(Copied pretty much
verbatim from my notes 50 years ago)
The uncountability of the real numbers would not seem to be
derived from “size” or “magnitude” discrepancies between the rationals /
integers –and the reals, but rather, would seem to boil down to a fundamental
question of order – i.e. the impossibility ,
(even in theory), of setting up a procedure, of setting up a continuing
intellectual (ordering) procedure which would present each and every real
number. They cannot all be presented serially -even in theory) –in spite of the
fact that we can (in theory) present in a serial list any given real,
and any list of reals.
Cantor’s proof displays a (variable) real number (via his
diagonal/slash procedure), appropriate to any serial list, (and continuing
intellectual procedure) which is not, and cannot be contained in that
list. Its construction is derived from
the specific serial nature of the particular list itself. It is also, of course, related to the serial
(decimal) presentation of any given real number itself.
Consider this alternative visualization:
(Cantor was allowed the same kind of freedoms in his
diagonal proof that I am claiming here.)
It seems I could set up a serial presentation of all reals
in this manner: set up a hypothetical
line segment of unit length one, (which, of course, is fully sufficient as this
segment can be shown to be in one to one correspondence with the whole real
line). Select a hypothetical dart with a
zero magnitude point, (equivalent in principal to Cantor’s writing out the
whole of a real as an infinite decimal –i.e. exactly, precisely). Throw the dart at the line, (axiom of
choice?), and the point hit is then r1.
Continue throwing the dart for r2, r3, etc. If the dart hits a previously speared point,
throw again for that member. No point is
privileged or exempt a priori! If I were infinitely lucky –which is theoretically
possible, (but infinitely improbable), so that I never duplicated points, it
seems that I might derive such a list in an infinite time. (Cantor gave himself an equal time in the
writing of his list!) Certainly,
though, this shows that the difficulty is not as usually thought and that such
a sequence would be possible except, I believe, for one fundamental
reason, not to do with “size” or “magnitude”.
Rather, I believe it derives from the inherent impossibility
of setting up such a procedure in the first place. Here, -of setting up an intellectual
procedure which will assure that every point on the line would definitely be accounted for –even
assuming infinite luck and time, (because the number of the throw is
always an integer value.) Thus, the
difficulty would seem to derive from fundamental differences in structure
of the reals and the integers/rationals –i.e. of the real and the rational
fields. You could not predetermine the
placing or even if there actually exists a definite placing –given a particular
real in the line –and this seems to be inherently so. This is contrary to the situation of the
rationals, (wherein a correspondence is easily demonstrable using the “zig-zag technique”), and, I think, is the essence of the
difference.
A reiteration on Cantor’s argument
(Much more recent):
“Infinite sets are unique in that they can be put into 1 to 1 correspondence with (some) subsets of themselves. (This is not to say that a given subset may not qualify –e.g. the rationals inside of the reals.) We may even leave out huge - infinite subsets. ( e.g.: 1-> 0, 2->4, 3->6, … -all the odd integers are left out of the second set of integers), but each of these sets is still infinite! “Largeness” is not the issue.
Consider Cantor’s discussion! Two infinite sets “are of equivalent size” precisely if they can be put into 1 to 1 correspondence with each other. O.K. so far- but suppose they can’t, (rejecting the “precisely”). Suppose we are not talking about “size”.
Now consider Cantor’s ‘diagonal slash’ argument. Suppose this reveals the fact that the
rationals and the reals cannot be put into 1 to 1 correspondence not
because they are of different sizes, but because the reals cannot be
ordered like the rationals. Suppose
this is an argument about possible ordering rather than about size, -
i.e. that the reals are incapable of a natural ordering! (Ordering”
had become a big word to me by that time as it became the focus of my
orientation of modern algebra which I saw as the progressive development of all
the possible orderings of ideal and abstract mathematical objects.)
It is certainly amazing that the whole of the rationals –and not just the integers- can be ordered countably as is clearly known and easily demonstrable, but it is an amazing fact nonetheless! But consider: between any two rationals there exists another rational. Between any two rationals there exists a real. But between any two reals their exists a rational as well. Are we talking about “size”? If this is, in fact, not a discussion of “size”, then there need exist only one “infinity” – one unbounded “quantity”-reflecting a statement about the delta/epsilon relationship; it is not a statement about magnitude! But then ordering and structure become the crucial issues!
Nowhere does Cantor’s diagonal argument have anything at all to do with “size”. Everywhere it does have to do with order and ordering. So why take the dubious, more complicated
conclusion over the leaner and clearly justifiable one? It is an argument against imposing a natural
ordering on the reals.
That two finite sets are equal
“in size” just in case they may be (“may be” = “can be”) put in one to one
correspondence with each other is clearly justifiable. But to make the same assertion for infinite sets does not seem to be anywhere near
as plausible. It is trivial, (and
definitional) that any infinite set may be set in one to one correspondence
with some proper subset of itself, (by definition). Are they then of “the same size”? Under the standard definition, (Cantor’s), of
course, they are because of the correspondence.
And yet the original set contains elements, (perhaps even an infinite
“number” of elements), not in its proper subset. It seems to be an equivocal assertion, then,
to assert the converse -that just because two (infinite) sets cannot be set in one to one
correspondence that they are therefore of “different sizes”. The simpler, (leaner), though more abstract
conclusion would seem to be the better one:
simply that they cannot be set in
one to one correspondence! (Occam’s Razor?) This
converts to a statement about the possibility of imposing an order. It seems to me that Cantor’s proof is a
profound revelation about “ordering” and about “correspondences”, not about
size. It elucidates not “size”, I
believe, but rather the impossibility of a natural ordering of the reals. (But whence then his
transfinite sets? Where have the “alephs” gone? DNE? - mathspeak for “Do not exist”?)