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

Ryan Reich wrote:
mikeh106@xxxxxxxxxxx wrote:
Thank you for your informative post.

However, despite your disagreement and having accused my understanding
of being "flawed," you have not offered a valid objection to my proof
of the meaninglessness of Goedel's unprovable statement, as I have
defined it.

My argument is simple. g sub g (or, as you call it, g_g) is not the
free variable.

g_g can be expanded as

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

Recall what this means: to take the second formula, find its Goedel
number, and replace all occurrences of y in the first formula with n.
Take note of my paragraph on the subtelties of the substitution
function, since it's very relevant here. There are actually two uses
back-to-back and they are confusing. The first use is the "lexical"
substitution: you take the Goedel number of the second formula and
physically insert it in place of all occurrences of y in the first
formula. The second is "numerical" substitution: that's what is
happening in the y_y there; after you have lexically substituted, this
is just

g(n) sub g(n)

a function of numbers and not formulae. You don't get y back in this
expression; you just get a number.

I believe you meant n(g) sub n(g)?

Let me reiterate how I defined x sub y on 2/15.

On 2/15, I wrote:
"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."

In my original post, I make it clear that x sub y *does* refer to a
number. Your comment about lexical and numerical substitution is a
straw man; I am referring to numerical substitution, and in fact I make
it clear throughout my entire original post that I am dealing with the
*Goedel numbers* of sentences, not with the sentences themselves, as if
their symbols were mathematical objects.

Now, let me reiterate another part of my post.

On 2/15, I wrote:
"A *Goedel number* [my emphasis] with a free variable, such as *that
of* [my emphasis] 'x is prime', is meaningless because it is not a
*statement about numbers*."

Here, I *clearly state* that I am referring to Goedel numbers with free
variables, not lexical formulae as you would object to my using.

[snip]

After substitution, we have

~(Ex)(Pf(x, ~(Ex')(Pf(x', y_y)) sub ~(Ex')(Pf(x', y_y)))).

Here, ~(Ex')(Pf(x', y_y)) sub ~(Ex')(Pf(x', y_y)) is Goedel's
unprovable statement. It *clearly* contains a free variable.

I'd like to note here that this statement does *not* clearly contain a
free variable. By definition of "sub", this expression actually means
that you take the Goedel number of the latter formula and substitute it
into all occurrences of the free variable in the former formula. The
free variables are eliminated. Now, the way you've written the
expression it looks like it has a free variable, but you haven't
written it in a logically formally correct way. When you do, the free
variable vanishes.

The statement ~(Ex')(Pf(x', y_y)) sub ~(Ex')(Pf(x', y_y)) is the mere
*formula* for the substitution; it is different from the *result* of
the substitution. Compare it with the statement "x = 2 + 2." This
statement obviously has a different Goedel number than the statement "x
= 4," now doesn't it?

It's clear that you didn't really understand what I was saying. It may
well be that the original formula,

g = ~(E x)( Pf(x, y_y) )

contains the free variable y. However, its Goedel number does not. It
does not even make reference to y. It is simply a number. Hence, when
n(g) is plugged into y in g, the free variable disappears. It may seem
that g_g contains a free variable when you unravel the Goedel numbers
in it, but that is a misuse of formal logic. g_g is simply a statement
about numbers.

This is the same straw man argument I pointed out earlier. I am dealing
strictly with Goedel numbers, not formulae, as I make clear in my
original post.

It should be obvious that when I speak of a Goedel number containing a
free variable, I am referring to the variable that occurs in the coding
of the Goedel number. It is a perfectly meaningful use of terms.

When you see a statement like that you try to understand it by reducing
it to simpler statements and understanding their relationships. In
doing this, you naturally are led to unravel the Goedel numbering,
back-substitute the corresponding formulae, and obtain what appears to
be an unresolved free variable. However, this is wrong in two ways:
first, in the way that I indicated above, that a Goedel number is not a
formula but a code for a formula, and that even if the formula it codes
has a free variable the Goedel number itself does not, being just a
number.

See above.

The second way it's wrong is much the same way that people who claim
that 0.999... and 1 are not equal are wrong. They often argue that
there is a slight difference, that 0.999... is just a little less "all
the way at the end". The thing is, you never get to the end. And when
you try to unravel the Goedel sentence you are trying to achieve the
same thing. You never actually see the free variable; it's "hidden" in
another expression, and in fact there's no way to write any stage of
the iteration so that y explicitly appears. At each stage, it's just a
reference inside another Goedel number that you must unravel to look
any farther. If you can't find it at any finite stage, however, it is
not there. Formulas cannot be infinite.

The free variable is never "hidden," except in lexical formulae that
use g in place of ~(Ex)(Pf(x, y_y)). The *Goedel number of g* still
contains a free variable y, and Goedel numbers can be said to have free
variables, as I have shown above.

_This is not circular_. There are no unresolved variables left in g_g;
the only free variable in g was 'y', and it got filled in with some
number when we made g_g.

You misunderstood my argument about the problematic nature of
statements that *mention* free variables. A statement can *mention* a
free variable without *containing* one. As I said before, g_g can be
expanded as

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

This statement does not contain a free variable, but it is *about* a
statement that *does*. The free variable is in the y_y of the second
statement that replaces.

This is different than what you seemed to be saying above.

I apologize. I meant to say that *after the substitution*, the sentence
does not contain a free variable. However, it still *mentions* one. I
meant to write the formula

~(Ex)(Pf(x, ~(Ex')(Pf(x', y_y)) sub ~(Ex')(Pf(x', y_y)))).

Would you say that the sentence

(E x) (x = 2)

is meaningless because if you write it like

(E x) (x = y) sub 2

I would say that before the substitution it is meaningless; after it is
performed it is meaningful.

The expression (Ex)(x = y) sub 2 is not a statement but a relation
between a Goedel number and the number 2, and it has a Goedel number
that is *different* from the Goedel number of (Ex)(x = 2). The first
contains a free variable, the second does not.

(where for simplicity I've chosen the Goedel numbering so that 2 is its
own Goedel number) it is "about" a statement with a free variable?
What about

(E x) (x = y) sub n(y)

where n(y) is whatever the Goedel number of the symbol y is. Actually,
let's say y has Goedel number 5. Then this sentence is

(E x) (x = 5)

which is again true, and meaningful, although it expands to an
expression that concerns free variables. Tell me how you can
distinguish between this and the situation with the Goedel sentence.

The substitutions in Goedel's unprovable statement, when carried out
indefinitely, lead to *paradoxes*; these do not.

The fact that this number is actually the
Goedel number of g itself

Please explain how g_g is the Goedel number of g. g_g is clearly a
*longer string of symbols* that *contains g*, so it must be larger.

I think you misunderstood me here. The Goedel number of g is what you
substituted for y within g; the resulting sentence is g_g.

Then g and g_g do not have the same Goedel number.

The same can be said about g_g in relation to ~(Ex)(Pf(x, g_g)). The
latter is clearly a longer statement containing the former, and thus it
has a larger Goedel number.

Statements can be equivalent while having different Goedel numbers.

However, I made no attempt to argue based on a relationship between
Goedel numbers and truth values of statements. My point was simply
that once you have replaced y with the Goedel number of g, the fact
that the expression you get has a number in it which codes a sentence
with a free variable is irrelevant; there is not actually a free
variable *in* the resulting expression.

There does not need to be. Read my post again:

On 2/15, I wrote:
"Although a statement can be about a meaningless Goedel number, it may
still be meaningful. However, the statement it is *about* is
meaningless."

.

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