Re: Goldbach Conjecture & the Foundation of First Order Logic.


Nam Nguyen wrote:

Let me rephrase it differently. If we talk about *theorems* of a T, we
*don't need* to talk about semantics, but the semantics is already
there: provided of course T is *consistent*. The *only time* we aren't
- and couldn't be - talking about semantics is when T is *inconsistent*.
One would of course remember that every formulae "follows from the
non-logical axioms by the first-order deductive system built into" an
inconsistent T. But then good FOL reasoning is not supposed to be
based on an inconsistent formal system. So if we care about truth,
then semantics; model; and interpretation could not be disregarded.

Of course we are interested in interpreted formalized theories. But so
what? The point remains that when we ask when a particular theorem
follows within a particular axiomatized theory, the issue of semantics
doesn't arise. We are asking whether the target theorem can or can't be
reached by deductive moves allowed by the syntactically described
proof-system from the axioms of the theory.

I think we all know what it means by Shoenfield's "A first-order
language is thus completely determined by its nonlogical symbols".
Therefore the 2 isomorphic languages

L'(0, + , <=) and
L''(0, U,included_in)

are different. Therefore the 2 isomorphic theories:

T' = {(Axy(x + y = y + x) /\ Ax(0 <= x))} and
T'' = {(Axy(x U y = y U x) /\ Ax(0 included_in x))}

are also different: and we know so because the lone axiom in T'
can be deduced from T' but not from T'', and vice versa.

Fair enough, thanks for the clarification. I'd wondered after I posted
if that's what you did mean, but I didn't follow it up because it
obviously doesn't help your argument. For suppose we go along with you,
and now identity Peano arithmetic with, not one particular formal
system but with a class of "isomorphic systems" got by re-lettering the
alphabet. Then the question whether a certain PA claim is a PA theorem
(in your sense) is still equivalent to the question whether, within
any one system in the family, the version of the claim follows by the
rules of that system from the axioms of that particular system -- for,
simply because the systems are isomorphic, what goes for one system in
the family goes for all. Whether you think of formal PA as one
particular exemplar system or a family of isomorphic systems makes no
difference at all to the point at issue re the possible provability or
otherwise of Goldbach's conjecture.

and you cited a definition of truths in PA:

<stuff cut>

I wonder then what would be the truth values of ~GC, and GC?

If GOLDSENT is the PA wff that expresses Goldbach's conjecture, then
the truth-definition delivers e.g.

|~GOLDSENT| is true iff it is not the case that every number greater
than four is the sum of two primes.

No problem there, then! Remember a definition of truth in the relevant
sense only delivers biconditionals. It doesn't tell us whether
|GOLDSENT| is actually true, unless we already know whether every even
number greater than four is the sum of two primes.

But
that's not the point. The point is since there are uncountably many
successor functions - each would correspond to one model out of
uncountable number of models (in the very same domain!) - we can't
assert in meta level that Val(|Sa|) = Val(|a|) + 1.

No. That's just confusing truth with truth-in-a-model. We've fixed on
an interpretation of the symbols of PA: in particular, "S" means "add
one".

The point becomes clear if you think about doing the logic of natural
language. We say e.g. "Socrates is wise and Plato isn't" logically
entails "Socrates is wise" because, vary the interpretation of
"Socrates is wise" and "Plato isn't" as you like, in the premiss is
true, so is the conclusion. But the fact that we do consider various
interpretation when we are thinking about the logical relations of
those sentences, of course doesn't imply that "Socrates is wise"
doesn't really mean that Socrates is wise. It is just the same with
interpreted formal languages. "S" in PA means successor (by our
stipulation). The fact that we consider different interpretations when
we are thinking about logical relations, etc. etc, doesn't impoly that
"S" doesn't really mean "add one"!

.

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