|
|
About three months ago, I posted, "How can the meaning of Goedel's
unprovable statement descend from infinity?" In that post, I tried to
show that Goedel's undecidable statement has no intuitive meaning
because it is a statement of the form: "The following is unprovable:
The following is unprovable: The following is unprovable: ..." I failed
to respect the fact that the arithmoquine (I will define this in a
minute) of any symbolic property is a meaningful statement about
numbers.
Although I clearly see the correctness of the proof, I have worked
obsessively since then to understand why our current theory of
arithmetic falls short of exhausting all true statements. Obviously,
although we know the undecidable statement is true, we *don't have
enough rules in logic to prove it*. It must therefore be some unique
style of reasoning that allows us to say that a theorem that states the
unprovability of its own symbolic representation is necessarily true.
But what is the nature of this style of reasoning? What is the "extra
thing" we do to show the truth of Goedel's undecidable statement?
Have any efforts been made to expand the technique of mathematical
proof to include such arguments?
Here is my work so far. First, I will give a proof of the
undecidability of a statement. I will copy some definitions from my
previous post and add some new ones.
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 n("Statement") be the Goedel number of the statement written
between the quotes.
Let N(number) be the symbolic representation of a number, called the
*numeral* of the number, to be used in a written statement. For
example, N(328) would become in the Goedel number of its statement,
through the identifier n, three primes with exponents representing the
symbols 3, 2, and 8.
Let Pf(x) be the property that there is a symbolic proof of the Goedel
number x. Thus, Pf(n("N(2) + N(2) = N(4)")). Notice the use of n and N
in the statement.
Let NPf(x) be the negation of the property Pf(x). Thus, NPf(x) is the
property that there is no proof of Goedel number x.
Let Aq(x) be a function on Goedel numbers with free variables -- which
may be rightly called properties -- to Goedel numbers of statements.
Aq(x) is the Goedel number of the statement that results when the
statement of x has replaced for its free variables the numeral of the
value of x. Thus, the Aq(n("x is prime")) = n("N(n("x is prime")) is
prime"). Again, notice the use of n and N.
Theorem: There is a statement in arithmetic that is neither provable
nor disprovable.
Proof: Consider the property "NPf(Aq(x))". It is true when there is no
proof of the arithmoquine of x, which may be a Goedel number with a
free variable. Now consider the *arithmoquine* of the property,
Aq(n("NPf(Aq(x))")). We see that
Aq(n("NPf(Aq(x))")) = n("NPf(Aq(N(n("NPf(Aq(x))"))))").
Thus, the arithmoquine of "NPf(Aq(x))" is actually equal to the Goedel
number of a statement about the unprovability of the arithmoquine of
"NPf(Aq(x))". In the right-hand expression, we see that a statement is
stating the unprovability of an expression for its own Goedel number.
If it were provable, the expression would be of an unprovable
statement, and -- because of equivalence in symbolic structure of proof
-- the statement itself would be, a contradiction. If it were *false*,
the expression would be provable, and again, by equivalence in the
symbolic structure of proof, the statement itself would be provable,
and thus true, another contradiction. Therefore the statement is *true*
but *unprovable*.
I have kept notes on this theorem and drawn diagrams showing the
"sameness relations" between symbols in the two expressions above: for
example, I have drawn lines between the Aq's of the first statement
above and the Aq's of the second. There seem to be two possible
associations between equal symbols: mathematical association and
self-referential association. The second Aq in the first statement is
mathematically associated with the first Aq in the second statement,
but *self-referentially* associated with the second Aq in the second
statement.
What I mentioned above, the "equivalence in symbolic structure of
proof", the idea that if I can prove a statement, I can find an
arithmetical proof of its corresponding Goedel number, and vice versa,
is also very important.
How do all these ideas function together in the "trick" of the proof?
What is the philosophical lesson of such a trick? Like I said, I
clearly see the proof's correctness, but I do not fully comprehend its
meaning. Even Wittgenstein called the proof a "logical conjuring
trick".
Thank you in advance for your responses.
|
|