Re: Which is 'this' sentence

On Feb 9, 8:37 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.

I have had this discussion before with Aatu Koskensilta also and he
said something similar to what you are saying. The difference in our
perceptions occurs at the above point. The objection is to
quantification over formulae like Phi with one or more free variables
(i.e., "For every formula Phi..." is objectionable in NAFL).

Classically, you view a formula Phi(x) with a free variable as a
finite entity, in the purely syntactical sense. In NAFL however, a
free variable x that ranges over, say, the natural numbers, has a
specific value when, and only when, the human mind specifies a value
for x. Thus x could have the value 10 only when the human mind
specifies a value (temporarily) of 10 for x, in which case Phi would
take on the value Phi(10). If no value is specified for x, x and
Phi(x) are considered to be in a superpositon state of all possible
values (i.e., <x=0, Phi(x)=Phi(0)> & <x=1, Phi(x)=Phi(1)> & .....),
whereas classically, Phi(x) is merely an uninterpreted syntactical
(and finite) entity. The superposition state is to be interpreted as
"The human mind has specified no value for x". Thus in NAFL, Phi(x),
for an unspecified x, is to be viewed as an infinite entity, something
like the class of all ordered pairs {(0,Phi(0)}, (1,Phi(1)), ...}. So
it is a function as far as NAFL is concerned. The bottom line is NAFL
will not permit you to quantify over these formulae.

In NAFL, the notion of provability is not formalizable, precisely
because you cannot quantify over formulae treated as purely
syntactical entities. An NAFL theory is a metamathematical object, not
formalizable as an object within NAFL theories. You may find this
unpalatable, but that is the way NAFL works.

This superposition state is similar to the Schrodinger cat's
superposition state of all possible classically permitted states,
namely, "alive and dead", when the cat's (classical) state is not
accessible (while it is in the box). The cat has a specific classical
state only when the box is opened and the human mind perceives that
state, according to the NAFL interpretation.


(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.

Newberry essentially stated the argument in terms of the fixed point
lemma and your "formulae" are my "functions".


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".

OK, let me think about this a little more and see if I can formulate
an objection from the NAFL point of view.

Regards, RS

.

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