On 23 Mar 2013, at 04:46, Stephen P. King wrote:
"In 1936 Tarski proved a fundamental theorem of logic: the
undefinability of truth. Roughly speaking, this says there's no
consistent way to extend arithmetic so that it talks about 'truth'
for statements about arithmetic.
That's not entirely correct. ZF is a consistent extension of PA, and
you can defined arithmetical truth in ZF. But you cannot defined "set
theoretical truth" in ZF.
The theorem says only that cannot defined arithmetical truth (or
Löbian truth) in arithmetic (or by the Löbian machine is the truth
encompass all what the machine can say).
Why not? Because if we could, we could cook up a statement that
says "I am not true." This would lead to a contradiction, the Liar
Paradox: if this sentence is true then it's not, and if it's not
then it is.
This is why the concept of 'truth' plays a limited role in most
modern work on logic... surprising as that might seem to novices! ..."
That's why Gödel avoided completely any reference to truth or
semantics in his proof of incompleteness. Today that avoidance of
truth is less important than in the beginning, as the notion of truth
and semantics are better circumscribed.
Note that Gödel found Tarski theorem too.
Machine's can prove their own Gödel's theorems, and there is some
sense in which they can proof their own Tarski theorem. This can be
used to refute all use of such incompleteness or undefinability
results for refuting mechanism, like Lucas and Penrose thought they did.
Tarski's theorem has been extended by Kaplan and Montague for the
undefinability of knowledge, making the Bp & p undefinable in the same
manner. This plyas a key role in the machine's knowledge theory.
Bruno
https://plus.google.com/u/0/117663015413546257905/posts/jJModdTJ2R3https://plus.google.com/u/0/117663015413546257905/posts/jJModdTJ2R3
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Onward!
Stephen
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