Re: Godel cant tell us what makes a mathematical statement true



Nam Nguyen wrote:
herbzet wrote:
Nam Nguyen wrote:

So, unlike what Baudouin said, whether 0=1, or 0=/=1,
is true does rely on some notion of truth!

What you've shown is that whether 0=1 (or 0=/=1) is true relies
on the structure in which it is interpreted. This, in itself,
relies on a model theoretic notion of truth.

That's right.

We're in simple agreement for once. :-)


There's no such thing
as a formula being true "just because it is true", and not relying
on a notion of truth.

A sympathetic reading of M. Le Charlier's assertion would be
that the correspondence of a statement to reality, or the
satisfaction of an interpreted formula in a structure, is
not a matter of opinion, or of proof, or of our knowledge
of such correspondence or satisfaction.

I think I understand what was stated; it's just I disagree
with the alleged *absolute* nature of it!

I didn't say that you didn't understand what was stated.

Of course you are right: to assert of any statement that
it "is true" or "is false" is to already have some notion
of what it means for a statement to "be true" or to "be false".

To think that something is true only because
it has been proven is the wrongest idea you can conceive.

Suppose for a given formal system T, we define true sentences
as the following:

- T(F) = true iff T |- F.
- T(F) = false otherwise.

What would you think as "wrong" with this definition?

Well, if T |/- F and T |/- ~F then both F and ~F are false.
Is that problematic for you?

Of course not. It's *not a perfect definition* but:

a) If a formula is not defined to be true w.r.t. T, then there's
nothing wrong to equate it with being false. In this way then:

- A consistent system is one in which one of {F, ~F} is true and
the other false.
- An inconsistent system is one in which both F and ~F are true.
- F is undecidable in T if both F and ~F are false (which only means
F and ~F aren't provable in T!)

If both F and ~F are false (which only means F and ~F aren't provable
in T) then F is undecidable in T (by your third clause) and T is
not a consistent system (by your first clause), since it will not
be the case that one of {F, ~F} is true (provable) and the other
false (not provable).

OK, so I typed it too quickly. What is meant here should have been typed:

- A consistent system is one in which one of {F, ~F} is true and
the other false, for an F.
- An inconsistent system is one in which both F and ~F are true,
for any F.
- F is undecidable in T if both F and ~F are false (which only means
F and ~F aren't provable in T!), for an F.

So we have that a system that is is not consistent is not necessarily
the same as a system that is inconsistent (by your second clause):
F and ~F are both true (provable).

I don't find this terminology quite satisfactory.

With the revised above, I hope you'd be convinced to change your mind.

It's pretty standard as amended, other than the fact that you
are identifying truth with provability.

Still, we have the situation that if T |/- F and T |/- ~F then
F and ~F are both false. This doesn't distinguish refuted
formulae (the negations of proven formulae) from undecidable
formulae.

Also, I think you have to think about the difference between the
concept of a formula F being true in a theory T (being true in
every model of T) and the the anterior notion of a formula F being
true in a given model (structure).

Nothing would seem "wrong" at all!

b) The standard definition of truth is *not perfect either*: there
are some T in which if T is consistent, there would be some formula
F which you can't tell whether or not F is true in any model
of T!

In the standard definition of truth, a formula F that is undecidable
in a theory T is true in some model of T. If F is false in every
model of T, then F is decidable in T: it is the negation of a
formula provable in T (F is refuted in T). I don't see that
as an imperfection.

You seem to wish to assert that the truth of a formula F relies
on our knowledge (whether we can "tell") of its truth in some model.

That's what I'd like to convey. The truth of a formula F is relative
to whatever definition of truth one is please. And in my case, I've
"relativized" it to that of syntactical proof.

What if a formula F has a proof but you don't know that it has
a proof? Is F false until you discover the proof? Or was it
true all along?

Commonly (but not universally) it is taken that the truth or
falsehood of a statement is not dependent on its epistemological
status of being known to be true or not. This is part of the
realist (Platonist) position that there is an objective reality
that is not dependent on our minds, or what we happen to know.

But the moment one is able to demonstrate logically that the formula
could assume the opposite truth value, this common would be on a shaky
ground!

I didn't say "formula", I said "statement". Of course a formula
can assume different truth values depending on the structure in
which you interpret it. This doesn't mean that you automatically
know its truth value in that structure.

Whether or to what degree mathematical reality partakes of this
objective status is, of course, a venerable debate.

(You have provoked in me some thought about how far the model
theoretic notion of truth is implicitly a realist position,
or not.)

All I know is if mathematical reasoning is about (or dependent on)
knowledge, then the reasoning can't go too far: either by means
of provability - or truth!

I don't know what this means.

But the point here is a formula can't be just simply true. You have to
*choose* *some* selected/defined notion of truth!

True. But once having chosen some notion of what it means
for a statement to "be true" or to "be false", then it
is no longer a matter of opinion as to whether a statement
is true or false -- it is only a question of whether the
statement is in accord with the notion adopted.

Of course, some notions of what "truth" is will be sillier than others.

You seem to have two issues here: that a formula must be interpreted
to have any meaning at all (much less be true or false),

No! Formula's semantic and truth don't have to be identical.

That's not what I said. An uninterpreted formula has no meaning
or truth value. Statements have truth values.

For example,
GC has some meaning but whether or not it's true or false (by whatever the
chosen underlying truth definition) is a different matter.

Well maybe you have a point here: Assume the standard semantics
for arithmetic, that is, assume, e.g., that Euclid's theorem
on the infinitude of prime numbers means what we usually take
it to mean. How do you vary its truth-value within the usual
semantics? I can weaken PA to where it can't prove Euclid's
theorem, but the statement of the theorem will still be true
in the standard model of PA, no?

and that a formula has to be known to be true (by some criteria of truth)
in order for it to be true (by that criteria).

That's I think is meant by something like "mathematical truth is relative",
and is what I'd like to convey. But that in itself is not an "issue", imho.

You wish to say that mathematical truth of a statement is
relative not only to the chosen criteria of truth, but
also to whether we know the statement meets the criteria?

IOW, a statement is false if it fails to meet the chosen
truth definition, but also if we don't know whether it meets
the truth definition or not?

--
hz
.

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