Science > Physics > Re: Meaning of Godel's incompleteness theorem in physics
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Science > Physics |
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"John Bailey" |
| Date: |
16 Apr 2007 07:23:05 AM |
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Re: Meaning of Godel's incompleteness theorem in physics |
On 15 Apr 2007 17:30:10 -0700, "Tony012" <dikim012@gmail.com> wrote:
I am novice in Godel's theorem and I am trying to understand its
implication in physics.
Let me write what I understand about Godel's theorem.
Godel's theorem states that there appear statements which are true but
not decidable in a given consistent axiomatic system.
Now, let us assume that this universe is [constructed with a] consistant
axiomatic system.
An excellent question. Ignore the snipers.
Here is my attempt to state the issue:
If we assume that all physical laws can be explained by their mapping
to a collection of axioms, then we are faced with a dilemna. (Call the
collection a theory.) Either the theory is incomplete or
inconsistent. If incomplete, there are some experiments the theory
cannot explain. If inconsistent, there are some experiments which the
theory describes in multiple ways. (Does this apply to situations
involving quantum mechanics and general relativity?)
Then, it seems that Godel's theorem suggests that there are two types
of motion in this universe:
First type is the motion which can be derived by the axioms. At the
same time, it seems that there exist the other type of motion which
can not be dervied by those axioms but the motions can exist because
they do not violate the axiom and are true.
IIRCC Godel allows a system of axioms to be extended until it begins
to lose consistency. I see no reason why motion would necessarily be
involved in the instances for which extensions result in
inconsistency. Most of his examples and proofs rely on situations
involving self reference. It seems more likely to me that situations
involving time (present, past, or future) or sentient consciousness or
creation (big bang) might involve self reference.
Is my reasoning correct?
What would Godel say?
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| User: "" |
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| Title: Re: Meaning of Godel's incompleteness theorem in physics |
16 Apr 2007 09:31:21 AM |
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In article <pfo623pba6sf84q46542s00cg6cos051ik@4ax.com>, John Bailey <john_bailey@rochester.rr.com> writes:
On 15 Apr 2007 17:30:10 -0700, "Tony012" <dikim012@gmail.com> wrote:
I am novice in Godel's theorem and I am trying to understand its
implication in physics.
Let me write what I understand about Godel's theorem.
Godel's theorem states that there appear statements which are true but
not decidable in a given consistent axiomatic system.
Now, let us assume that this universe is [constructed with a] consistant
axiomatic system.
An excellent question. Ignore the snipers.
Here is my attempt to state the issue:
If we assume that all physical laws can be explained by their mapping
to a collection of axioms, then we are faced with a dilemna. (Call the
collection a theory.)
Godel's theorem does _NOT_ say that every theory is either
incomplete or inconsistent. There are other requirements that must
be met first.
Godel's argument depends on the ability (roughly speaking) of the
system to contain am embedded model of itself. This means that
the system needs to be both powerful and describable. A system can
escape Godelization by being either too simple to contain itself or by
being too complex to fit into itself.
Turing may be a better fit than Godel for giving an argument that any
physical theory must be inconsistent or incomplete...
Claim: Any physical theory whose predictions can be correctly
evaluated using a Turing machine in finite time and which is powerful
enough to model arbitrary Turing machines must neccessarily be unable
to correctly solve the halting problem.
Note well the caveats in that claim.
o The theory must be able to be emulated on a TM.
o The theory must be able to model the behavior of an arbitrary TM.
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| User: "Aatu Koskensilta" |
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| Title: Re: Meaning of Godel's incompleteness theorem in physics |
16 Apr 2007 09:38:22 AM |
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wrote:
Godel's argument depends on the ability (roughly speaking) of the
system to contain am embedded model of itself. This means that
the system needs to be both powerful and describable. A system can
escape Godelization by being either too simple to contain itself or by
being too complex to fit into itself.
I'm afraid your comments offer but a marginal improvement over those by
Tony012 and John Bailey.
--
Aatu Koskensilta (aatu.koskensilta@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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| User: "" |
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| Title: Re: Meaning of Godel's incompleteness theorem in physics |
16 Apr 2007 02:11:01 PM |
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In article <ITLUh.37486$4S5.19756@reader1.news.saunalahti.fi>, Aatu Koskensilta <aatu.koskensilta@xortec.fi> writes:
briggs@encompasserve.org wrote:
Godel's argument depends on the ability (roughly speaking) of the
system to contain am embedded model of itself. This means that
the system needs to be both powerful and describable. A system can
escape Godelization by being either too simple to contain itself or by
being too complex to fit into itself.
I'm afraid your comments offer but a marginal improvement over those by
Tony012 and John Bailey.
OK. I'm sure you're right. You've demonstrated significant expertise
in the area. Unfortunately, your remarks leave me guessing on where I
went wrong.
I suppose one answer is that it doesn't matter much whether the theory
in question can directly encode a model of itself. What matters is
whether it can encode a model of a particular subset of arithmetic.
Then, as best as I can fill in the details from various popularizations,
what matters is whether the encoded model of arithmetic can be used to
encode a model of the original theory in such a way that provable
statements in the original theory become provable statements in the
doubly-encoded version of the theory and vice versa.
With those two pieces in hand, one can demonstrate the existence of a
well formed formula G which can be interpreted as asserting its own
unprovability.
[I'm not clear on whether the existence proof for G is constructive.
The popularization I'm looking at right now does not spell that part out]
Am I back on track now? Or still heading into the weeds?
Assuming I'm on the right track...
It doesn't matter whether the theory is "too complex" by reason of
not being able to fit into itself. It does matter whether the theory
is too complex to permit a finite encoding of the theory in
the language of arithmetic. In particular, the theory must not
contain an infinite number of axioms unless all but a finite subset
of those are redundant and can be eliminated.
It doesn't matter whether the theory is "too simple" by reason
of not being able to contain itself. It does matter whether the theory
is "too simple" in the sense of not being able to encode a model of
an appropriate subset of arithmetic.
Or am I still missing something major?
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| User: "Aatu Koskensilta" |
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| Title: Re: Meaning of Godel's incompleteness theorem in physics |
18 Apr 2007 07:06:28 AM |
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On 2007-04-16, wrote:
OK. I'm sure you're right. You've demonstrated significant expertise
in the area. Unfortunately, your remarks leave me guessing on where I
went wrong.
My remarks were unnecessarily snarky. Your comments do make sense, provided
one already understand the conditions under which the incompleteness theorem
applies to a formal system. Alas, talking about "embedding" a theory into
itself and so on, does nothing to explain these conditions to someone not
already familiar with the subject, and, as they have many meanings and
connotations in ordinary language, are extremely likely to engender
confusion, especially in context where someone puts forth some "application"
of the incompleteness theorem.
So what are the relevant conditions on the formal theory? Formal theories
and the related notions of derivability, consistency, completeness, and so
on, are defined using such mathematical notions as finite trees, sequences,
and so on. The incompleteness theorem applies to formal theories the
language of which is capable of expressing elementary statements about
these. (Elementary in the sense that they can be expressed without any
reference to infinite sets and abstract stuff like that). It turns out that
elementary statements about finite structures, sequences, and the like, are
mathematically equivalent to elementary statements formulated using just 0,
1, addition and multiplication. The first incompleteness theorem tells us
that for any formal theory T in which elementary statements about finite
sequences, trees, and so on, are expressible - or, equivalently, in which
elementary arithmetical statements about 0, 1, addition and multiplication
are expressible - and in addition certain basic mathematical principles
concerning these - or, equivalently, certain basic arithmetical truths - are
formally derivable, we can mechanically construct an elementary arithmetical
sentence G[T] of the same logical complexity as Goldbach's conjecture or
Fermat's last theorem with the property that
If T does not for some sentence A formally prove both A and not-A, i.e. if
T is consistent, then G[T] is not formally provable in T, and furthermore,
G[T] is true.
(G[T] has the form "for all natural numbers n, P(n)" where P is a
mechanically checkable property of naturals, and that G[T] is true means just
that, in fact, all naturals have the property P).
An additional condition is hidden in the phrase 'formal theory'. The first
incompleteness theorem, in its original formulation, applies to 'formal
theories', that is formally specified mathematical systems in which the set
of axioms, on which the mechanical rules of inference may be applied to derive
further theorems, of the theory is recursive (or, equivalently, recursively
enumerable). This means that it must be a mechanical matter, when given a
string of symbols, to determine whether the string is an axiom or not.
Unlike often stated, there is on requirement that the set of axioms be
finite; many of the most important formal theories studied in logic, such as
Peano arithmetic or Zermelo-Fraenkel set theory, are, in fact, provably not
axiomatizable using a finite set of axioms.
It is not uncommon to find Gödel's theorems applied in all sorts of
fantastic ways to physical theories, the universe, the human mind, and so
on, and the idea that the theorem shows that there can be no ultimate theory
of physics, or that some physical questions are necessarily unanswerable is
a common one -- Freeman Dyson, for example, has argued that there can be no
"Theory of Everything" because of the incompleteness theorem. This is a
rather confused idea.
First, it is patently obvious that physical theories are not formal theories
in the sense outlined above; it makes no sense to speak of the "formal language
of General Relativity" (in the technical sense of a mathematically defined set
of strings), or the "rules of inference of General Relativity" (in the technical
sense of mechanical rules allowing one to derive a string from a given finite s
et of strings), and so on. Assuming that the universe is a "formal theory"
is even more non-sensical.
Secondly, the incompleteness theorem tells us that any formal theory meeting
certain criteria is incomplete with respect to arithmetical statements of
certain form. Even if physical theories were formal theories, it would not
be very likely that physicists would be unduly concerned if every true
statement of the form "the Diophantine equation D(x_1, ..., x_n) = 0 has no
solutions" was not formally derivable in their Theory of Everything. (The
idea that such statements are provable or refutable in, say, Quantum
mechanics is itself rather baffling).
--
Aatu Koskensilta (aatu.koskensilta@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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| User: "Aatu Koskensilta" |
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| Title: Re: Meaning of Godel's incompleteness theorem in physics |
16 Apr 2007 08:35:22 AM |
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John Bailey wrote:
What would Godel say?
Gödel, when asked about Eduard Wette's "proof" of inconsistency of
elementary arithmetic (by means of producing a consistency proof),
remarked that "it would be interesting to publish it". Alas, Tony012's
post was just your ordinary run-of-the-mill Gödel waffle, and lacked a
colour-coded chart. Gödel wouldn't have been impressed.
--
Aatu Koskensilta (aatu.koskensilta@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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| User: "Rock Brentwood" |
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| Title: Re: Meaning of Godel's incompleteness theorem in physics |
17 Apr 2007 06:34:33 PM |
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On Apr 16, 7:23 am, John Bailey <john_bai...@rochester.rr.com> wrote:
On 15 Apr 2007 17:30:10 -0700, "Tony012" <dikim...@gmail.com> wrote:
I am novice in Godel's theorem and I am trying to understand its
implication in physics.
Contrary to the usual take on the issue: Goedel's theorem does NOT
state the incompleteness of Arithmetic, per se. Arithmetic has a
complete formulation -- in second order logic.
What it states is far more general and universal; namely that second
order logic, itself, is incomplete. That is: it cannot be reduced to
first order logic by any finite set of postulates, nor even by any
recursively specifiable infinite set of postulates.
That was the point of Goedel proving the completeness of first order
logic a few years before setting out to resolve the main theorem. He
needed that as the background on which to state incompleteness of
second order logic (otherwise, one could have argued that second order
logic was complete too, but merely that first order logic was
incomplete!)
The only ramification of this result is that some theories are
inherently second order theories. So, a hierarchy is established
between first order theories (which essentially captures the essence
of "algebraic" or "combinatorial") vs. second order theories (those
that are essentially "analytical").
There's nothing more profound to Goedel's result than that. Too much
has been made of Goedel's result almost like some kind of GEB
Hofstadter cult of cliched coffee-house intellectual mysticism that
even Hofstadter himself has long ago already distanced himself from.
So, that pretty much pulls the rug out from under any question about
what "physical import" it has.
Goedel tried to prove a physical "Goedel's theorem" back in the 1930's
or 1940's or so. As part of the effort was his construction of the
solution to the Einstein field equations which has closed timelike
curves (and in which the entire universe is rotating -- thereby also
refuting any connection of General Relativity to Mach's Principle).
Obviously that effort went nowhere.
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| User: "Aatu Koskensilta" |
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| Title: Re: Meaning of Godel's incompleteness theorem in physics |
18 Apr 2007 06:17:43 AM |
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On 2007-04-17, Rock Brentwood wrote:
Contrary to the usual take on the issue: Goedel's theorem does NOT
state the incompleteness of Arithmetic, per se. Arithmetic has a
complete formulation -- in second order logic.
Second order arithmetic is complete in a different sense of completeness
than that of the incompleteness theorems. Second order arithmetic, however
formulated as a formal system, i.e. by specifying a recursive set of axioms
and mechanical rules of inference, is just as incomplete as first order
arithmetic. Second order arithmetic is complete only if we define
completeness for second order theories in terms of logical consequence
rather than formal derivability. In some contexts this might be the
appropriate, but in relation to the incompleteness theorems it's just
confusing.
What it states is far more general and universal; namely that second
order logic, itself, is incomplete. That is: it cannot be reduced to
first order logic by any finite set of postulates, nor even by any
recursively specifiable infinite set of postulates.
That second order logic is incomplete, again in a different sense of
completeness than that of the incompleteness theorems, is not what the first
incompleteness theorem states; rather, it is a corollary of the first
incompleteness theorem.
That was the point of Goedel proving the completeness of first order
logic a few years before setting out to resolve the main theorem. He
needed that as the background on which to state incompleteness of
second order logic (otherwise, one could have argued that second order
logic was complete too, but merely that first order logic was
incomplete!)
This is pure fantasy, and has nothing to do with how Gödel actually
discovered the incompleteness theorems. Indeed, one looks in vain for any
statement of the incompleteness theorem in terms of incompleteness of second
order logic in Gödel's original paper.
Goedel tried to prove a physical "Goedel's theorem" back in the 1930's
or 1940's or so. As part of the effort was his construction of the
solution to the Einstein field equations which has closed timelike
curves (and in which the entire universe is rotating -- thereby also
refuting any connection of General Relativity to Mach's Principle).
Obviously that effort went nowhere.
Gödel's rotating universe solution to the field equations of General
relativity has absolutely nothing to do with the incompleteness theorems.
Correcting misconceptions about the incompleteness theorems is probably not
most effectively done by replacing them with different misconceptions.
--
Aatu Koskensilta (aatu.koskensilta@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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| User: "Don Stockbauer" |
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| Title: Re: Meaning of Godel's incompleteness theorem in physics |
17 Apr 2007 09:46:36 PM |
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On Apr 17, 7:34 pm, Rock Brentwood <markw...@yahoo.com> wrote:
On Apr 16, 7:23 am, John Bailey <john_bai...@rochester.rr.com> wrote:
On 15 Apr 2007 17:30:10 -0700, "Tony012" <dikim...@gmail.com> wrote:
I am novice in Godel's theorem and I am trying to understand its
implication in physics.
There's nothing more profound to Goedel's result than that. Too much
has been made of Goedel's result almost like some kind of GEB
Hofstadter cult of cliched coffee-house intellectual mysticism that
even Hofstadter himself has long ago already distanced himself from.
Nope. Hof's still hanging in there with the his so-called "cult of
cliched coffee-house intellectual mystics." See his latest "I am a
strange loop."
Godel's Inc theorm is basically a very complicated version of the
Epimenides sentence "this statement is false" What it all tells you
is that self-reference can get you in trouble, and it can also be the
pathway to self-awareness. So one detects these traps and navigates
around them (cybernetics = navigation).
IF (self-reference encountered)
then beware of paradoxes.
I think.
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| User: "Aatu Koskensilta" |
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| Title: Re: Meaning of Godel's incompleteness theorem in physics |
18 Apr 2007 06:21:51 AM |
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On 2007-04-18, Don Stockbauer wrote:
Godel's Inc theorm is basically a very complicated version of the
Epimenides sentence "this statement is false" What it all tells you
is that self-reference can get you in trouble, and it can also be the
pathway to self-awareness. So one detects these traps and navigates
around them (cybernetics = navigation).
A fine piece of "clichéd coffee-house intellectual mysticism".
--
Aatu Koskensilta (aatu.koskensilta@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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