Re: OUTGOEDELING A HUMAN?

abo says...

On Feb 21, 3:22 pm, stevendaryl3...@xxxxxxxxx (Daryl McCullough)
wrote:
Aatu Koskensilta says...


The same argument applies to any
explication for truth in context of statements such as the liar,
demonstrating that 'truth' is an "indefinitely extensible" concept.

Right. I think that resolves all the semantics paradoxes nicely.


Well it would if one could make sense of "indefinitely extensible"
concept.

The way I understand it is this: Suppose we have any language L
with a mathematically precise semantics. Then we can make sense
of the notion of truth for sentences of L. However, no matter
what language L we started with, "truth for sentences of L"
does not exhaust the concept of truth: there will be true
sentences which are not covered by L.

For a specific example, pick a Goedel coding for all
sentences in first-order logic. Let L be some countable
first-order language extending Peano arithmetic such that
L has a well-defined model. Then we can axiomatize "truth
in L" by introducing a new predicate true_L(x) and for each
sentence Phi in the language of L add the axiom

true_L(#Phi) <-> Phi

where #Phi is the Goedel code of Phi. Then using Godel's
trick, we can come up with a new sentence G such that

G <-> not true_L(#G)

This new sentence G cannot be a sentence in the original
language L. So it is not true_L. But it *is* true in the
extended language (L extended with the predicate true_L).
So we have extended the notion of truth to our new language
(we've come up with a new sentence that is true that was
not covered by the notion of "truth in L").

--
Daryl McCullough
Ithaca, NY

.

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