FOM: "conservative extension"?

[Robert Tragesser]

        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 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 "contrdiction" 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 in terest and
relevance.

robert tragesser
westbrook,  connecticut