|
Hi Bill:
Let me begin by copying a part of an
article on Godel's theorem, and underlining some important points:
"Gödel’s Theorems
Kurt Gödel is most famous for
his second incompleteness theorem, and many people are unaware that,
important as it was and is within the field of mathematical
logic and beyond, this result is only the middle movement, so to speak, of a
metamathematical symphony of results stretching from 1929 through
1937. These results are: (1) the Completeness Theorem; (2) the
First and Second Incompleteness Theorems; and (3) the consistency of the
Generalized Continuum Hypothesis (GCH) and the Axiom of Choice (AC) with the
other axioms of Zermelo-Fraenkel set theory. These results are discussed in
detail below.
Gödel was born in 1906 in the Austro-Hungarian province of
Moravia. He finished his doctoral thesis (the completeness theorem) at the
University of Vienna in 1929, and commenced work on the incompleteness results
while an unpaid lecturer there from 1930 to 1933. During this period he was a
frequent participant in the “Vienna Circle,” an informal group of thinkers which
included Karl Menger and Rudolph Carnap, among many others. He also came to know
the Polish logician Alfred Tarski and the Hungarian mathematician John Von
Neumann, both of whom took an avid interest in his work. Gödel first visited the
Institute for Advanced Study in Princeton in 1933-34, lecturing on his new
incompleteness results. (The institute had just been founded in 1930 with Albert
Einstein and Oswald Veblen as its first professors, and soon included Von
Neumann, James Alexander, and Hermann Weyl.) Gödel suffered from both depression
and exhaustion in the next several years, and spent at least two periods in a
sanitarium between 1934 and 1937. He visited the institute again in 1938-39, and
then finally, under pressure from the new Nazi regime in Austria, came to the
Institute permanently in 1940. Due to the war, this last journey had to be taken
through Asia, and required several months. Gödel never returned to Europe.
At
the institute, although he joined in a small social circle (and he and Einstein
became fast friends), Gödel was very reclusive and never supervised the work of
students directly. He was paranoid about his health and about the intentions of
others. His most important work in logic and metamathematics was essentially
completed by 1940, although he did some interesting work in relativity theory
for many years, creating a cosmological model that permitted “time travel” into
the past. He wrote prolifically, however, criticizing the work of Bertrand
Russell and others, and promulgating a strong Platonist philosophy of
mathematics at a time when Platonism was all but abandoned by most
thinkers.
Gödel’s health was poor from 1960 on, and his depressions returned.
He developed fears about being poisoned, and would not eat. He died in Princeton
Hospital in 1978 of malnutrition.
THE COMPLETENESS THEOREM (1929)
In 1928, David Hilbert and Wilhelm Ackermann
published Grundzüge der theoretischen Logik, a slender but potent text on the
foundations of logic. In this text they posed the question of whether a certain
system of axioms for the first-order predicate calculus is complete, i.e.,
whether every logically valid sentence in first-order logic can be derived from
the axioms and rules of inference of the system. Gödel’s Completeness Theorem,
which he presented as his doctoral dissertation at the University of Vienna in
1929, showed that first-order logic is indeed complete in this sense. In other
words, every valid sentence in first-order logic can be derived in first-order
logic.
THE INCOMPLETENESS THEOREM(S) (1933)
This question of completeness of theories was
central to the larger metamathematical program, initiated by David Hilbert, of
determining a complete axiom set for mathematics as a whole that would
be:
Finitely given,
Consistent, and
Complete.
By finitely
given, we mean that the set of assumptions must be describable in a finite
way. (Without this condition, one could take as one’s axioms the
set of all true propositions of mathematics – a vacuous supposition!) By
consistent, we mean that the axioms, together with the rules of inference, must
not allow the deduction of a contradiction, such as 1 = 2. (Math is doomed if we
can’t satisfy this condition.) The third condition – completeness – is the crux
of the matter: We want, given any true mathematical statement, the
assurance that our axioms and rules of inference are strong enough to give a
proof of the statement. (We might not, for a given
statement, actually know the proof, but we would like to be assured that,
supposing the statement is really true, a proof exists.) This was the Holy
Grail, and it was this that Gödel showed was
unattainable.
Gödel’s first effort was in the realm of number
theory, and specifically addressed whether the accepted axiomatization of
arithmetic (the Peano axioms) are complete. Through an ingenious device known as
Gödel numbering, Gödel found a way to assign natural numbers in a unique way to
the statements of arithmetic themselves, effectively turning statements about
numbers into numbers – numbers moreover about which new statements could be
made. He made numbers talk about themselves, in other words. This permitted him
to construct a Gödel sentence, a logical sentence in the language of number
theory which amounts to the statement, “this theorem is not provable in number
theory.”
What does this achieve? If the Gödel sentence is false, then it is
necessarily provable in arithmetic – a contradiction. If this contradiction
stands then arithmetic is inconsistent. But there are very strong reasons for
believing arithmetic is consistent, so we must suppose that the Gödel sentence
is a true theorem of arithmetic, and therefore arithmetic itself can never prove
it. (You see why this is called metamathematics!) So, arithmetic is necessarily
incomplete.
This result stunned the mathematical world, but Gödel wasn’t
finished yet. For he realized at once that the only requirement for this result
was that the axiom system in question be strong enough to permit one to do
arithmetic. Consequently, any mathematical theory (axiomatic system) which meets
this criterion is likewise incomplete. Set theory, algebra, analysis – indeed
the whole of mathematics is incomplete, assuming that the axioms are finitely
given and consistent (which are conditions mathematics cannot forego), and
robust enough to account for arithmetic. This means that there are true
statements of mathematics (theorems) which we can never formally know to be
true. We are barred from achieving complete knowledge of mathematical
truth.
It is this second incompleteness result which is generally
meant when people talk about “the Gödel Incompleteness Theorem.” It is widely
believed to have important repercussions beyond mathematics, since it is really
a statement about the limits of formal knowledge, that is, knowledge which
depends upon the rigors of logic. However, one must be careful here
to remember what the entire edifice rests upon: numbers. The incompleteness
theorems should not be made to do service in areas outside of mathematics unless
that crucial link to formal logic and robust axioms can be made. That Gödel
should have found his result by discovering the potential for self-reference in
arithmetic should not surprise us, in retrospect, because the importance of
paradoxes which arise in this way have been know since ancient times. (Consider,
for example, the so-called liar paradox.)
There are other results
which Gödel’s incompleteness theorems made possible. These include the fact
(which he formulated) that no theory can prove its own consistency.
(Arithmetic, for example, can only be proved consistent within a larger
theoretical framework such as Zermelo Fraenkel set theory – but then set theory
can’t prove it’s own consistency.) Another closely related statement
is Alfred Tarski’s result that “truth” in a formal system or model cannot be
defined within that system or model."
Godel's theorem is actually about
mathematics, but it didn't take long for logicians to extend it to all axiomatic
systems. I have no desire to quibble over debates about "what Godel's
theorem says", because in reality, his theorem only talks about mathematical
systems, but in essence it can be extended to any axiomatic system.
You've made reference to two criteria
for truth.
1) The truth of a
statement
and
2) Conformity between the statement
and reality.
I will look at both.
1) The statement : "A
unicorn is a horse with a horn" is true by definition.
However; I don't think hardly anybody thinks that unicorns actually
exist. This is in response to your statement that we know alot of things
about the concept of infinity. Well, we know alot of things about
unicorns, but I don't think any of them are "true" because in my opinion,
unicorns don't exist (and neither does infinity in my opinion).
2) If the truth is conformity
between thought and being, then the truth becomes an approximation (i.e. a
limit).
"What then is immediacy? It is
reality itself. What is mediacy? It is the word. How does the
one cancel the other? By giving expression to it, for that which is given
expression is always presupposed.
Immediacy is reality; language is
ideality; consciousness is a contradiction. The moment I make a statement
about reality, contradiction is present, for what I say is ideality."
(Soren Kierkegaard, "Philosophical Fragments)
In other words, contradiction is
present in any statement I make about reality, because reality and ideality are
two separate things, and if you are going to wait for "the Truth to be revealed"
in hopes of conformity between ideality and reality, you will be waiting for
infinity, and you will never get there.
That's why I asked the question,
"When is infinity?".
Christians believe that God is the
word, and the idea of Incarnation is that the Word was made Flesh. In
other words, ideality and reality did become one - not in infinity, but at a
point in time. That is the idea of Christ.
No amount of rationalization will
ever get you to know the mind of God. It will always be beyond
reach.
BTW, thank you for the debate.
I'm sorry I don't have more time to spend on it as I'm busy at
work.
Take care,
Jim
----- Original Message -----
Sent: Wednesday, August 31, 2005 10:37
AM
Subject: Re: [socialcredit] The Mind of
God
> Quoted from the link: *Although this theorem can be
> stated
and proved in a rigorously mathematical way,
> what it seems to say is
that rational thought can
> never penetrate to the final ultimate
truth...*
> ---------------------------------------
>
-------------------------------------
> [Reply] This is a very perverse
spin to the theorem.
> It's not what it "seems to say" to me at
all. When
> stated in the form of the Inclusive Logistic
>
Progression, it points to God as theoretical limit, or
> end-point -- so
is inferential proof of the existence
> of God.
>
---------------------------
> ----------------------------
>
-------------------------
>
> I want to expand on this a bit
further. Zeno's
> paradox proved to Zeno (or should have) that
a
> mathematics existed in principle that was not
> available to him
in his time that would resolve the
> paradox. The actual mathematics
was not discovered
> until Newton came along 2000 years later. It
could
> have been discovered earlier, or later, or not at all
> (if
a man of the intellectual capability of Newton had
> never come along) --
which does not invalidate the
> proof that the mathematics existed in
principle.
> Opening the paradox supplies the motivation for
>
finding its resolution. Do you see the point?
>
> It is
possible to admit the absolutely certain
> existence of something we do
not understand at all.
> Over time we may learn more and more aspects of
that
> something that absolutely exists, without coming to
> the
full comprehension of that something in logical
> whole, as did Newton's
predecessors, the "giants" on
> whose shoulders he stood.
>
-
>
> __________________________________________________
> Do
You Yahoo!?
> Tired of spam? Yahoo! Mail has the best spam
protection around
> http://mail.yahoo.com
>
---------------------------------------------------------------------
>
Some introductory materials to the discussion topic of this list are at
>
http://www.geocities.com/socredus/compendium
> You're subscribed to this list with the email jschroeder@shaw.ca
> For more
information, visit http://www.eListas.com/list/socialcredit
> |
Re: [socialcredit]
Reply to this message
