[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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