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

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.

Let me give a short, artificial example of what's going on here. Let

p = (E x) (x = y)

which is obvious and also not of interest in the same way as the Goedel
sentence, but it does contain a free variable. It's got a Goedel
number, say N, and using this we can form p sub p:

p sub p = (E x) (x = N).

This does not contain a free variable. It does not even make sense to
look for one by "expanding" N to get back p, like

(E x) (x = (E x) (x = y))

since it is meaningless for a variable to equal a sentence; this is no
longer a "well-formed" formula (this is discussed in any introductory
text on formal logic).

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.

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.

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.

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.

_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. Would you
say that the sentence

(E x) (x = 2)

is meaningless because if you write it like

(E x) (x = y) sub 2

(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 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.

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.

If you want to argue that to assign meaning to a sentence you must
unravel any codes in the sentence and see if they are secretly
meaningless, you could easily conclude that no sentence has meaning,
since it's trivial to invent a code that turns any meaningless
statement you like into any expression you want. The point of formal
logic is to make manipulations that do *not* take internal structure
into account, but merely proceed by manipulation of formal expressions.
In this sense, the brilliance of Goedel's proof is that it shows that
despite this fact, you can perform this manipulation in the context of
arithmetic, which hides the "metalanguage" of formal logic behind still
more formal manipulations. Once you can put the metalanguage, the
language of talking about proofs, into the machinery of doing proofs
within the formal language itself, you can write a formal expression
whose very existence takes the indirect route through arithmetic to
have meta-meaning.

--
Ryan Reich
ryan.reich@xxxxxxxxx

.

Relevant Pages
Quantcast