On 25 Mar 2013, at 11:10, Alberto G. Corona wrote:
I suspect that this impossibility is because math uses concept of a
model, while truth refers to the match of facts of the model with
facts of the reality . Or at least to facts of a metamodel outside
of the model. That is AFAIK the Tarsky idea.
Tarski is with an "i". I change the title of the thread accordingly.
Tarski is talking about mathematical truth, and this is the match with
some model (in the logician's sense).
Logician does not assume a notion of reality, per se, but in applied
logic we can "model" some reality by models (mathematical structure
with a notion of satisfying formal formula).
For example the meaning of "ExP(x)" is that it exists an object m in
the model such that it is the case that P(m)".
Tarski theorem is a consequence of the diagonal lemma. That is: the
fact that for all formula F in the language used by the theory, you
can find a sentence S provably (by the theory) equivalent with F('S')
where ' ' represents some fixed coding of S in the language of the
theory. So if we could defined a predicate defining "truth of
sentence" (and thus non-truth of a sentence) in the theory, we would
find a sentence K provably equivalent with NOT-TRUE (K). But for a
truth predicate we ask that the theory proves, for all sentence X,
that X is equivalent with TRUE('X'), and so any reasonably rich theory
would prove that the K above is both true and false. Like Stephen
said, it is really a form of Epimenides "paradox". Gödel's theorem is
also an easy consequence of the diagonal lemma, with "not-provable" in
place of not-true. Yet "provable" and "not-provable" are definable in
the theory, and this entails that if the machine is consistent (does
not prove the false), the fixed point "K" will be true but not
probable in the theory/by the machine.
All Löbian machines satisfy the diagonalization or diagonal lemma.
They cannot define an all encompassing notion of truth for all the
sentences that they can talk about, but they can still define very
large notions of truth, restricted to very big subset of their
language. Peano Arithmetic can already define a notion of truth for
any sigma_i sentences, for example.
Bruno
For example, "All men are mortal is true" . Here suppose that "all
men are mortal" is a fact of a model that admit inductive silogisms.
True in this case can express a match of the fact "All men are
mortal" with reality. or a match with a metamodel in which the
trueness of the model "all men are mortal" is assumed as an axiom.
In both cases the whole statement "all men are mortal is true" is
outside of the model in which the statement "all men are mortal" is
However I can make an ordinaty mathematical model of matches between
models and metamodels. That is the boolean logic.
2013/3/23 Stephen P. King <stephe...@charter.net>
"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. 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! ..."
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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