The Platonist response to the flaw in Gödel’s proof

Page last updated 07 Sept 2023

Various people have tried to get round my demonstration of the flaw in Gödel’s Proposition V of Gödel’s original incompleteness proof by claiming that the R(x₁ … x_n) in the proposition is not actually a number-theoretic relation, but is only an expression that refers to an independently “existing” number-theoretic relation that does not belong to any language. This of course is a Platonist notion that has no evidence nor logic to support it. Nevertheless, it is easily shown that this notion makes no difference whatsoever to the demonstration that Gödel’s proof is irredeemably flawed, as shown below.

So, following that Platonist notion, we will suppose that an expression that includes numbers and variables for numbers cannot itself be a number-theoretic relation, but simply an expression that corresponds to a number-theoretic relation that “exists” independently of any language. That being the case, then it must also follow that there are valid expressions of the formal system that correspond to such independently “existing” number-theoretic relations.

That being so, let’s look at Gödel’s Proposition V in Gödel’s original incompleteness proof. Clearly, for any expression R(x₁ … x_n), there is also a corresponding formula f (x₁ … x_n) of the formal system that corresponds to the same independently “existing” number-theoretic relation that R(x₁ … x_n) corresponds to. That being the case, then there is no reason why we should not use this formula f (x₁ … x_n) rather than R(x₁ … x_n) to refer to that independently “existing” number-theoretic relation. This gives us, for Gödel’s Proposition V, for a number-theoretic relation with one free variable:

For every recursive relation f (x₁) … :

f (x₁) ⇒ Bew{Sb[r ^u₁/_Z(x1) ]}  …

This is a proposition, and x₁ is a variable with the domain of natural numbers, whereas u₁ is not a variable whose domain is natural numbers. Gödel refers to it as a free variable, but what one must recognize is that when Gödel uses word free variable in italics (in my online translation, and in Meltzer’s translation; but in Van Heijenoort’s and Hirzel’s translations capital letters are used for the same purpose) (Footnote: The online translation is recommended as it has clickable cross-reference links, but there are other English translations available. A translation by Meltzer was freely available online for many years but it now seems that many sites want you to pay for it, and hide it behind a pay wall, but you may find it online at PDF AltExploit - Meltzer’s translation and PDF Minnesota State University - Meltzer’s translation. Note that there are some minor errors in Meltzer’s translation which are not in the original German text. Other translations of “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme” are available; one that is often referred to is one by Jean Van Heijenoort. It is not available online; it can be found in the book: From Frege to Gödel: A Source Book in Mathematical Logic, publisher: Harvard University Press. There are arguments as to which English translation is the ‘best’ and most faithful to the German original (which you can see at PDF Gödel’s original proof in German), but the reality is that the differences between this translation and Meltzer’s are minor and inconsequential. Both of these use the same names for relations/ functions as in Gödel’s original paper. Another English translation by Martin Hirzel is available online at PDF Hirzel’s translation of Gödel’s Incompleteness paper; however, it is not recommended for reading in conjunction with my step-by-step guide to Gödel’s paper, since in Meltzer’s translation, in van Heijenoort’s translation, in the original German text, and in the guide the names for the various relations and functions are all the same, but they are different in Hirzel’s translation.) that actually means the specific number that is defined to correspond to that free variable - so in this case, the free variable u₁ is not any number, but some specific prime number greater than 13. Gödel states that this prime number can be arbitrarily chosen - in other words, we can choose any variable of the formal system, and there will be a prime number greater than 13 that corresponds to it. That number is given by a function that Gödel describes in his paper, and we will refer to this function by U(X), (Footnote: Gödel did not give this function a name, but his description clearly describes a function.) where X is a variable whose domain is variables of the formal system, i.e: x₁, x₂, … , x_n and where U(X) evaluates as a prime number greater than 13. We will choose x₁ to be the variable of the formal system. This now gives us for Proposition V:

For every recursive relation f (x₁) … :

f (x₁) ⇒ Bew{Sb[r ^U(x₁)/_Z(x1) ]}  …

But now we have a major problem. In the expression:

f (x₁) ⇒ Bew{Sb[r ^U(x₁)/_Z(x1) ]}  …

the x₁ is a free variable, with the domain of all natural numbers. But the function U is only defined for values that are variables of the formal system, and it is not defined if an attempt is made to substitute a numerical value for the x₁ in U(x₁).

Note that an attempt to claim that you can substitute the expression U(x₁) by the number 17 does not obviate the problem, since in mathematics, it is fundamental that if U(x₁) = 17, then that equality always exists, and the two expressions must be always interchangeable. And if they are interchangeable, then x₁ is a free variable wherever it occurs in the expression:

f (x₁) ⇒ Bew{Sb[r ^U(x₁)/_Z(x1) ]}  …

or, if you insist on Platonist notions, the x₁ corresponds to an independently “existing” free variable in the independently “existing” relation that corresponds to the above expression. But it should be abundantly clear by now that such Platonist notions only add verbiage that has no substance.

Yet again, no matter which way you look at Gödel’s Proposition V, if you analyze it logically, you find that there is always a logical absurdity within it.

See also the page Gödel’s Proposition V. For more on Platonism see Platonism, The Myths of Platonism, Platonism’s Logical Blunder, Numbers, chairs and unicorns and the posts Moderate Platonism and Descartes’ Platonism.