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

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)).

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. Because
statements with free variables are meaningless, this statement is
meaningless. It is not interesting that there is no proof or disproof,
any more than it is that there is no proof or disproof of the statement
"x is prime."

It is not even interesting when one makes an additional substitution,
which gives us

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

This statement is read, "There is no proof that there is no proof of
~(Ex'')(Pf(x'', y_y)) sub ~(Ex'')(Pf(x'', y_y))." Because its core
statement, ~(Ex'')(Pf(x'', y_y)) sub ~(Ex'')(Pf(x'', y_y)), *contains a
free variable*, it is only a meaningful statement about a meaningful
statement about a *meaningless* statement. Statements about the
nonexistence of proofs or disproofs of meaningless statements are not
interesting.

I will comment on your post below.

Ryan Reich wrote:
mikeh106@xxxxxxxxxxx wrote:

<snip long post; summary follows>

As I understand it, your argument is as follows: although Goedel did in
fact prove that there exist statements which can be neither proven nor
disporven, the statement he actually produced with this property is
completely uninteresting. The reason this is so is that rather than
being a complete logical sentence it contains "free variables", and
therefore does not express a property of numbers. You justify this by
considering the formula g_g, where g is the formula

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

so that when "expanded", g_g means

g_g = ~(E x)(Pf(x, g_g))

which requires further expansion to expunge the "free variable" g_g,
and so on, so that the statement does not arrive at a precise meaning
until it has been "expanded infinitely", at which point you claim the
resulting statement is "paradoxical".

It is more correct to say that the free variable is *in* g_g. Other
than that, you have understood me completely up to now.

I first of all disagree with you that the statement is meaningless, and
I think the reason your understanding is flawed is that you do not
quite understand the real necessity of the Goedel numbering system to
the proof. This may itself have something to do with the fact that you
are misusing the phrase "free variable": in logic, a variable is a
symbol, and a free variable is a symbol which is not quantified within
its scope. The sense you are using it seems to be that g_g is a "free
variable" in the above expression because it appears in its own literal
expansion, rather than being defined purely by basic logical entities.
This apparent self-reference is exactly the reason Goedel numbers are
necessary to make the proof work, and when they are used correctly, the
definition of g_g is not in fact self-referential: you don't need to
"substitute" g_g into itself to get an arithmetic statement.

It is not my argument that g_g is self-referential. I claim that g_g
contains a free variable and is therefore meaningless.

[beginning of proof snipped]

Now the sentence

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

which has one free variable, can be written down. That's the hard
part. What we accomplish by doing this is to obtain a formula which,
if we insert any Goedel number into the free variable 'y', tells us (by
the truth or falsity of the resulting sentence) whether or not the
sentence of which 'y' is the Goedel number, has a proof. Now I repeat;
g is a _formula_ of logic, merely a string of symbols. It has a Goedel
number, and that's an integer. When we form g_g we replace y by g in
the above expression, which means first we compute its Goedel number,
then we substitute that number into g in all the places that y occurs
as a free variable (which are hiding inside the very complicated
arithmetic operation y_y), at the end of which we have no free
variables left, just a lot of symbols and numbers; this gives us g_g.
Now, g_g is also a sentence, since in arithmetic, all numbers are
logical objects. And as you noted, g_g states that it does not have a
proof, and also implies that it does not have a disproof.

_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.

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.

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.

is not relevant if we are simply wondering
whether g_g is a sensible statement; it is, because it consists only of
logical symbols and numbers. It is the intricate web of relations that
tie together the Goedel-numbering scheme, the Pf function, and the
actual logical meaning of a sentence, which ends up leading to g_g
talking about its own unprovability.

That's as much as I can say on the subject. Goedel numbering is like
two people talking behind each others' backs: person A says "Person B
never lies" and person B says "Person A is an idiot", and the net
result is that Person A insults himself, without ever knowing that's
what he's doing. Actually, though, Goedel's proof is more airtight
than this, since in addition it has a way of talking about what it
means for a person to lie without actually having to talk about people.

Well, there is one more thing I can say. You are right in that the
sentence Goedel wrote is pretty useless, as well as almost impossible
to identify in practice. In later years people have identified actual
statements which are independent of a logical system (like the
continuum hypothesis).

--
Ryan Reich
ryan.reich@xxxxxxxxx

.

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