On 17 Dec 2012, at 22:02, meekerdb wrote:
On 12/17/2012 11:47 AM, Bruno Marchal wrote:
On 16 Dec 2012, at 20:28, meekerdb wrote:
On 12/16/2012 2:31 AM, Bruno Marchal wrote:
No. With the CTM the ultimate truth is arithmetical truth, and we
cannot really define it (with the CTM). We can approximate it in
less obvious ontologies, like second order logic, set theory,
etc. But with CTM this does not really define it.
Don't confuse truth, and the words pointing to it. Truth is
always beyond words, even the ultimate 3p truth.
What would it mean to 'define truth'? We can define 'true' as a
property of sentence that indicates a fact.
That's the best definition of some useful local truth. But when
doing metaphysics, you have to replace facts by "facts in some
model/reality".
OK. But then it's "True relative to the model." and it's not
necessarily The Truth.
Indeed. But for arithmetic (or Turing equivalent), "The Truth" = true
in the standard model (learned in high school).
But I'm not sure how to conceive of defining mathematical 'true'.
It is the object of model theory. You always need to add more axiom
in a theory to handle its model. You cannot define the notion of
truth-about-set in ZF, but you can define truth-about-set in ZF in
the theory ZF +kappa (existence of inaccessible cardinals).
PA can define all the notion of truth for the formula with a
bounded restriction of the quantification.
So what is that definition?
It is long and has to be defined by induction on the complexity of
formula. Like "ExP(x)" is true if it exists a n such that P(n), etc.
Does it just mean consistent with a set of axioms,
No. That means only having a model. true in some reality. But for
arithmetic "true" means satisfied by the usual structure (N, +, *).
i.e. not provably false?
How is not provably false different from 'satisfied by the usual
structure'? Can you give an example?
Well the most famous example is "provable "0=1". This is not provably
false (as PA cannot prove ~Bf), but is false in the standard model.
That just consistent.
I would think it was incompleteness. Consistency means not being
able to prove every proposition.
or ~Bf
But in a consistent system there can be propositions that are
neither provable nor disprovable. Are those true?
Some are, some are not. Bf is not provable and false. Dt is not
provable and true. All arithmetical interpretation of any formula of
G* minus G are true but not provable. Their negations are false and
not provable.
Bruno
Brent
True entails consistency, but consistency does not entail truth.
Bruno
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