[FOM] Is Godel's Theorem surprising?
joeshipman@aol.com
joeshipman at aol.com
Sun Dec 10 02:55:23 EST 2006
-----Original Message-----
From: rda at lemma-one.com
>The commentary says that von Neumann was looking for a "minimal model
of set
>theory that could be uniquely characterised". Perhaps, the intuition
was that
>that minimal model would come equipped with or maybe even be defined
by a
>:complete logic.
Interestingly, there is such a model M, first defined by Shepherdson,
and used to great effect by Cohen. It consists of what Cohen calls the
"strongly constructible sets". There is a sense in which no set outside
of M need exist (indeed, one may view it as a class rather than a set,
and deny its countability; if M is a set, then every element of M can
be "named"), but each set in M must exist in every standard model of
set theory (compare: each set in L must exist in every standard model
of set theory which contains enough ordinals).
In one sense, it's not such a strong assumption to deny M's
countability and view it as a proper class. This is the same as denying
that ZF has a STANDARD model (Godel's arguments show that if ZF is
consistent a model must exist, but the proof gives a nonstandard model)
-- but although it is intuitive to me that ZF is consistent, it is not
intuitive that there MUST be a "standard model" that isn't already all
of V. In another sense it's a very strong assumption since it denies
that an inaccessible exists, while set theorists all supposedly believe
in inaccessibles -- but V=L denies the existence of measurables and
it's still considered plausible by some; V=M has exactly the same
proof-theoretic status (indeed, the proofs of the consistency of AC and
GCH work the same way using M instead of L).
-- JS
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