Re: Well Ordering the Reals


Tony Orlow wrote:
> The way I understand inductive proof, if one fact about a member of a set, due
> to its nature, logically implies the same fact regarding the successor to that
> member, then this forms an unending chain of logical implications from member
> to member in the recursively defined set, so that the fact is proven true for
> all members of the set.

For all members which can be reached by a chain of successor
relationships from elements. The problem is that you can't realize this
doesn't get you to things like aleph_0 which
(a) isn't a member of the set N, and
(b) is certainly not reachable by a chain of successor
relationships,
a chain which you agree will NEVER GET TO AN INFINITE VALUE
if it starts from 0.

A chain which NEVER GETS TO AN INFINITE VALUE does not have
an infinite value in the chain.

Yet you believe that there are chains which have start at 0, never get
to an infinite value, and nevertheless contain aleph_0.

There's an old cartoon with a spoof of a math professor. At the top of
the blackboard is a couple of equations, at the bottom is "QED", and
in the middle is "and then a miracle occurs". This seems to be your
method of proof. If there's no way to get from the beginning to the
end, you just drop in 3 dots and claim that the jump happens somewhere
in those 3 dots.

Hence, 0, 1, 2, ... and then a miracle occurs, ... aleph_0 -1, aleph_0.

- Randy

.

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