FOM: Re: Independent axiomatizations

Sam Buss sbuss at
Wed Dec 17 10:06:33 EST 1997

Neil Tennant proposed a stronger set of criteria for 
independent axiomatizaions.

The independent axiomatizations of PA from my earlier email and
from Harvey's mail certainly can meet this criterion -- this is
obvious from the form of the axioms.

On the other hand, one can formulate independent axiomatizations which
meet the letter of Neil's criterion, but fail to meet the spirit:
for instance, take your favorite independent axioms T, and let A
be fixed logicaly valid sentence.  For each axiom in T, replace
every atomic proper subformula B of the axiom (A and B).
The resulting set of axioms is clearly equivalent to the old axioms
(on a one-by-one basis);  but it badly fails the spirir of the
proposed criteria.

My general impression is that any search for a hard and fast rules
on what constitutes a "desirable" independent axiomatization are doomed to 
failure.  I missed some the fom contributions on this subject, but
I also fail to see why an independent axiomatization is preferable
to a non-independent axiomatization in all circumstances.  The main
example of this being PA, whose natural axiomatization is not
independent.  A second example is the ZF formulation of set theory.
Both of these examples are reflexive theories of course, which might
be a reason why the natural axiomatizations are not independent.

 ---- Sam

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