Re: Well Ordering the Reals

Randy Poe said:
>
> 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.

What you and Daryl have failed to grasp is that aleph_0 and omega mean
extremely little to me. They are very central to the standard theory, but in my
mind they do not exist as valid concepts. There are no more smallest infinites
than there are largest finites. For me, despite the fact that one cannot name
any finite step where a successor to a finite is infinite, nevertheless an
infinite number of finite steps will achieve infinity, and so we have infinite
natural numbers. So, I do not agree that the chain starting at the origin will
"NEVER GET TO AN INFINITE VALUE", given an infinite amount of "TIME".

Therefore, given that the fact proven inductively does not have a limit of
FALSE at n=oo, it is true for the infinite case.

>
> A chain which NEVER GETS TO AN INFINITE VALUE does not have
> an infinite value in the chain.
That's true. And if the value changes a constant finite amount with each link
in the chain, you will also never have an infinite number of links 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.
Aleph_0, as a concept of some smallest infinity, doesn't really exist. But, I
do believe that, despite the paradox of the unidentifiable point of change,
that an infinite number of increments will produce an infinite value froma
finite starting point.
>
> 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.

That's about right. There is no finite explanation, and we are finite
creatures. There is no identifiable point, but an infinite expanse of points,
where the transition from finite to infinite occurs. We cannot measure
infinity, but we can talk about what different patterns over the expanse look
like in relation to each other. I can see that this is as intuitively
unsatisfying for you as the disappearing vase balls and the multiplying Banach-
Tarski balls, and the equal proper subsets, and the dense sets equal to sparse
ones, etc., are to me. I am comfortable with the Twilight Zone, and can see
that these gulfs exist in many places between 0 and 1 and between 1 and oo. So,
maybe it's unsatisfying to not have some clear concept of this boundary, but
since there is no boundary, I'd rather live with it than fight it.

>
> Hence, 0, 1, 2, ... and then a miracle occurs, ... aleph_0 -1, aleph_0.
Yes, something we cannot imagine. What is the first point which is a finite
distance, and hence an infinite number of points, from 0?
>
> - Randy
>
>

--
Smiles,

Tony
http://www.people.cornell.edu/pages/aeo6/WellOrder/
.

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