Re: Which is 'this' sentence

On Feb 9, 3:37�pm, stevendaryl3...@xxxxxxxxx (Daryl McCullough) wrote:
R. Srinivasan says...

No. The informal statement is "This sentence is unprovable". But how
do you *formalize* it, i.e., how do you translate it into the language
of a first-order theory (say, Peano Arithmetic)? This is what Godel
achieved. The Godel sentence for PA is a purely arithmetical sentence
and can be thought of as a translation of the informal statement "This
sentence is unprovable". But in order to achieve this translation
Godel had to quantify over functions -- see for example the diagonal
argument given in the recent sci.logic thread "Goedel's proof" started
by Newberry.

That doesn't sound correct. Goedel's translation doesn't quantify over
functions. There are several parts to Goedel's translation:

(1) Associating natural numbers with the formulas of PA.
(2) Defining a formula in PA (call it Theorem(x)) such that
for each natural number n, PA proves Theorem([n]) if and only
if n is the code of a theorem of PA. (where [n] means the
numeral corresponding to number n.
(3) Proving the fixed point lemma.
This says that for every formula Phi of PA, there is a
natural number n that is the code for a formula Psi such
that PA proves Psi <-> Phi' where Phi' is the result of
substituting [n] for all free variables occurring in Phi.
(4) Applying the fixed point lemma to the formula not Theorem(x)
gives Godel's theorem.

In other words, Godel had to consider infinitely many
functions fq(n), where each function f1(n), f2(n), ... is an infinite
entity.

No, he doesn't do that.

Informally, my objection to "This sentence is not true" is that when
we utter "sentence", we have no construction in mind for any sentence.

That's an inessential feature of the liar paradox. Instead of using
the phrase "this sentence", you can introduce fixed points in some
other way. For example,

If S is any string, define the "quotation" of S to be the result of
replacing all occurrences of '\' in S by '\\', and replacing all
occurrences of '"' in S by '\"', and then placing '"' at the beginning
and end. For example, the quotation of

     hello

is

    "hello"

and the quotation of that is

    "\"hello\""

and the quotation of that is

    "\"\\\"hello\\\"\""

Now define the "fixed point term" for any string S to be
the string "The fixed point of" followed by the quotation
of S. So the fixed point term of

     hello

is

     The fixed point of "hello"

Finally, define the fixed point of any string S to be the
result of replacing all occurrences of the string
"[fill in the blank]" in S by the fixed point term of S.
So the fixed point of

    hello

is just

    hello

(since "hello" doesn't contain the string "[fill in the blank]") while
the fixed point of

    Hello, [fill in the blank].

is

    Hello, The fixed point of "Hello, [fill in the blank].".

So far, there is nothing *meaningful* about any of these operations.
They are just operations on meaningless strings that produce other
meaningless strings. However, now consider the string, (call it
L0)

    [fill in the blank] is not true.

This is still a meaningless string. But now let's take the
fixed point of L0. It's the following string (call it L).

    The fixed point of "[fill in the blank] is not true." is not true.

L appears to be a meaningful sentence. It seems to be claiming
that the fixed point of some particular string is not true.
That string is actually the string L0. So L seems to be claiming
that the fixed point of L0 is not true. But the fixed point of L0
is L. So L seems to be claiming that L itself is not true.
This is accomplished without actually using the construction
"This sentence".

--
Daryl McCullough
Ithaca, NY

As I said before I think, you cannot present a self-reference by
referencing the marks it makes. That is a ploy that fails to work.

.

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