Newberry says...

Let's break it down.
Step 1:
We will say that a sentence is neither true nor false if it and its
negation are not provable or if the proof is infinitely long.
According to this definition G is neither true nor false. The system
is semantically complete because there is no true but unprovable
senetnce. The system is still capable of expressing all p.r. functions
and all r.e. sers.

Do you find any problem with this approach?

Yes. It doesn't make any sense. Whether a sentence is provable
or disprovable depends on what axioms you are assuming. So your
definition of "neither true nor false" doesn't make any sense
until you say what your axioms are.

--
Daryl McCullough
Ithaca, NY

Newberry says...

Let's break it down.
Step 1:
We will say that a sentence is neither true nor false if it and its
negation are not provable or if the proof is infinitely long.

So you are defining truth in terms of provability? Then it is
indeed a trivial consequence that everything true is provable.

However, if your theory T is consistent (and its theorems
are r.e., and it is powerful enough to define all primitive
recursive functions, etc.) then there is a formula Phi
such that

T does not prove Phi

but that fact is not provable in T. I consider that
fact to be meaningful, whether or not T can prove it.

According to this definition G is neither true nor false. The system
is semantically complete because there is no true but unprovable
sentence. The system is still capable of expressing all p.r. functions
and all r.e. sers.

Do you find any problem with this approach?

It's not an approach at all. It's just sticking your head in the
sand. I can just as easily prove that I'm all-powerful:

Definition: A person is all-powerful if he can do
anything that is worth doing.

Definition: We say that a task is "worth doing" if
I am capable of doing it.

Conclusion: I am all-powerful.

I can similarly prove that I am all-knowing.

--
Daryl McCullough
Ithaca, NY

Daryl McCullough says...



Newberry says...

On Apr 30, 9:26=A0am, stevendaryl3...@yahoo.com (Daryl McCullough)
wrote:

Yes. It doesn't make any sense. Whether a sentence is provable
or disprovable depends on what axioms you are assuming. So your
definition of "neither true nor false" doesn't make any sense
until you say what your axioms are.

We will get to that in a minute, but have we lost any p.r. functions
thus far?

Let me just say further how much I object to defining truth as provability.. With
that definition of "truth", why not just take an *arbitrary* collection of
axioms, and use that as your theory. You can make it complete by adding the
inference rule: "If you'd like to conclude Phi, then go ahead". Then you can
make it consistent by adding an axiom saying that this very theory is
consistent. There it is, the ultimate mathematical theory, the most powerful
theory imaginable, and it's *provably* consistent (you can prove it in this very
theory).

However, the reason that people originally cared about mathematics was because
of applications. They wanted to be able to use mathematics to answer such
questions as: If I have a room that is 32 meters by 76 meters, how many square
meters of carpeting do I need? Or: If a rock falls so that each second its
velocity increases by 9.8 meters per second, then how far does it fall in 10
seconds?

You'd like to be able to have a process for translating such questions into your
theory and get a "correct" answer. Saying that by definition, the answer is
correct if and only if it is derivable by your theory isn't very comforting.

You yourself have pointed out that truth involves correspondence with reality.

If G says that it is not provable in theory T, and G is actually *not* provable
in theory T, then G is true.

the whole point of talking about truth in the first place. (Yes, it would be
*nice* if everything true were provable, but it's kind of pointless to say that
that follows by definition of "true".)

On May 1, 6:37 am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote:

However, the reason that people originally cared about mathematics was because
of applications. They wanted to be able to use mathematics to answer such
questions as: If I have a room that is 32 meters by 76 meters, how many square
meters of carpeting do I need? Or: If a rock falls so that each second its
velocity increases by 9.8 meters per second, then how far does it fall in 10
seconds?

Any suggestion what to use |R| > |N| for?

On Apr 30, 9:09=A0am, stevendaryl3...@yahoo.com (Daryl McCullough)
wrote:

So you are defining truth in terms of provability? Then it is
indeed a trivial consequence that everything true is provable.

However, if your theory T is consistent (and its theorems
are r.e., and it is powerful enough to define all primitive
recursive functions, etc.) then there is a formula Phi
such that

T does not prove Phi

but that fact is not provable in T. I consider that
fact to be meaningful, whether or not T can prove it.

~(Ex)Pxm, where m = #~(Ex)(Px & Qxy)
is provable in T,

On Apr 30, 9:26=A0am, stevendaryl3...@yahoo.com (Daryl McCullough)
wrote:

Yes. It doesn't make any sense. Whether a sentence is provable
or disprovable depends on what axioms you are assuming. So your
definition of "neither true nor false" doesn't make any sense
until you say what your axioms are.

We will get to that in a minute, but have we lost any p.r. functions
thus far?

Newberry says...

On Apr 30, 9:26=A0am, stevendaryl3...@yahoo.com (Daryl McCullough)
wrote:

Yes. It doesn't make any sense. Whether a sentence is provable
or disprovable depends on what axioms you are assuming. So your
definition of "neither true nor false" doesn't make any sense
until you say what your axioms are.

We will get to that in a minute, but have we lost any p.r. functions
thus far?

Does everybody agree that

"All panduks are green"

is meaningless?

Any suggestion what to use |R| > |N| for?

If G says that it is not provable in theory T, and G is actually *not*
provable in theory T, then G is true.

But G does not say that it is provable.

It does not say anything. It is "vacuously true"
and hence meaningless.

Newberry wrote:

Does everybody agree that

    "All panduks are green"

is meaningless?

I think it will turn out to be meaningful.  You are seeking to make some
point.  If you convey meaning then the constituents of what you write
are meaningful.

you are conveying meaning.

It may be that my reply relies on a confusion of use and mention.

--
Remove "antispam" and ".invalid" for e-mail address.

Newberry says...

Any suggestion what to use |R| > |N| for?

Most of calculus and probability theory would be inconsistent
if this were not the case.

If G says that it is not provable in theory T, and G is actually *not*
provable in theory T, then G is true.

But G does not say that it is provable.

Not for T, since your theory T is a meaningless theory.
But if, instead of T, we have a theory such as PA,
then we can see that G is true as a statement about
the natural numbers if and only if G is not provable
in PA.

It does not say anything. It is "vacuously true"
and hence meaningless.

Maybe G for your theory T is meaningless, but the
corresponding G for PA is perfectly meaningful.

If your G is meaningless, its because your theory
T is meaningless.

--
Daryl McCullough
Ithaca, NY