FOM: contradiction-free vs consistent

Stephen G Simpson simpson at
Mon Feb 15 16:41:15 EST 1999

A reply to Robert Tragesser's postings of 29 Nov 1998 08:02:39 and
1 Dec 1998 09:55:59.

 > Topic: the identification of consistency with
 > freedom-from-contradiction seems now to be rather appalling, hiding
 > a comos, or a cosmology, of powerfully significant problems and
 > phenomena.

The background here is that it's usual in f.o.m. since Hilbert to
identify consistency with freedom from contradiction.

 > For example,  it is absurd to say that contradiction-free-ness 
 > entails being, ...

Absurd?  What about G"odel's completeness theorem?  This basic theorem
of mathematical logic says that a theory T in the predicate calculus
is consistent (i.e. contradiction-free) if and only if there exists a
model of T.  In this sense, contradiction-free-ness *does* entail
being.  It appears that if Robert wants to dispute the identification
of consistency with freedom from contradiction, then he needs to
somehow dispute G"odel's completeness theorem.  Robert, what do you

In Robert's defense, it occurs to me that one could try to dispute
G"odel's completeness theorem on Brouwerian grounds, because the
models of T that are produced are not always `unique' or
`mathematically specific' or `constructive' in various senses.  For
instance, even if we assume that T is recursively axiomatizable and
consistent, then it is well known (Kleene) that there always exists a
Delta^0_2 definable model of T, but there does not always exist a
Delta^0_1 (i.e. recursive or computable) model of T.  Furthermore,
it's easy to see that the set of sentences of predicate calculus that
are true in all recursive models is not only not recursively
axiomatizable but not even arithmetically definable (I think Vaught
published a note proving this many years ago).

Robert, does the previous paragraph address your concerns in some

 > Of course, what I driving me is the great discomfort I've always
 > felt with mathematics that is too logic driven, perhaps because I
 > place a great value on understanding in some full or perhaps
 > melodramatic sense.

Shades of postmodernism.  I truly don't understand this remark.
Mathematics is too logic-driven???  It seems to me that logic greatly
enhances our understanding of mathematics.  Logic is certainly *not*
an attempt to evade the need for `full' or even `melodramatic'
understanding of whatever topic is under discussion.

-- Steve

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