Need example from number theory
Timothy Y. Chow
tchow at math.princeton.edu
Sat Aug 28 22:40:07 EDT 2021
If I understand what Joe Shipman is asking, the responses so far don't
really answer his question. But I think that it's not so easy to come up
with an example. The trouble is that "number theory" at the undergraduate
level deals primarily with the natural numbers, which of course comes
equipped with a well ordering. That well ordering means that we rarely
have to invoke the axiom of choice for anything.
If we broaden our scope from elementary number theory to finitary
combinatorics, then we have a better chance of sneaking in AC. We can
find examples of statements that are equivalent if you assume AC, but
which are distinct if you don't assume AC. For example, consider the
"chromatic number of the plane," meaning the chromatic number of the graph
whose vertices are the points of R^2 and in which two vertices are
adjacent if and only if they are a unit distance apart. If AC is in play
then we have compactness, and the chromatic number of the plane is the
same as the maximum chromatic number of any finite subgraph. There is a
paper by Shelah and Soifer (J. Combin. Theory Ser. A 103 (2003), 387-391)
that shows that the chromatic number of the plane depends on your
set-theoretic axioms.
But one could argue that this type of example is cheating, and that the
"real" combinatorial question here is the finitary question about finite
graphs, where the irrelevance of AC is clear.
Along similar lines, one can phrase some theorems in terms of *unlabeled*
combinatorial objects, and then AC may sneak in if you assume that you can
simultaneously label a countably infinite family of unlabeled objects.
But again, arguably the "real" statement is the statement about the
labeled objects, and so AC isn't really needed.
I honestly can't think of anything at the undergraduate level that "feels
like number theory" and yet where the invocation of AC doesn't feel like a
red herring.
Tim
More information about the FOM
mailing list