Re: Courage?

Lawrence House wrote:
> For finite sets we can decide whether they have an equal number by
comparison. If we match the fingers of the left hand with those in the
right hand we say both have the same "number" of members.
> This same approach can be done with infinite sets. If we can match
the elements of one set with another we say both sets have the same
"cardinal". By "match" we mean there is a 1:1 mapping of one set ONTO
the other. For example the positive even integers have the same
cardinal as the positive integers because the map n->2n is a 1:1 map
from the positive integers onto the positive even integers.
> Any set that has the same cardinal as the positive integers is
called "countable" so clearly the positive even integers are countable.
It is a little harder to see that the rationals are countable but they
are. The reals are NOT countable however. Even the positive reals
between 0 and 1 are not countable. The proof of this is to imagine
they are countable by putting them, using their decimal expressions, in
a countable list in which the first is the one associated with 1 the
second with 2 etc. By then writing the number whose first digit
differs from the first digit of the first number and whose second digit
differs from the second digit of the second number number etc. we will
have found a number not included in the list.
> This contradiction proves the reals are not countable.

Have you heard of leading zeros?

The idea with that is to prepend to each expansion a zero. Then, the
everywhere-non-diagonal is never an element of the range.

In unary, where necessarily the non-integer portion of the real number
is represented as integer iota-multiples, the antidiagonal is always on
the list via induction. Using "base infinity", the non-integer portion
is again represented as an integer iota-multiple, in one digit, and an
antidiagonal is again on the list via induction.

In base two, or binary, there is one specific anti-diagonal, and
variously you might consider dual representation, or that it shuffles
off of the range. That's particularly so where the bijective mapping
as well satisfies the requirements of the Cantor/Megill style nested
intervals, "Cantor's first proof", as would any well-ordering of that
set of real numbers. The same holds true in trinary, base three.

In base four and higher for finite radix n, that appears to be a strong
argument. While that may be so, it is contrary to the notion that the
set is infinite, inexhaustible and unbounded. About the numbers being
the same in any base, besides that you can add leading zeros, the
numbers are not necessarily the same, at a raw, structural level,
particularly with the numbers as primary objects in the theory.

About something like the set of all sets, and the powerset result via
antidiagonalization of coded binary of the sets' elements, that leads
to conflation, or dual meaning, of some element(s).

With the real numbers, it seems that an enumeration or well-ordering
can only be considered as monotone progression, or piecewise
composition by disjoint segments of monotone progressions. Another
notion is of the zeros of a space filing curve, interleaving of those
elements of the monotone progressions.

If your intellectual thought exercise is to consider a bijection, a
1-to-1 function, between the naturals and reals, I encourage you to
consider this expression of a definite integral:

b
S x dx
a

That equals b - a. Now, here's the part that is bad: when you were
learning calculus and were told over and over that the integral is not
the sum of infinitesimally small areas, that's slightly incorrect and
it is. That's the sum from zero to infinity of dx, iota, times b-a.

It's taught that that explanation is not to be used as the mechanism
behind the analytical integral of standard classical analysis, and it's
perhaps good that that is so, because it took a hundred years for
textbook authors to agree on the limit or delta-epsilon. In more
modern times, there are some texts that are a return to the use of
infinitesimals in the integral calculus directly.

So anyways, for the antidiagonal argument in finite base n greater than
3, I don't yet have a generic response.

Infinite sets are equivalent. There are different meanings of
quantization than rounding.

Regards,

Ross F.

.

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