XPost: sci.logic, comp.lang.prolog
On 11/28/2022 3:46 AM, Julio Di Egidio wrote:
On Monday, 28 November 2022 at 01:20:41 UTC+1, olcott wrote:
"Gödel sentence in the 1931 incompleteness proof
is not a truth bearer thus simply untrue"
Prove it... Moron.
Corollary: model theory my ass.
Julio
Prolog detects [and rejects] pathological self reference in the Gödel
sentence
https://www.researchgate.net/publication/350789898_Prolog_detects_and_rejects_pathological_self_reference_in_the_Godel_sentence
I already proved my point to everyone knowing Prolog. I reiterated this
same point in Minimal Type Theory in my above paper.
*Here is how Wittgenstein ties to model theory*
<Wittgenstein>
8. I imagine someone asking my advice; he says: "I have constructed a proposition (I will use 'P' to designate it) in Russell's symbolism, and
by means of certain definitions and transformations it can be so
interpreted that it says: 'P is not provable in Russell's system'. </Wittgenstein>
The following says that:
there exists an φ such that φ is neither provable nor refutable in T:
The conventional definition of incompleteness:
Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))
When we see that the following Prolog expressions satisfy the above
definition of incompleteness then we can see that they are equivalent to
the Gödel sentence in the 1931 incompleteness proof.
?- G = not(provable(F, G)). % G = ¬(F ⊢ G)
?- G = not(provable(F, not(G))). % G = ¬(F ⊢ ¬G)
When we test the above pair of expressions we find that neither of them
are provable in the Prolog formal system: (SWI-Prolog (threaded, 64
bits, version 7.6.4)
?- unify_with_occurs_check(G, not(provable(F, G))).false.
?- unify_with_occurs_check(G, not(provable(F, not(G)))).false.
Thus fulfilling the conventional definition of incompleteness, and
proving equivalence to the 1931 Gödel “Incompleteness” sentence. The
1931 Gödel Incompleteness theorem correctly concludes that neither G nor
¬G are provable in F.
The key detail that it leaves out is that neither G nor ¬G are provable
in F because both are erroneous cyclic terms that cannot be resolved in
any formal system what-so-ever.
Gödel analyzed within Wittgenstein's controversial formalization of true
and false: (thus also defeating the Tarski Undefinability theorem)
'True in Russell's system' means, as was said: proved in Russell's
system; and
'false in Russell's system' means: the opposite has been proved in
Russell's system.
Then the above minimal essence of Gödel's logic sentence: φ is construed
as *neither true nor false* thus (*like the liar paradox*) simply not a
truth bearer. Gödel would construe this same case as Incomplete(T)
--
Copyright 2022 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
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