GRGR(12)NOTES(11)

[Terrance F. Flaherty]

GR.V.275.25-26 Murphey's Law
restatement of Godel's Theorem 
A humorous axiom of engineers and lots of other folks,
Murphy's law (origin unknown) holds that
" If anything can go wrong, it will" or "when nothing can go
wrong something will."
 "If there is a possibility of several things going wrong,
the one that will go wrong first will be the one that will
do the most damage."
"Left to themselves, all things go from bad to worse."
" Nature always sides with the hidden flaw."

Gödel also spelled GOEDEL, Austrian-born U.S. mathematician,
logician, and author of Gödel's
proof, which states that within any rigidly logical
mathematical system there are propositions (or
questions) that cannot be proved or disproved on the basis
of the axioms within that system and that,
therefore, it is uncertain that the basic axioms of
arithmetic will not give rise to contradictions. This
proof has become a hallmark of 20th-century mathematics, and
its repercussions continue to be
felt and debated.  A member of the faculty of the University
of Vienna
from 1930, Gödel was also a member of the Institute for
Advanced Study, Princeton, N.J. (1933,
1935, 1938-52); he emigrated to the United States in 1940
(naturalized 1948) and from 1953 served
as a professor at the institute.  Gödel's proof first
appeared in an article in the
Monatshefte für Mathematik und Physik, vol. 38 (1931), on
formally indeterminable propositions of
the Principia Mathematica of Alfred North Whitehead and
Bertrand Russell. This article ended
nearly a century of attempts to establish axioms that would
provide a rigorous basis for all mathematics,
the most nearly (but, as Gödel showed, by no means entirely)
successful attempt having been the
Principia Mathematica. Another well-known work is
Consistency of the Axiom of Choice and of the
Generalized Continuum-Hypothesis with the Axioms of Set
Theory (1940; rev. ed., 1958), which
has become a classic of modern mathematics. 

According to Weisenbergur,  Godel's incompleteness theorem
is a hopeful sign. When it crops up again (GR.V.320.19), it
is in the context of an infinite postponement of suicide,
because something might be missing from one's catalog of
worldly disgusts and denials. In formal logic and
mathematics the incompleteness theorem establishes that
formal closure, completeness, and the internal consistency 
of any complex, logical system may all be pipe dreams. As
such, it makes a telling background to Pynchon's
representations of closed versus open fields, of being "shut
in by words" )GR.V.339.36-37) as opposed  to breaking free
by means of them. Some twenty pages back Slothrop though
himself: "Free? What's free?" (GR.V256.33). As metaphor,
Godel's theorem---in its vernacular version, Murphy's
Law---begins to answer that question and sends us tripping
In the Zone of Part 3.  

The first and second incompleteness theorem

Gödel's first incompleteness theorem, from 1931, stands as a
major turning point of 20th-century
logic. It states that no finitely axiomatizable theory
sufficient to derive the Peano postulates is both
consistent and complete. (How Gödel proved this fascinating
result is discussed more extensively in
the article mathematics, foundations of.) In other words, if
we try to build a theory sufficient for a
foundation for mathematics, stating the axioms and rules of
inference so that we have stipulated
precisely what is and what is not an axiom (as opposed to
open-ended axiom schemata), then the
resulting theory will either (1) not be sufficient for
mathematics (i.e., not allow the derivation of the
Peano postulates for number theory) or (2) not be complete
(i.e., there will be some valid proposition
that is not derivable in the theory) or (3) be inconsistent.
(Gödel actually distinguished between
consistency and a stronger feature, - [omega-] consistency.)
A corollary of this result is that, if a
theory is finitely axiomatizable, consistent, and sufficient
to derive the Peano postulates, then that
theory cannot be used as a metalanguage to show its own
consistency; that is, a finitely axiomatized set
theory cannot be used to show the consistency of finitely
axiomatized set theory, if set theory is
consistent. This is often called Gödel's second
incompleteness theorem.  These results were widely
interpreted as a blow to both the logicist and formalist
programs. Logicists seemed to have taken as their goal the
construction of rigorously described theories that were
sufficient for deriving mathematics and also consistent and
complete. Gödel showed that, if this was their goal, they
would necessarily fail. It was also a blow to the
longer-standing axiomatic, or formalist, program, since it
seemed to show that precise
axiomatic descriptions of valuable domains like mathematics
would also necessarily fail. Gödel
himself eventually interpreted the result as showing that
there exist entities with well-defined
properties, namely numbers, that are beyond our ability to
describe precisely with standard logical
tools. This is one source of his inclination toward what is
usually called mathematical Platonism.

GR.V.275.34-35  legend of the black scapeape
tallest
erection in the world
King Kong, where the great Ape was machine gunned from NYC's
Empire State Building in the closing scenes of Marian
Cooper's film.  Rocco Siffredi, who "crossed over from porn
to well, not quite porn in Director/Screenwriter Catherine
Breillat's latest film "Romance", will play the Empire State
Building in her soon to be released (though not in NYC
theaters) cross over version of the Cooper classic. Pop Corn
boxes will be equipped with  Slothrop's easy punch out penis
hole in the bottom.