[FOM] typo, choice, and more reals
Andreas Blass
ablass at umich.edu
Fri Jul 18 14:12:00 EDT 2003
First, let me correct an error in my last posting. I wrote that
"the double power set of any Dedekind infinite set (or indeed of any set
that isn't finite) includes a copy of the natural numbers." Here
"Dedekind infinite" should have been "infinite Dedekind-finite".
Second, let me briefly reply to Todd Eisworth's question "what
happened that makes the Axiom of Choice seem so much more reasonable to
mathematicians now than 100 years ago?" For some mathematicians, what
happened was probably the discovery of a lot of proofs that use the axiom
of choice. For me, and probably for most set theorists, what happened was
a clarification of the notion of set. 100 years ago, very few (if any)
people clearly understood the distinction between arbitrary sets and sets
that are somehow definable. I think a lot of the arguments against AC in
the early days depended on an intuition that a set should be definable,
while the arguments in favor of AC depended on an intuition of completely
arbitrary sets. After a while, it became clear that the notion of
definable set would be rather complicated to treat with precision --- one
needs to say in what language the definitions take place, one needs a
formal semantics for that language, etc. So it became natural to accept,
as the customary notion of set, the version that required no definability;
the definable versions were then expressed by saying what sort of
definability is involved. This decision was probably also influenced by
the observation that, given a notion of arbitrary set, one can explain ---
by formalizing a language, etc. --- what a definable set is, but there
seems to be no way to go in the other direction. With the acceptance of
the notion of arbitrary set came the acceptance of AC as a (partial)
expression of that arbitrariness.
Third, I'd like to use Adrian Mathias's posting of June 25 as an
excuse for telling people about another way to construct the reals, which
differs in interesting ways from the Dedekind and Cauchy constructions.
(By the way, I've been told that the use of Cauchy sequences to construct
the reals was first proposed by Cantor. So maybe I should call it the
Cantor construction.)
Adrian wrote that "R is the product of a geometric intuition".
Some years ago Steve Schanuel told me about the following construction of
the reals, which looks (to me) a lot less geometric than the familiar
ones. Schanuel calls this construction the Eudoxus reals.
The construction is based on functions f mapping the integers to
the integers. Recall that such a function is called a homomorphism (more
precisely an additive homomorphism) if the expression f(x+y)-f(x)-f(y)
vanishes identically. Call f an "almost homomorphism" if this expression
is bounded. That is, as x and y range over all the integers, the values
of f(x+y)-f(x)-f(y) range over only a finite set. Call two almost
homomorphisms f and g "almost equal" if their difference is bounded. That
is, as x ranges over all the integers, f(x)-g(x) ranges over only a finite
set. It's easy to check that almost equality is an equivalence relation.
Define R to be the set of equivalence classes. These equivalence classes
serve as the reals in this construction. The algebraic operations are
defined as follows. To add [f] (by which I mean the equivalence class of
f) and [g], just add the functions f and g (i.e., (f+g)(x)=f(x)+g(x) for
all x) and form the equivalence class [f+g]. The product of [f] and [g]
is [fog], where o means composition of functions (i.e., (fog)(x)=f(g(x))).
The ordering can be defined from the field structure (the non-negative
elements are the squares) or by setting [f]<[g] if, for every constant c,
all sufficiently large positive x satisfy f(x)+c<g(x).
The isomorphism from this R to the familiar field of reals is
given by sending [f] to the limit, as x approaches +infinity, of f(x)/x.
Of course some effort is needed to show that this works --- beginning with
showing that this limit exists.
There are several things that I should point out about this
construction. First, it looks remarkably like algebra, rather than
analysis or geometry. The Dedekind and Cantor constructions are obviously
connected to familiar formulations of the completeness property of the
reals as "every bounded nonempty set has a least upper bound" and as
"every Cauchy sequence converges". But there doesn't seem to be anything
in elementary analysis (e.g., in a standard textbook for advanced
calculus) that similarly corresponds to the construction of the Eudoxus
reals.
Also, this construction doesn't presuppose the construction of the
rational numbers. It proceeds directly from the integers to the reals.
(I believe this is the reason Schanuel used the name "Eudoxus reals";
apparently the ancient Greek tradition was to explain things in terms of
integers rather than fractions.)
Furthermore, it uses remarkably little about the integers. It
uses their additive structure, but not their multiplicative structure.
You don't need to understand multiplication of integers in order to
understand this construction of the reals. (You do need multiplication
and rational numbers to understand the proof that these reals are
isomorphic to the usual ones. But you need them to understand the uusal
reals in the first place.)
Finally, a curious technical point: Composition of functions is
not commutative. Yet it gives the multiplication operation for the
Eudoxus reals, which is commutative. This works because, even though fog
and gof can differ, the difference is bounded whenever f and g are almost
homomorphisms. So [fog]=[gof].
Andreas Blass
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