Gödel’s Proposition V
In Gödel’s proof of incompleteness (see online English translation of Gödel’s original proof) 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 not 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 the 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 ):
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(sf), where sf 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(V) to it, where V is a variable of the formal system.
Gödel’s Proposition V is (here in terms of relations with only one free variable):
For every recursive relation R(x1) there corresponds a relation-string r (with the free variable u1) such that for every x1 the following holds:
The terms relation-string and free variable have special meanings in Gödel’s proof. The term relation-string r means that r is a number where there is a formal system formula sf, where r = Φ(sf), and the term free variable u1 means that u1 is a number where u1 = β(V).
So we can write Gödel’s Proposition V with these terms explicitly defined, as follows:
For every recursive relation R(x1) there corresponds a number r and a number u1, where sf is a formula of the formal system with one free variable V, and r = Φ(sf) and u1 = β(V), such that for every x1 the following holds:
Now, take any R(x1) 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(x1) 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 (x1), where x1 is the free variable of that formula. So that formula satisfies Gödel’s definition of recursive, and has one free variable x1 and (3) and (4) above hold for that formula, which gives:
For the recursive relation Rf (x1) there corresponds a number r and a number u1, where sf is a formula of the formal system with one free variable V, and r = Φ(sf) and u1 = β(V), such that for every x1 the following holds:
And, according to Gödel, the number u1 may be chosen arbitrarily (as long as it corresponds to a variable of the formal system by the function β ). So we can choose 17 as the value, where this corresponds to the variable x1 of the formal system by the equality 17 = β(x1).
Applying the above in the statement of the proposition we obtain:
For the recursive relation Rf (x1) there corresponds a number r and a number 17, where sf is a formula of the formal system with one free variable x1, and r = Φ(sf) and 17 = β(x1), such that for every x1 the following holds:
This is of course, ridiculous, since in the above “proposition”, in (5) and (6), the term 17 is supposedly a constant, but at the same time it is defined as being equal to β(x1), and in the proposition x1 is a variable, so the term β(x1) cannot be a constant. (Footnote: Note that choosing 19 as the value for u1 so that 19 = β(x2) does not circumvent the problem, since we would then have, within the same proposition, one variable of the formal system that is a variable, and another variable of the formal system that is not a variable, but a constant - which is equally absurd.)
This demonstrates the absurdities present in Gödel’s Proposition V, a proposition for which he declined to actually furnish a proof (amazingly, his paper was accepted despite this obvious mathematical no-no). This shows why it was absurd to simply assume that it was a valid proposition. It isn’t, it’s nonsensical due to a confusion of language and meta-language. See also Gödel’s contradiction.
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 pretend that no formula of the formal system is allowed to be a “number-theoretic relation ”. Furthermore, since any formal system may be chosen as the subject of Gödel’s proof, that would mean that no formula of any formal system is a “number-theoretic relation ”. And that would mean that “number-theoretic relations” cannot ever be expressions that belong to a well-defined formal system - so that would mean that they must be informal things that cannot ever belong to a fully defined system. Welcome to the land of mathematical absurdities.
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 examines Gödel’s Proposition V in more detail.
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