How can the meaning of Goedel's unprovable statement descend from infinity?

(Note: The author is aware that solutions of certain Diophantine
equations are said to be mechanically uncheckable. This post focuses
strictly on Goedel's Incompleteness Theorem, as it is proved by J.N.
Crossley in "What Is Mathematical Logic?")

Don't you find it curious that the proof of Goedel's Incompleteness
Theorem relies on "free variables"? When do these "free
variables" ever appear in axioms or theorems in arithmetic? Why are
they so important in proving the existence of a statement that cannot
be proved or disproved?

I have thought hard about Goedel's Incompleteness Theorem since I
read a proof at age fifteen. My goal was to intuitively understand the
meaning of the unprovable statement. I now believe that while the
theorem is true, the statement is meaningless. It is just not the kind
of statement we would wish to prove in arithmetic. In this post, I will
attempt not a disproof of Goedel's theorem itself, but a proof of the
"meaninglessness" of the unprovable statement.

First, a little background information.

The symbols of arithmetic (+, -, =, logic symbols, etc.) may be
assigned certain numbers, so that statements in arithmetic may be coded
into a "Goedel number" by making the assigned numbers exponents of
the primes, in the order they occur.

We can define what it means for a number to be the "Goedel number of
a proof": we make the Goedel numbers of sequentially derived
statements exponents of the primes, in the order they are deduced. It
is a proof if each deduction is valid, and this can be checked
mechanically.

Let "x sub y" be the Goedel number that results when all the free
variables of the statement of the Goedel number x are replaced with the
Goedel coding of the number y. In essence, x sub y refers to a function
by which the value of y is "filled in" for the free variables in x.

Let Pf(x, y) be *not* the property that x is the Goedel number of a
proof of Goedel number y, but the *Goedel number of the statement* that
x is the Goedel number of a proof of the Goedel number y.

I give a brief outline of Crossley's proof here:

Let g be the Goedel number of the statement ~(Ex)(Pf(x, y sub y)). This
is read, "There is no proof of y sub y." It contains one free
variable, y, twice. Now consider the statement "g sub g." One can
easily see from the definition of g that it is precisely, "There is
no proof of g sub g." Thus, we have found a statement that is
equivalent to the nonexistence of a proof of it. If it were provable,
then it would be true, and a proof would not exist -- a contradiction.
If it were disprovable, then there would be a proof -- another
contradiction. It is therefore neither provable nor disprovable. Q.E.D.

What I will attempt to show is that Goedel's unprovable statement is
not a statement about meaningful propositions in arithmetic; rather, it
is a statement about a meaningless statement about the substitution
into itself of a Goedel number with two free variables. I will argue
that it is meaningless precisely because it -- and all the equivalent
statements that result when the repeated substitutions are carried out
-- mention free variables in their nested statements. When the
substitutions are carried out indefinitely, we arrive at a paradox of
meaning I would call "descent from infinity." For Goedel's
unprovable statement to be a *meaningful statement in arithmetic*,
there must be no free variables in any of the nested statements, and
therefore the substitutions must be carried out indefinitely. But then
we get a statement of the form: "The following theorem is not
provable: The following theorem is not provable: The following theorem
is not provable: ..." and so on, to infinity, which is clearly
meaningless.

First, I will make a few points.

Goedel numbers can be about other Goedel numbers, which in turn may be
about other Goedel numbers, and so on.

A Goedel number with a free variable, such as that of "x is prime",
is meaningless because it is not a *statement about numbers*. It is
only a statement about a possible number x. It is at best a
"property"; it is not meaningful until we replace x with something.
It should be remembered that there is not a single theorem in
mathematics that explicitly contains a free variable.

Although a statement can be about a meaningless Goedel number, it may
still be meaningful. However, the statement it is *about* is
meaningless. For example, the statement, "the Goedel number of 'x
is prime' is neither provable nor disprovable" is a meaningful
statement about a meaningless Goedel number.

Now let me explain why Goedel's unprovable statement is meaningless.

As in Crossley's proof, define g to be the Goedel number of the
statement

~(Ex)(Pf(x, y sub y)).

The unprovable statement is g sub g, and it is supposedly meaningful.
What does it mean? Let us examine it more closely:

g sub g = Goedel number of ~(Ex)(Pf(x, g sub g)).

From here on I will drop the words "Goedel number of" for the sake
of clarity. Notice that g sub g is still about the statement g, and
more importantly, it *still mentions a free variable* (hidden in g).
What, then, does *this* statement mean? To decide its meaning, we must
remove the free variable, and to do that, we must find the equivalent
statement in which the substitution is already performed:

g sub g = ~(Ex)(Pf(x, g sub g)) = ~(Ex)(Pf(x, ~(Ex)(Pf(x, g sub g))))

But this statement still mentions a free variable! And so on, to
infinity.

We are left with two choices. We must either admit that g sub g refers
ultimately to a meaningless statement, or carry out the substitution
process indefinitely, resulting in a paradoxical statement which is
also meaningless. This proves the meaninglessness of Goedel's
unprovable statement.

All objections are welcome. No personal attacks.

.

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