FOM: Conservative extensions and consistency: reply to Tragesser

[Joe Shipman]

Tragresser:
>>--- I feel in somewhat the position of someone who can only explain
'consistency' as "freedom from contradiction",  but who also sees (but
without benefit of a clear and distinct idea) that the notion of
consistency is far richer and more resonant and multifaceted than
"freedom from contradiction" could possibly suggest,  and indeed that
there are important contexts in which "consistency" is sought but
"freedom from contradiction" is a mere side issue if not of little
interest and
relevance.<<

"Consistency" is nowadays officially defined as "freedom from
contradiction", a negative attribute, but it also has the positive
connotation of "being instantiated (or satisfied) in a model".  Since
Godel's completeness theorem showing the equivalence of noncontradiction
and satisfiability for the predicate calculus, it has been easy to
ignore this distinction.

>>The notion "conservative extension" has acquired the reality of
centrality and powerful significance in logical investigations in the
foundations of mathematics.   To this outsider ranging widely through
the literature,  'conservative extension',  like 'consistency',  seems
to be a
highly resonant and multifaceted notion,  with different implications in

different contexts,  but which I,  an outsider,  find difficult to draw
together.  Could someone explain/expose something of the history and
more particularly something approaching the full sheaf of various
"implications" of this "property" [as a foundational concept]?<<

I won't attempt a full answer to this, but in my view the essence of
"conservative extension" is "relative consistency" where "consistency"
is viewed in a positive sense.  A conservative extension result takes
the form "axiomatic system X+Y proves the same S-sentences as system X",
where S is a syntactic criterion.  If S is perfectly general (all
sentences are S-sentences) then this just amounts to saying that Y is
already derivable from X.  On the other hand, if S is the simplest
possible type of sentence, namely a logical term ("true" or "false")
this is simply the relative consistency result "if X is consistent so is
X+Y".   But relative consistency proofs usually show more; thus Godel
showed "ZF+AC+GCH proves the same arithmetic-sentences as ZF" and we say
"ZF+AC+GCH is a conservative extension of ZF *with respect to
arithmetic-sentences*".  This is more interesting and informative than
"ZF+AC+GCH is consistent if ZF is".  Cohen showed that in this case you
can't replace the conservative extension result with a full derivability
result (replace "arithmetic-sentences" with "sentences").  (Although you
can indeed derive AC from ZF+GCH).

The classical type of conservative extension Hilbert had in mind was the
"introduction of ideal elements" and he justified the free use of these
extensions by their conservativeness over e.g. arithmetic.  This allowed
theorems to be proved by the method "extend the system with ideal
elements, prove the theorem in the new system, and apply the metatheorem
to convert the proof to one acceptable in the old system if anyone
insists".  One need not have the same ontological commitment to theorems
of the expanded system which talk about the fictitious new elements.

Can anyone suggest a non-contrived example of two incompatible
conservative extensions?  (By non-contrived I mean that both of them
should allow for the easier derivation of some sentences in the base
theory.)

-- Joe Shipman