On 12/18/2018 6:20 AM, Bruno Marchal wrote:
By definition; soundness means that it reflect reality.
You're now messing with words. What does "reflect reality" mean? It
looks like an appeal to correspondence theory to truth. But that
means to know a theory is sound you need to know what is true.
There are two definition of soundness.
Arithmetical soundness: it means that the theorem are true in (N, 0, +, *)
Exemple if the theory proves ExEyEx(x^3 + y^3 + z^3), if the theory is
sound, it means that there are (standard) natural numbers n, m, r such
that their sum of cube gives 33.
Soundness (in general): it means that what is proved in the theory is
true in all models/intepretations of the theory.
Completeness is the reverse of that one; a theory is complete if what
is true in all models is provable.
That's ambiguous. Do you mean a theory is complete if whatever is true
in some model is provable, or do you mean that whatever is true in every
model is provable? Since there are in general arbitrarily many models,
is completeness only knowable by a proof in a metatheory?
Not to confuse with Arithmetical Completeness: a theory is
arithmetically complete if it proves all truth in (N, +, *).
PA is a complete theory, like all first order theories (Gödel 1930)
PA is arithmetically incomplete (Gödel 1931).
You can prove that PA is sound using a bit of set induction, like in
Analysis. Humans tend to trust Analysis, which assumes much more than
arithmetic.
Sound just means its theorems are tautologies, i.e. they are valid
inferences from the axioms.
You might makes number theorists a bit nervous if you say that Fermat
is a tautology. We use just “theorem”, and use only tautology for
classical propositional logic.
But they are conceptually the same.
Brent
Sound means true in all models (arithmetically sound means true in the
standard model (N, +, *).
Soundness implies consistency. But consistency does not imply
soundness. The robot describing the Venus of Milo in front of
another sculpture is consistent, but unsound.
All the machines I am talking about are supposed to be
arithmetically sound.
Meaning that what they "believe" are theorems.
No, meaning that the theorem/beliefs are true in the structure (N, +,
*). It means that if PA would prove ExEyEx(x^3 + y^3 + z^3), then
there are really numbers n, m, r, having their cube added to 33. With
PA + []f, you might be able to prove ExEyEx(x^3 + y^3 + z^3), but
still not know if such standard n, m, and r exist, because x, y and z
could be non standard natural numbers. PA + []f is typically unsound.
It asserts that there is proof of the false, but can prove that it not
0, nor 1, nor 2, nor 3, nor 4, etc. It is consistent, but not
omega-consistent.
Not that they "believe" everything that is true. In fact you have
proven anything about what a person might or might not believe. Your
ideal machines do not include any concept of acting on "beliefs"
which is the real test of beliefs.
I have no clue why this could be true, except contingently, as I use
all this to derive physics from arithmetic, not for doing machine
acting in our local neighbourhood.
Bruno
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