Re: My investigations into Godels Incompleteness Theorem

Alan Smaill wrote:
"Charlie-Boo" <shymathguy@xxxxxxxxx> writes:
Chris Menzel wrote:

Here is a central point where your ignorance betrays you. Notably,
aside from the previously mentioned fact that the Gödel sentence, unlike
the Liar, is not literally self-referential

"The analogy of this result with Richard's antinomy is immediately
evident; there is also a close relationship with the Liar Paradox, for
the undecidable proposition [R(q);q] says that q belongs to K, i.e.
according to (1), that [R(q);q] is not provable." - Kurt Godel, 1931

Yes, there is an *analogy*.
No, it is *not* *literally* self-reference.

Is there a difference between literal and analogical use
of language, in your view?

I'm not sure what you mean by "literal use" and "analogical
use", but if you are referring to the distinction that is called
"use or mention" elsewhere in this thread, then sure. In fact,
this is just another case of a principle that has never been fully
understood or explained in the literature.

The distinction is really that of the difference between a program and
its output. Here, a string of characters in quotes is the program, and
the string of characters (without quotes added) is the output. In CBL
it is represented as PROGRAM # OUTPUT where output is any (recursive)
function, normally illustrated as the general (using variables)
expression M # f(I).

When CBL is used to formalize the Liar Paradox, we get all sorts of
variations with quote marks to an unlimited depth. We can formalize
statements such as:

It is true. M # ts(I)
This is true. M # ts(M)
This is false. M # fs(M)
"This is true." is true. M # ts(M) + N # ts(M)
"This is true." is false. M # ts(M) + N # fs(M)
"This is true of it." is false of "It is true of this."
M # ts(s11(M,I)) + N # ts(s11(I,N) + O # ts(s11(M,N))
"This is true of 'It is true.' " is false of "It is true of
'This is true.' ".
M # ts(I) + N # ts(s11(N,M)) + O # ts(O) + M4 # ts(s11(I,O)) + M5
# fs(s11(N,M4))

where ts(I) means I is a true sentence, fs(I) means I is a false
sentence, and s11(I,J) is the result of replacing the first pronoun
("it") in I with J. (This is a sibling to Recursion Theory. I
once posted 500 of these generated by my software. Search for "500
Paradoxes".)

You seem to be making a distinction between "This is false." and
the Godel Sentence because of the word "This", no? The Godel
Sentence with provability replaced by truth, and Predicate Calculus
replaced by English, is " 'It is false of it.' is false of 'It
is false of it.' " which is equivalent to "This is false." To
represent "This is false." in Predicate Calculus would require an
addition to Predicate Calculus for "this" - a symbol that
represents the wff in which it appears e.g. @ and "This is false."
would be ~@.

So the distinction to which you refer is only incidental, not inherent.

BTW the premises are:

TS / TS Liar (Truth is expressible in English.)

~PR / TW Godel's 1st. Incompleteness Theorem based on soundness
(Unprovability is expressible in Logic.)

DIS / TW Smullyan's Dual Form Theorem
(Refutability is expressible in Logic.)

DIS / PR Rosser 1936
(Refutability is representable in Logic.)

following Smullyan et al's terminology "expressible" and
"representable". In the case of a paradox the set of premises is
collectively false, or is unmentioned and inconsistently referenced
within the description of the paradox (e.g the unexpected hanging and
the God paradox.)

It's the same process each time using different sets and different
systems of representation. (CBL is the formalization of the common
principle independent of these differences, and by referring to
particular sets and systems of representation we formalize these
various results of Computer Science.)

C-B

--
Alan Smaill

.

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