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Gödel’s Proposition V

In Gödel’s proof of incompleteness (see online English translation of Gödel’s original proof or English translation of Gödel’s original proof, PDF file) there is an arbitrary distinction between number-theoretic relations and formulas of the formal system.


In the proof, a symbol sequence such as “x + SSSSSS0 > SSS0” is simply a sequence of symbols that belongs to the formal system, and “x + SSSSSS0 > SSS0” is neither a number-theoretic relation, nor can it be a name that refers to an abstract number-theoretic relation.


But on the other hand an expression such as “x + 6 > 3” is either a number-theoretic relation, or else a name that refers to the abstract concept that is an abstract number-theoretic relation.


However, Proposition V in Gödel’s proof refers in a vague way to some sort of isomorphism between abstract number-theoretic relations and formulas of the formal system, by simply asserting that there is a “correspondence” between every number-theoretic relation that is recursive and some formula of the formal system. Of course there is no essential difference between “x + 6 > 3” and “x + SSSSSS0 > SSS0”; they simply belong to slightly different systems. And that simply means that, given a language that is a meta-language to both systems, there can be a translation function in that meta-language that translates the symbol sequences of of system to symbol sequences of the other system.


And that means that if “x + 6 > 3” is simply a name that refers to the abstract concept that is an abstract number-theoretic relation, then “x + SSSSSS0 > SSS0” is also a name that refers to the very same abstract concept.


Or if the symbol sequence “x + 6 > 3” satisfies the definition of a number-theoretic relation, then the symbol sequence “x + SSSSSS0 > SSS0” also satisfies that definition.


Gödel’s paper doesn’t define “number-theoretic relation”; and Gödel’s perfunctory outline of how a proof of his Proposition V might be created relies on the pretense that formulas of the formal system are not number-theoretic relations, nor are they names for the same abstract concept as number-theoretic relations. Without that pretense, his proof outline results in a blatant contradiction as shown below (see also the paper The Fundamental Flaw in Gödel’s Proof of the Incompleteness Theorem PDF ):


Gödel’s Proposition V is expressed in terms of variables x1, x2, …, u1, u2, … . But, as for any other mathematical proposition, one can use different variable names in a proposition, provided they do not clash with variable names that are already in that proposition.


In Gödel’s proof we have the Gödel numbering function defined as Φ(s2), where s2 is a symbol sequence of the formal system. He also defines a variable numbering function which assigns a number to every variable of the formal system. Gödel does not give this function a name, but for convenience we shall assign the name β(s1) to it, where s1 is a variable of the formal system.


Gödel’s Proposition V is (here stated in terms of relations with only one free variable):



  R(x1) ⇒ Bew{Sb[r, u1, Z(x1)]}

¬R(x1) ⇒ Bew{Neg Sb[r, u1, Z(x1)]}


The terms relation-sign and free variable have special meanings in Gödel’s proof. The term relation-sign r means that r is a number where r = Φ(s2), where s2 is a formal system formula. The term free variable u1 means that u1 is a number where u1 = β(s1) , where s1 is a formal system variable. So, including these definitions explicitly in the statement of the proposition does not change the assertion of the proposition in any way:



  R(x1) ⇒ Bew{Sb[r, u1, Z(x1)]}

¬R(x1) ⇒ Bew{Neg Sb[r, u1, Z(x1)]}


Now, since there is no variable v1 already in the proposition, we may use v1 instead of x1 as the variable name in the above, giving:



  R(v1) ⇒ Bew{Sb[r, u1, Z(v1)]}

¬R(v1) ⇒ Bew{Neg Sb[r, u1, Z(v1)]}


Now, take any R(v1) as referred to by the proposition, and translate it into a formula of the formal system. Either that formula is a recursive relation, or else it is a name that refers to the abstract concept that is the same abstract number-theoretic relation as the R(v1) referred to originally.


So there will be such formulas of the formal system for which Proposition V holds. Let’s call one such formula Rf (v1), where v1 is the free variable of that formula. So for that formula, we have:



  Rf (v1) ⇒ Bew{Sb[r, u1, Z(v1)]}

¬Rf (v1) ⇒ Bew{Neg Sb[r, u1, Z(v1)]}


Now, according to Gödel, the number u1 (where u1 = β(s1)) may be chosen arbitrarily (provided that s1 is a formal system variable). So, let’s choose the number 17, which corresponds by Gödel variable numbering to the variable v1 of the formal system. This gives us:



  Rf (v1) ⇒ Bew{Sb[r, u1, Z(v1)]}

¬Rf (v1) ⇒ Bew{Neg Sb[r, u1, Z(v1)]}


This is, of course, ridiculous, since in the above proposition, we have that v1 is both a constant and a variable at the same time; it is a constant in 17 = β(v1) since 17 is a constant - but everywhere else it is a variable. This is an absurd contradiction which completely invalidates Gödel’s proof.


This show the nonsensical nature of Gödel’s Proposition V, and shows why it was absurd for Gödel to simply assume that it was a valid proposition, and to assume that it could be easily proved. It isn’t a valid proposition, it’s complete nonsense where there is a confusion of language and meta-language.


The language confusion arises because the proposition implies a function that translates from “number-theoretic relations” to formal system formulas (the correspondence referred to in the proposition), but if such a function is used, then that function is a meta-language to both “number-theoretic relations” and formal system formulas. And that means that in that function all symbols and symbol sequences of both “number-theoretic relations” and formal system formulas are objects as seen by that function, and therefore they must be objects in the proposition, and hence they cannot be variables in the proposition.


The only way to attempt to get around this absurdity is to fudge the issue, and to pretend that there is some distinction of language level between number-theoretic relations and formal system formulas, even though a translation function prevents that being possible.


See also the page The Platonist response to the flaw in Gödel’s proof. The paper The Fundamental Flaw in Gödel’s Proof of the Incompleteness Theorem PDF examines Gödel’s Proposition V in more detail.



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Previous Blog Posts  




For convenience, there are now two pages on this site with links to various material relating to Gödel and the Incompleteness Theorem


– a page with general links:

Gödel Links


– and a page relating specifically to the Gödel mind-machine debate:

Gödel, Minds, and Machines


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