On 16 Oct 2011, at 06:48, Stephen P. King wrote:
Hi,
I ran across the following:
http://en.wikipedia.org/wiki/Tarski%27s_indefinability_theorem
"Tarski's undefinability theorem, stated and proved by Alfred Tarski
in 1936, is an important limitative result in mathematical logic,
the foundations of mathematics, and in formal semantics. Informally,
the theorem states that arithmetical truth cannot be defined in
arithmetic."
Where then is it defined?
You can defined arithmetical truth in second order arithmetic; or in
set theory. I have often mentioned Tarski non definability of truth to
explain that a machine cannot define its own knowledge notion,
including in sane04 and almost all the papers of mine that I cite. We
cannot define Kp by Bp & True(p) because the machine cannot define, in
its own language "True(p)". By the diagonalization lemma we would be
able to construct Epimenides' paradoxical "I am not true" sentence. It
might be possible to define knowledge (S4 modal operator) in some
other way, but this has been shown impossible to by Kaplan & Montague.
Of course Gödel knew that Bp cannot be a knowledge operator. Amazingly
it acts like a belief operator, confirming that the ideal science of
the correct machine leads only to beliefs (roughly speaking).
But we can define a knowledge operator at the metalevel, and this in
the Theaetetus modus operandi, just by modeling the truth of an
arithmetical proposition p by itself. This is the origin of the Bp & p
hypostase. It works fine and lead to the S4Grz logic.
Also, I have often mentioned that, although a machine cannot compute
nor even defined all its theology (the 8 hypostases), a rich Löbian
machine, like ZF can define and study the whole theology of a simpler
Löbian machine. Then, the rich LUM can lift that theology on itself,
by betting on its own correctness, but at his own risk and peril: such
an operation should not lead the machine to use its correctness as a
brute fact (axiom), as this would make the machine unsound and
inconsistant.
I sometimes, when I want to be short, refer to a machine theology by
describing it by the label "Tarski minus Gödel", that is the truth on
the machine minus what the machine can prove. Incompleteness means
that it is quite different.
Note that Löbian machines can still prove a Truth operator for each of
all sentences with a bounded number of quantifier, and can approximate
truth in many other ways, which I will not detail here. Tarski theorem
is a fundamental component in the unravelling of the machine's
theology. See my paper on Plotinus. The reason why the "p" hypostase,
and the "Bp & p" hypostase play the role of the "ONE" and the "SOUL"
respectively, is that the introspecting machine can discover them
despite not being able to give them a name, and so it matches the main
defining attribute used for them by the neoplatonist inquirers.
Bruno
Onward!
Stephen
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