Page 164 - Concise Encyclopedia of Robotics

Incompleteness Theorem
because society is not the same as it was two centuries ago. Values have
changed. People have different priorities and beliefs.
When history is put down in books, or told as stories, much of the in-
formation is simply lost, never to be recovered. Computers, however, can
keep data indefinitely. To some extent, computers can interpret data as well
as store it.Some scientists think this will reduce the rate of change of human
thought modes over long periods of time. It might also act to cause
people in different parts of the world, and in different cultures, to think
more and more alike.
Computers will make little details in information (and misinforma-
tion) more permanent. If carried to the extreme, computers will give
humanity knowledge that lasts essentially forever. This has been called
immortal knowledge. The data stored in any medium can be backed up to
prevent loss because of computer failure, sabotage, and aging of disks
and tapes. Every fact, every detail, and possibly all the subtle meaning,
too, can be passed along unaltered for century after century.
Some engineers argue that computerization might have a detrimental
effect on the preservation and accumulation of human knowledge. Com-
puter data is easier than hard copy (such as books,scrolls,and other written
documents) to tamper with on a large scale. It is not inconceivable that a
few brilliant humans with nefarious intent could literally rewrite history,
and no one, generations later, would be the wiser.
INCOMPLETENESS THEOREM
In 1931,a young mathematician named Kurt Gödel discovered something
about logic that changed the way people think about reality. The incom-
pleteness theorem demonstrated that it is impossible to prove all true
statements in a first-order logical system. In any such system of thought,
there are undecidable propositions.
In mathematical systems, certain assumptions are made. These are
called axioms or postulates. Logical rules are employed to prove theorems
based on the axioms. Ideally, there are no contradictions; then we have a
consistent set of axioms. If a contradiction is found, we have an inconsistent
set of axioms.
Generally, the stronger the set of axioms—that is, the greater the
number of implied statements based on them—the greater is the chance
that a contradiction can be derived.A logical system with a set of axioms that
is too strong literally falls apart, because once a contradiction is found,
every statement, no matter how ridiculous, becomes provable. If a set
of axioms is too weak, then it does not produce much of anything
meaningful. For centuries, mathematicians have striven to build “thought
universes” with elegance and substance, but free of contradictions.

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