[FOM] a question about conservativeness
Robert Black
Mongre at gmx.de
Sun Aug 7 10:19:08 EDT 2005
In _On the Infinite_ Hilbert cites Kummer's theory of ideals as
'perhaps the most brilliant (genial)' application of the principle of
the introduction of ideal elements (other examples being complex
numbers and points at infinity). Later in the article it is made
clear that the introduction of ideal elements is only legitimate when
the resulting theory is conservative over the original theory.
But is it in this case? As I understand it, Kummer's theory of ideal
numbers was introduced to get new results in number theory (in
particular Fermat's last theorem, which of course he didn't quite
get). It's hardly obvious that these new results would have to be
also in principle gettable without the theory of ideals. (This of
course with hindsight: we know, as Hilbert didn't, that PA is
incomplete.)
So: is the theory of ideals conservative over, say, first-order PA
for arithmetical sentences? I'm aware that this is a vague question,
since I haven't said what the 'theory of ideals' is, but the many
people on this list who could answer it could also hopefully make it
precise. (For all I know it could even be the case that Kummer's
original version is conservative but the modern Dedekind version
where an ideal over a ring is conceived of as a certain subset of the
ring is not.) Is there some result in revere mathematics that says
just how strong the 'theory of ideals' is and over what and for what
kinds of sentence it is conservative?
Robert
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