On Mar 25, 9:26*am, (Daryl McCullough)
wrote:
says...

Perhaps you would like to give some examples of Gs that are provably
unprovable that don't involve self-referentiality?

Godel described how to construct a sentence G and gave
an argument that this G is not provable (under the assumption
that the axioms are all consistent). His proof is completely
*mechanical* in that given any proof of G, we can convert
that into a proof of a contradiction.

As to the nature of the sentence G, it is just an ordinary
statement of arithmetic. G is completely written in the
language of the axioms I gave: It only uses +, *, 0, 1, =,
together with logical operators. It doesn't directly involve
self-reference at all.

It is possible to *interpret* G as "talking about itself",
but G is perfectly meaningful as an ordinary statement of
arithmetic, as well. It can be cast into the form of
a claim that a certain polynomial equation with integer
coefficients has no integer solutions.

Banishing self-reference in mathematical proofs doesn't
make G become provable. It just makes it more difficult
for you to *see* that it is unprovable. It is like putting
a blindfold on to keep from seeing something you don't
want to see.

--
Daryl McCullough
Ithaca, NY

Daryl,

Godel's example of a non-provable is self referential. That was my
point. Without the self referentiality (expressing statements about
arithmetic as arithmetic) there is no proof by Godel.

Edgar

On Mar 24, 3:53*pm, Tom Roberts wrote:
Michael Helland wrote:
Newton's law of gravity didn't last forever.
If GR gets over turned in the next century, does that mean it wasn't a
law of nature?

No modern scientist discusses "Laws of Nature", because we understand
that Nature is inscrutable to humans. What we do is construct MODELS of
the world. If GR gets overturned in the next century, that will occur
because a better MODEL is discovered, with presumably a wider domain of
applicability. That will not affect GR's applicability within its domain.

There are Laws, Truth, out there, but we don't know them.

Doubtful, HIGHLY doubtful. There is Nature out there, but there's no
requirement whatsoever that She behave as you want Her to behave (by
following "Laws" or implementing some "Truth").

What we know are hypotheses and theories, which were invented.

Sure. They are MODELS OF THE WORLD, not "Laws", not "Truth".

Tom Roberts

Tom,

Nonsense. There are many scientists that discuss the laws of nature.
Of course they are not the same as human approximations to those laws.
That is precisely the point.

Edgar

says...

Godel's example of a non-provable is self referential.

No, it's not. It's an ordinary statement of arithmetic.
It's a statement of the form: "Such and such polynomial
equation has no integer solutions". The *proof* that it
is unprovable uses self-reference, but the statement itself
is just arithmetic.

If you banish self-referential arguments, then you lose
the ability to prove that G is unprovable, but that doesn't
make it any more provable.

--
Daryl McCullough
Ithaca, NY

On Mar 25, 11:24 am, Tom Roberts wrote:
Michael Helland wrote:
On Mar 24, 12:53 pm, Tom Roberts wrote:
Sure. They are MODELS OF THE WORLD, not "Laws", not "Truth".

I agree.
But does that mean there is no truth in science?

We humans have no hope of every knowing any sort of "ultimate truth". We
are humans, not Gods.


Sure.


This is just a minor point, but if you define truth as:
"Absolutely and perfectly true"
Then the models are not "true"

Models are either valid or not valid; "true" has nothing to do with it.
The concept of validity includes a domain of applicability, which varies
for each theory (model).


Only if you define "true" as "ultimate truth".

Thanks to Karl Popper, in science, truth is a conjecture that has not
been refuted, or a hypothesis that hasn't been falsified.