Here's a funny way to construct the reals (or rather a redundant
version of the reals, together with an extra element called
plus-or-minus-infinity):
First, given a sequence of subsets A_1,A_2,... of some set X,
say that the sequence "converges" if, for all x in X, *either*
x belongs to A_n for all n sufficiently large *or* x belongs
to X\A_n for all n sufficiently large. We define the limit
of such a sequence of sets to be the set of all x in X for
which x belongs to A_n for all n sufficiently large. That
is, we are using the discrete topology on the power set 2^X.
Now let S be the set of rationals strictly between -1 and 1.
Then, for any sequence of rational numbers r_n, we can look
at the translates S+r_n = {s+r_n: s in S}; if this sequence
converges (in the sense defined above), then it converges
to some set T. The sets T constitute our initial model of
the extended real numbers, which we can then fix by throwing
out a countable set of redundancies (much as we fix the decimal
system by throwing out .999..., or by throwing out 1.000...,
or by decreeing them to be the same).
Putting it differently: the extended real line (with some
redundancy) can be identified with the orbit-closure of S
under the additive action of Q on 2^Q, using the discrete
topology on 2^Q. (Plus-or-minus-infinity is the empty set.)
If we put in a kludge by modifying the definition of convergence
so as to require that at least one element of X belongs to A_n
for all n sufficiently large, then we effectively prevent the
sets A_n from wandering off to plus-or-minus infinity, so we
end up constructing a redundant version of the real numbers,
without an extra element.
Alternatively, we can let X = Q/Z instead of Q, with S equal to
the set of elements of Q/Z with fractional part strictly between
0 and 1/2 (say). Then we don't have to worry about the translates
of S wandering away, so the orbit-closure construction constructs
R/Z from Q/Z (with some redundancy but with no extra points "at
infinity"). Then we can construct R from R/Z (as a wreath-product).
(1) Is this orbit-closure point of view useful for anything? (And
is it new?) You can use it to define the p-adic completion of Q
if you change the setup slightly (replacing the set S by a function,
namely the p-adic valuation function, and by letting Q act on this
function in the obvious way). You can also try this approach when
Q is replaced by any group. Can other interesting completions of
groups be gotten this way?
Also:
(2) Has anyone ever needed to use a number system that contains
two or three copies of each rational number and one copy of each
irrational number? (You can get such a system from Dedekind cuts
in a fairly obvious way, but if you want your cuts to be closed
under algebraic operations, you need to enlarge your notion of a
cut to include pairs in which the left-set includes everything
less than some rational number q, and the right-set includes
everything greater than q, but q itself is in neither half.)
I can't think what this number system is good for, but it must
be good for describing something.
(3) Order ideals (sets that are closed under "under") in Q that
aren't the empty set, or all of Q, come in three flavors:
ideals of the form {r: r < q} for some rational number q;
ideals of the form {r: r \leq q} for some rational q; and
ideals of the form {r: r < x} for some irrational number x.
Non-trivial order ideals in R come in two just flavors:
those of the form {x: x < y} for some y, and those of the
form {x: x \leq y} for some y.
Are there any ordered sets (or "generalized ordered sets" of
some kind --- I choose to be purposefully vague here) that
look something like the continuum, but have the property that
non-trivial order ideals come in only one flavor?
If there were such a continuum, it might be weird in some
respects, but it might in other ways match the naive properties
of the physical world better than the real numbers do. After
all, when we cut a piece of paper into two pieces, it makes no
sense to ask whether the dividing line got included in the left
piece or the right piece!
(4) How does one's picture of the real line change if one adopts
some version of constructivism? (Does the constructivists'
real line have the sort of properties I am looking for in (3)?)
Jim Propp
