Re: naive question from a non-mathematician

"Gene Ward Smith" <genewardsmith@xxxxxxxxx> writes:

Denis Feldmann wrote:
....
Now I define Q as the
intersection of all subfields of C (or R.)

There is a much simpler way : Q is the field generated by 1.

True, but less in the spirit of the thing.

Waitaminnit.

There are many kinds of structure, of which "field" is just one,
for which "the substructure of structure F generated by the subset
[or perhaps `sub<structure of another kind>'] X of F" can be *defined*
as "the intersection of all substructures of F containing X".
Certainly it *happens* (i.e., there is a theorem which asserts
correct) that--for fields--this definition is equivalent to
a definition of the form "the set (which turns out to be a
substructure) of all elements of F that can be obtained by
finitely many applications of certain operations characteristic
of the kind of structure in question." But I wouldn't concede
that one of these kinds of definition is better than the other
in general, and I'm not at all sure that a dedicated nitpicker
should concede it even for fields, if we're going to bring up
"the spirit of the thing".

Perhaps it would be entertaining to digress by considering for
which species of structure, in which contexts, which way of
definining "generated by" is better.

Lee Rudolph
.

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