers, which applies if and only if:
- there is some formula φ(y) such that:
- b = ⌐φ(y)¬ and
- ¬B(a, c) is true, where a = ⌐A¬ and c = ⌐φ ( ⌐φ(y)¬ )¬ (Footnote: Note that this implies that A and φ ( ⌐φ(y)¬ ) are expressions of the formal system, since they are members of the domain of the Gödel numbering function.)
This corresponds to the crucial part of Gödel’s paper, a combination of Gödel’s Proposition V and Proposition VI, yet Pierce provides no indication whatsoever of how it might be proved that there must be a formula Q(a, b) with the properties Pierce claims for it. Perhaps one might not be surprised, since Gödel himself didn’t prove this part of his paper either. (Footnote: Gödel only provides a brief outline of how one might create a proof, saying in his paper: “We content ourselves here with indicating the proof of this proposition in outline, since it offers no difficulties of principle and is somewhat involved.” See Gödel’s Proposition V.)
But that fact is an important point that should be remarked upon, and the correct thing for Pierce to say at this point would be to say something such as:
At this point Gödel assumes that it would be possible to prove that there is a formula Q(a, b) with the properties…
But no, Pierce makes no mention of Gödel’s assumption; there is no mention of the fact that Gödel didn’t actually prove this crucial step. Pierce isn’t unusual in this respect, most mathematicians and logicians who discuss Gödel’s paper also gloss over this fact. What is wrong with these people? Pierce proudly proclaims on his website that he is a professional mathematician. Is this what professional mathematicians do? Cover up the assumptions of other mathematicians who simply assume part of a proof in order to produce the desired result? It is a story state of affairs when there is a continuing failure to acknowledge that Gödel didn’t prove the crucial step of his “proof ”, and mathematicians and logicians continue to present his result as though he had actually fully proven it.
The ridiculously misplaced hero worship of Gödel and his incompleteness “proof ” should be called out everywhere for what it is, and I shall continue to do so. The mathematicians of the future will look back on this sorry saga with astonishment.
What is the point of claiming that you are explaining/
A common retort is that it’s obvious that there is a formula Q with the properties described - thus resorting to a plea to intuition. It’s rather ironic that on his web-page Mathematics and Logic Pierce refers to confirmation bias, and refers to the attempts to replicate the Millikan Oil Drop Experiment, and how researchers, when finding discrepancies between those results and Millikan’s, considered that it was so obvious that Millikan must have been correct that they invented false reasons to explain why their results were different. It was only later they discovered that Millikan’s result was actually incorrect. Pierce goes on to quote Richard Feynman: (Footnote: From a speech by Feynman at Caltech, California, USA, 14 June 1974.)
“The first principle is that you must not fool yourself.”
Now consider a similar case, where J. R. Lucas pointed out in 1961 in his article Minds, Machines and Gödel that Gödel’s result indicated that it must be the case that humans could “perceive” Gödel’s result but that no machines could ever do so, and so he postulated that humans must be using something special that machines can never access. And since then all mathematicians and logicians have attempted all sorts of convoluted responses to explain away this difficulty, just like the researchers trying to replicate the Millikan Oil Drop Experiment. Even the renowned Roger Penrose fell into the fashionable trap of claiming that - if we can’t explain something, then the explanation must lie in the quantum realm - and he proclaimed that there could be some quantum process occurring in the human brain that would explain the conundrum posed by Lucas. (Footnote: For more details see the web-page Man versus Machine.) But of course this is an absurdity - it doesn’t even begin to explain why, even if such quantum process might be involved in the human brain, why no machine could ever be built that could use those same quantum processes.
This and other similarly absurd circumlocutions should have made people sit up and take notice and suspect that perhaps there was something not quite right with Gödel’s “proof ”, and on that account engage in a scrupulously careful analysis of it to ascertain if there might be any logical error in it. But no, mathematicians and logicians have been so certain that Gödel’s intuitive assumption must be correct that they have never actually subjected it to a fully detailed logical analysis. The standard approach is to deal with Gödel’s crucial assumptive step by carefully inserting ones head into the sand.
It’s precisely because - when I searched for a fully detailed proof of Gödel’s outline regarding his Proposition V and could not find any such thing - I decided to take the time and effort to fully flesh out Gödel’s outline regarding his Proposition V, as is detailed in my paper PDF The Fundamental Flaw in Gödel’s Proof of his Incompleteness Theorem. And when you do so, you discover that Gödel’s naive intuitive assumption was incorrect, with the result that his proof has no logical validity.