Godel can use proof to prove his incompleteness of proof without a
problem since his conclusion is not that any particular proof, or
proof itself, is invalid, only that some valid propositions cannot be
proven.

What needs to be understood is to understand the precise hidden
assumptions that Godel's incompleteness are based upon. I believe
these include infinities and recursion. When these are removed then I
believe we can have logically complete systems. My premise is that
there is an actual physical mathematics in which this is true (laws of
nature) and that human mathematics is an abstract approximation to the
actual physical mathematics. The human version makes a number of
unwarranted extensions into infinities based on a view that nature is
continuous. Eg. that there are an infinite number of real numbers
between any two integers. If nature is discrete, not continuous then
this view fails and Godel's incompleteness no longer applies. We then
have recovered the possibility of completeness.

Thus we need to formulate what this actual physical mathematics might
entail. It seems to me that what we get is a fuzziness at extreme
scales, and the notion that the number system must be scaled somehow
to physical reality.

It also may be that this fuzziness of physical math at the quantum
scale might itself be capable of explaining certain 'fuzzy' aspects of
the quantum world as mathematical rather than physical. For example
uncertainty might understandable as an imprecision in a physically
scaled number system rather than as a strictly physical quantum
characteristic.

Edgar

says...

Godel can use proof to prove his incompleteness of proof without a
problem since his conclusion is not that any particular proof, or
proof itself, is invalid, only that some valid propositions cannot be
proven.

What needs to be understood is to understand the precise hidden
assumptions that Godel's incompleteness are based upon. I believe
these include infinities and recursion.

Not really. Godel's theorem applies to the following
very simple axiom system:

1. forall naturals x: x+1 is not equal to 0
2. forall naturals x: if x is not equal to 0, then there is a
natural y such that y+1 = x
3. forall naturals x and y: if x+1 = y+1, then x=y
4. forall naturals x: x+0 = x
5. forall naturals x and y: x+(y+1) = (x+y)+1
6. forall naturals x: x*0 = 0
7. forall naturals x and y: x*(y+1) = (x*y)+x

Which axiom do you think involves infinity or recursion?

--
Daryl McCullough
Ithaca, NY

wrote:
What needs to be understood is to understand the precise hidden
assumptions that Godel's incompleteness are based upon. I believe
these include infinities and recursion. When these are removed then I
believe we can have logically complete systems.

Well known -- a FINITE system is not "interesting" enough for Goedel's
theorem to apply. Recursion is irrelevant AFAIK.


My premise is that
there is an actual physical mathematics in which this is true (laws of
nature)

You seem to be confusing mathematics and physics.


The human version makes a number of
unwarranted extensions into infinities based on a view that nature is
continuous. Eg. that there are an infinite number of real numbers
between any two integers. If nature is discrete, not continuous then
this view fails

You also confuse "reality" with MODELS of the world we inhabit. The
models we use are mathematical, and usually include real numbers. Nature
quite clearly does not use any kind of numbers.

Also: Nature _IS_ discrete, in that no continuous structures have ever
been observed. But our MODELS of the world include a spacetime manifold
which is continuous, and on which spatial intervals act as if they are
continuous down to thousands of times smaller than nuclei. Basically,
the model is that discrete objects move on a continuous underlying
spacetime manifold.


BTW there is no hope of physics ever being "complete" (for any
reasonable meaning of the word). This is completely independent of
Goedel's theorem, which is mathematics and does not apply to physics.


Tom Roberts

On Mar 21, 12:40*pm, (Daryl McCullough)
wrote:
says...



Godel can use proof to prove his incompleteness of proof without a
problem since his conclusion is not that any particular proof, or
proof itself, is invalid, only that some valid propositions cannot be
proven.

What needs to be understood is to understand the precise hidden
assumptions that Godel's incompleteness are based upon. I believe
these include infinities and recursion.

Not really. Godel's theorem applies to the following
very simple axiom system:

1. forall naturals x: x+1 is not equal to 0
2. forall naturals x: if x is not equal to 0, then there is a
natural y such that y+1 = x
3. forall naturals x and y: if x+1 = y+1, then x=y
4. forall naturals x: x+0 = x
5. forall naturals x and y: x+(y+1) = (x+y)+1
6. forall naturals x: x*0 = 0
7. forall naturals x and y: x*(y+1) = (x*y)+x

Which axiom do you think involves infinity or recursion?

--
Daryl McCullough
Ithaca, NY

Daryl,

It is not the axioms of arithmetic that involve recursion but the
methodology he uses in his proof which is recursive since it requires
the representation of every formula of an arithmetic as (Godel)
numbers in that arithmetic.

Edgar

On Mar 21, 1:12*pm, Tom Roberts wrote:
wrote:
What needs to be understood is to understand the precise hidden
assumptions that Godel's incompleteness are based upon. I believe
these include infinities and recursion. When these are removed then I
believe we can have logically complete systems.

Well known -- a FINITE system is not "interesting" enough for Goedel's
theorem to apply. Recursion is irrelevant AFAIK.

My premise is that
there is an actual physical mathematics in which this is true (laws of
nature)

You seem to be confusing mathematics and physics.

The human version makes a number of
unwarranted extensions into infinities based on a view that nature is
continuous. Eg. that there are an infinite number of real numbers
between any two integers. If nature is discrete, not continuous then
this view fails

You also confuse "reality" with MODELS of the world we inhabit. The
models we use are mathematical, and usually include real numbers. Nature
quite clearly does not use any kind of numbers.

Also: Nature _IS_ discrete, in that no continuous structures have ever
been observed. But our MODELS of the world include a spacetime manifold
which is continuous, and on which spatial intervals act as if they are
continuous down to thousands of times smaller than nuclei. Basically,
the model is that discrete objects move on a continuous underlying
spacetime manifold.

BTW there is no hope of physics ever being "complete" (for any
reasonable meaning of the word). This is completely independent of
Goedel's theorem, which is mathematics and does not apply to physics.

Tom Roberts

Tom,

I believe the laws of nature including number are in fact part of
nature and that human world views, mathematics and physics are
approximations of those rules in human terms. If the laws of nature
were not part of nature then how would nature know to produce which
result in any event? The rules have to be part of process itself.
Think of nature as computational. The rules as well as what they
operate on must be present in the computer itself for anything to
happen.

Edgar

On Mar 21, 2:58*pm, Sam Wormley wrote:
wrote:
I believe the laws of nature including number are in fact part of
nature and that human world views, mathematics and physics are
approximations of those rules in human terms. If the laws of nature
were not part of nature then how would nature know to produce which
result in any event? The rules have to be part of process itself.
Think of nature as computational. The rules as well as what they
operate on must be present in the computer itself for anything to
happen.

Edgar

* *Nature didn't need humings to exist... We have devised models
* *that are good approximations making predictions about how nature
* *behaves or probabilities about natures behavior. This idea of rules
* *and laws is a human construct.

* *Edgar... to say that "nature as computational" is likely misleading
* *you.

Sam,

Then how does nature know what results to produce for each event in a
consistent manner if it is not embodying some rules that determine the
results? Nature obviously follows its own rules. Things don't just
happen, they happen according to rules. Nature has been doing this
since the universe began. These events happen in a consistent way
according to the laws of nature. Therefore humans could not have
invented the laws of nature, they must be part of nature itself.
Humans discover the these rules the best they can.

Edgar

says...

On Mar 21, 12:40=A0pm, (Daryl McCullough)
wrote:
says...



Godel can use proof to prove his incompleteness of proof without a
problem since his conclusion is not that any particular proof, or
proof itself, is invalid, only that some valid propositions cannot be
proven.

What needs to be understood is to understand the precise hidden
assumptions that Godel's incompleteness are based upon. I believe
these include infinities and recursion.

Not really. Godel's theorem applies to the following
very simple axiom system:

1. forall naturals x: x+1 is not equal to 0
2. forall naturals x: if x is not equal to 0, then there is a
natural y such that y+1 = x
3. forall naturals x and y: if x+1 = y+1, then x=3Dy
4. forall naturals x: x+0 =3D x
5. forall naturals x and y: x+(y+1) = (x+y)+1
6. forall naturals x: x*0 =3D 0
7. forall naturals x and y: x*(y+1) = (x*y)+x

Which axiom do you think involves infinity or recursion?

It is not the axioms of arithmetic that involve recursion but the
methodology he uses in his proof which is recursive since it requires
the representation of every formula of an arithmetic as (Godel)
numbers in that arithmetic.

What difference does it make what "methodology" he used? The
point is that, given the above axiom system, one can construct
a sentence G such that G is not provable from the above axioms.
Do you dispute that? You think that there is no such sentence G?
Or you think that G is actually provable from the above axioms?

--
Daryl McCullough
Ithaca, NY

On Mar 21, 5:51*pm, (Daryl McCullough)
wrote:
says...





On Mar 21, 12:40=A0pm, (Daryl McCullough)
wrote:
says...

Godel can use proof to prove his incompleteness of proof without a
problem since his conclusion is not that any particular proof, or
proof itself, is invalid, only that some valid propositions cannot be
proven.

What needs to be understood is to understand the precise hidden
assumptions that Godel's incompleteness are based upon. I believe
these include infinities and recursion.

Not really. Godel's theorem applies to the following
very simple axiom system:

1. forall naturals x: x+1 is not equal to 0
2. forall naturals x: if x is not equal to 0, then there is a
natural y such that y+1 = x
3. forall naturals x and y: if x+1 = y+1, then x=3Dy
4. forall naturals x: x+0 =3D x
5. forall naturals x and y: x+(y+1) = (x+y)+1
6. forall naturals x: x*0 =3D 0
7. forall naturals x and y: x*(y+1) = (x*y)+x

Which axiom do you think involves infinity or recursion?
It is not the axioms of arithmetic that involve recursion but the
methodology he uses in his proof which is recursive since it requires
the representation of every formula of an arithmetic as (Godel)
numbers in that arithmetic.

What difference does it make what "methodology" he used? The
point is that, given the above axiom system, one can construct
a sentence G such that G is not provable from the above axioms.
Do you dispute that? You think that there is no such sentence G?
Or you think that G is actually provable from the above axioms?

--
Daryl McCullough
Ithaca, NY

Daryl,

His proof of that depends on self referentiality (redundancy) as I
explained above. If self referentiality is disallowed his proof fails.
His proof is not some absolute, it is a conclusion based on a set of
premises that must be assumed to be true. If we remove one of those
necessary premises the proof fails. In that case we may not be able to
find a G such that G is unprovable. My point is that I see no
necessity for that premise in actual descriptions of the real physical
world.

Perhaps you would like to give some examples of Gs that are provably
unprovable that don't involve self-referentiality? Any such examples
should be carefully analyzed as to their characteristics and
relevance. That would provide some test of the actual working
relevance and importance of Godel's theorem.

Edgar

On Mar 21, 4:35*pm, Sam Wormley wrote:
wrote:
On Mar 21, 2:58 pm, Sam Wormley wrote:
wrote:
I believe the laws of nature including number are in fact part of
nature and that human world views, mathematics and physics are
approximations of those rules in human terms. If the laws of nature
were not part of nature then how would nature know to produce which
result in any event? The rules have to be part of process itself.
Think of nature as computational. The rules as well as what they
operate on must be present in the computer itself for anything to
happen.
Edgar
* *Nature didn't need humings to exist... We have devised models
* *that are good approximations making predictions about how nature
* *behaves or probabilities about natures behavior. This idea of rules
* *and laws is a human construct.

* *Edgar... to say that "nature as computational" is likely misleading
* *you.

Sam,

Then how does nature know what results to produce for each event in a
consistent manner if it is not embodying some rules that determine the
results?

* *Maybe it doesn't "know".... it just is.... We make up rules.... and
* *when we find the rules are wrong... we make up new rules.... We're
* *get'n pretty good at it!

* *Quantum Mechanics, QED, QCD, etc.
* *General Relativity
* *etc.

Sam,

Well of course it doesn't 'know' in the sense of consciously deciding.
The point is that everything that occurs occurs according to rules
therefor those rules must be part and parcel of the occurrence itself.
Thus the laws of nature must be part of nature.

EDgar

On Mar 21, 12:50 pm, wrote:
On Mar 21, 2:58 pm, Sam Wormley wrote:



wrote:
I believe the laws of nature including number are in fact part of
nature and that human world views, mathematics and physics are
approximations of those rules in human terms. If the laws of nature
were not part of nature then how would nature know to produce which
result in any event? The rules have to be part of process itself.
Think of nature as computational. The rules as well as what they
operate on must be present in the computer itself for anything to
happen.

Edgar

Nature didn't need humings to exist... We have devised models
that are good approximations making predictions about how nature
behaves or probabilities about natures behavior. This idea of rules
and laws is a human construct.

Edgar... to say that "nature as computational" is likely misleading
you.

Sam,

Then how does nature know what results to produce for each event in a
consistent manner if it is not embodying some rules that determine the
results? Nature obviously follows its own rules. Things don't just
happen, they happen according to rules. Nature has been doing this
since the universe began. These events happen in a consistent way
according to the laws of nature. Therefore humans could not have
invented the laws of nature, they must be part of nature itself.
Humans discover the these rules the best they can.

Edgar


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?

What makes you sure the next evolution of Nature's Law of Gravity
after GR would be the right one, and not just another invention?


The answer has been known for 2500 years, since Xenophanes.

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

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