[Avodah] Godel's incompleteness theorems -- Proof of G-d's existence

[Micha Berger]

On Mon, Nov 15, 2010 at 10:40:06AM -0500, Hankman wrote to Areivim:
: I suspect (speculate, I am no expert here) that Godel's incompleteness
: theorems could be at the bottom of the reason why absolute proofs of
: G-d's existence are hard (impossible by the theorm) to come by. It
: asserts that not all things that are true can be proven to be true (no
: consistent system can prove its own consistency - thus as proof of an
: all powerful deity would show the consistency of a deity based theology
: [logical construct] no such proof is possible). Am I off base here? See
: his theorems below:

: Kol Tuv
: Chaim Manaster

: Goedel's incompleteness theorems

: It is not strictly science, but rather a very interesting set
: of mathematical theorems about logic and the philosophy that is
: definitely relevant to science as a whole. Proven in 1931 by Kurt
: Goedel, these theories say that with any given set of logical rules,
: except for the most simple, there will always be statements that are
: undecidable, meaning that they cannot be proven or disproven due to the
: inevitable self-referential nature of any logical systems that is even
: remotely complicated. This is thought to indicate that there is no grand
: mathematical system capable of proving or disproving all statements. An
: undecidable statement can be thought of as a mathematical form of a
: statement like "I always lie." Because the statement makes reference to
: the language being used to describe it, it cannot be known whether the
: statement is true or not. However, an undecidable statement does not need
: to be explicitly self-referential to be undecidable. The main conclusion
: of Goedel's incompleteness theorems is that all logical systems will have
: statements that cannot be proven or disproven; therefore, all logical
: systems must be "incomplete."

: The philosophical implications of these theorems are widespread. The
: set suggests that in physics, a "theory of everything" may be impossible,
: as no set of rules can explain every possible event or outcome. It also
: indicates that logically, "proof" is a weaker concept than "true"; such a
: concept is unsettling for scientists because it means there will always be
: things that, despite being true, cannot be proven to be true. Since this
: set of theorems also applies to computers, it also means that our own
: minds are incomplete and that there are some ideas we can never know,
: including whether our own minds are consistent (i.e. our reasoning
: contains no incorrect contradictions). This is because the second of
: Goedel's incompleteness theorems states that no consistent system can
: prove its own consistency, meaning that no sane mind can prove its own
: sanity. Also, since that same law states that any system able to prove
: its consistency to itself must be inconsistent, any mind that believes
: it can prove its own sanity is, therefore, insane.

Goedel's proof involves self-referential statements. The only particular
statement that he shows can't be proven within system X is "This
statement can't be proven from system X", I statement I'll call GX
(the Goedel statement for X) or one that maps 1:1 to that statement --
a GY in system Y that maps 1:1 with a "I can't be proven" GX in system X.
(Where Y is to GY as X is to GX.)

If GX is true, then it's true that it can't be proven, so X must be
incomplete -- we know of something true that can be expressed in X's
language that can't be proven within X.

If GX is false, then we are saying it CAN be proven -- and yet it's
false. That would make X inconsistent -- it can prove a falsehood.

So while "an undecidable statement does not need to be explicitly
self-referential to be undecidable" it must be implicitly self-referential
-- ie map 1:1 to a self-referential statement to be provable to be
undecidable.

Second, I would not assume without proof that our minds are formal
systems. Consciousness, unlike math formulae and algorithms, might well
be exempt from Goedel's analysis.

However, I think Goedel offers a good mashal for halakhah...

Eilu va'eilu divrei E-lokim Chaim (DEC) -- the halachic process is
inconsistent, it allows proof of both Beis Hillel's position and Beis
Shammai's

vehalakhah keBH -- the halakhah pesuqah is not a closed finite system.
The intial is incomplete, and we grow the halakhah based not only on
the halachic system itself, but also on matters like the middos of BH,
their numbers, and other new information.

So, the DEC level is intentionally inconsistent. At any one moment, the
halakhah pesuqah level is incomplete -- there are always new situations
not yet encountered and not yet pasqen-ed upon. And that incompletion
is closed using a broader process that isn't a closed finite system. In
no sense of the word "halakhah" do Goedellian limitations set in.

Tir'u baTov!
-Micha

-- 
Micha Berger             A cheerful disposition is an inestimable treasure.
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Fax: (270) 514-1507         - R' SR Hirsch, "From the Wisdom of Mishlei"