Aaron Goldsmith 1

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Oct 10, 2013 (3 years and 10 months ago)

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Aaron Goldsmith
1

Gödel's Incompleteness Theorems


David Hilbert put considerable effort into finding a complete set of axioms for all of
mathematics.
In fact, this was Hilbert's second problem, one of the ten presented at the
International Congress in 1900.
Such a set wo
uld unify mathematics and place everyone on the
same page as to what is assumed to be true without proof. A logician named Kurt Gödel,
howe
ver, foiled Hilbert's program with two "incompleteness theorems" in 1931.

These theorems
say that any formal system

that is advanced enough to prove
Peano
arithmetic is either
inconsistent or incomplete. That is, either you arrive at a contradiction, or there exists a true
statement which cannot be proved in the system.


Alfred North Whitehead and Bertrand Russell set

out in "The Principia Mathematica" to
complete the foundation of mathematics, when around the same time, Gödel proved that they
were wasting their time. Sadly, when Gödel's paper was first published, it made no sense to even
most mathematicians, but Whit
ehead and Russell's was not so esoteric. The logic and reasoning
in Gödel's proof at the time was so novel that only those specialized with the highly technical
literature could understand the full implications

[3]
.


Basically, Gödel used half his proof t
o show that the four axioms for
second
-
order

arithmetic were incomplete (there exists a true statement that cannot be proven), and from there
set a correlation between Peano arithmetic and
the symbols of the language
. Gödel line
d

up all
the symbols, opera
tors, axioms, and everything about the system, then assigns positive integers
to each one. This means any formal proof can be represented with a string of integers

[
1
]
. The
other half of the proof involves

finding a statement that could not be proved or
disproved. It
turns out that he used a self referential statement similar to, "I am lying," though this is not
directly feasible in a formal proof.
In this way, he showed that no finite set of axioms was
Aaron Goldsmith
2

sufficient for proving all true statements in any
system that
invokes
Peano

arithmetic
.
Ironically,
the very cornerstone of Gödel's proof looks similar to Russell's Paradox, from the same Russell
that was
pursuing the opposite result as Gödel. Russell's Paradox
is like
a town with a barber
that shaves p
recisely those that do not shave themselves. The barber must and cannot shave
himself, a more problematic situation than when we can't decide on which pizza place to support.


This inconsistency theorem is
sometimes

used to say the H
ebrew

Bible

as well as

any
other

is either incomplete or inconsistent (contradictory), by regarding the Bible as a finite set of
axioms. This is a completely unnecessary argument. Göde
l's theorems are from

formal logic
and should only be used in formal mathematical systems.
The Bible does not claim to decide
whether irrelevant statements are true or not. Thus, incompleteness is not as applicable as many
think. For example, if I asked the Bible to decide whether or not the statement "Peter had a large
mole on his left cheek"

is true, it would give me nothing. That is not what the Bible concerns.
The same argument is used for the United States Constitution, the English language, and many
other documents

[4]
. And, the same rebuttal applies. There is also much debate over wh
ether
Gödel's first inconsistency theorem means that artificial intelligence will never reach the level of
the human brain

[3]
. This assumes that the human brain is
capable of knowing truths without
proof

and that artificial intelligence uses sufficiently

advanced mechanisms to invoke Göd
el's
Theorem.


Clearly, Gödel rocked the

mathematical

world with his discoveries. When he was
awarded an honorary degree from Harvard University in 1952, his work was described as "on
e

of
the most important adva
nces in lo
gic in modern times"

[3].



Aaron Goldsmith
3

Bibliography

[1] David Hofstadter. Gödel, Escher, Bach: An Eternal Golden Braid. Basic Books
, New York,
20th Anniversary Ed, 1999.

[2] Raymond Smullyan. Gödel's Incompleteness Theorems. Oxford UP, New York, 1992.

[3] Ernest
Nagel and James R. Newman. Gödel's Proof. New York UP, New York, 1958.

pp 3,
100.

[4]
http://en.wikipedia.org/wiki/Incompleteness_theorem#Discussion_and_implications