Re: infinity ...

sci.math_20040413:
About the leading zeros, maybe it's one way to illustrate that the
antidiagonal process as applied to an infinite list that is supposed to
be a list of all real numbers is a contradiction in terms, because each
element of the set of reals is distinct from each other, because a set
is not a multiset. Here we are seeing that an antidiagonal different
in representation than each element of an infinite list is not an
element of that list, nor of the range of the function.

sci.math_20040415:
It's kind of interesting that all rational numbers have at least dual
representation as simple continued fractions that are finite. As well,
it appears that an element of a continued fraction might be an
arbitrarily large number. Is it not so that a rational number can be
represented with an infinite continued fraction, in a similar way to
how a root of a power series might be an integer?

sci.math_20040415:
Consider other ways to represent each of the set of real numbers as a
sequence, besides trivially representing each real number as the
sequence (0), representations that give each real number at least one
unique identifying sequence. Congratulations. Most of the sequences
are infinite and most of the arguments about binary integer modulus
representation apply.

sci.math_20040415:
I model only binary numbers where there is exactly and only one
antidiagonal of the matrix of elements thus constructed, all elements
of the unit interval of reals as infinite bitstrings with a beginning,
and dual representation allows the existence of an antidiagonal,
another extension.

sci.math_20040415:
The Equivalency Function is defined in a way thus that all values of
EF(n) except for n=0 are indefinite, and greater than zero, and less
than or equal to one. This is where the set of integers is an infinite
set.

sci.math_20040415:
I might suggest a completely different tack on the antidiagonal: not
that it shows that there is no mapping between a set and its powerset,
but that infinity is dually represented as zero.

sci.math_20040417:
I think more about the antidiagonal of an infinite list of particularly
all possible sequences of binary elements.

sci.math_20040417:
Thus, either a does not exist or a is dually represented. Why would
that not be so? Alternatively, as the reals are uncountable, you might
say, or infinite, x_n, is never the last element of the reals to affect
the antidiagonal, yet x_n is always itself.

sci.math_20040418:
I consider the Equivalency Function, and its list of values, and what
its binary antidiagonal would be. It would start with infinitely many
1's, then, at some point there might be some values that are not 1's,
for around EF^-1(1/2), then, I'm not quite sure what happens. An
alternative consideration is that the antidiagonal would be all ones,
then perhaps some ones and zeros, and eventually all zeros. Another
consideration is that it is ill-defined as all the elements are
indefinite, or that the antidiagonal is .111....

sci.math_20040418:
So say you want to add that to the list and generate a new
antidiagonal. The list elements are sorted in ascending order, so you
have to put any list element you consider in the correct location. If
it's .111..., that's greater than or equal to any element of the range,
and the infinite list has no end, so there's no place to put it.

.

Relevant Pages

  • Re: Courage?
    ... > This same approach can be done with infinite sets. ... The reals are NOT countable however. ... By then writing the number whose first digit ... Have you heard of leading zeros? ...
    (sci.math)
  • Re: Corrective interpretation of real numbers
    ... So it is an exact number without exact representation ... I did not denote infinite periodic repetition but I ... Cantor did so when he presented evidence concerning the reals. ...
    (sci.logic)
  • Re: Is 0.123... a member of the set {0, 0.1, 0.12, 0.123, ...}?
    ... representation of the infinite approach of 0.123... ... > You can define B* to be the set containing the limit of the sequence, ... When an infinite set contains an infinite sequence of points (quasi ... Similarly Pi, sqrt2, and all other irrationals ...
    (sci.math)
  • Re: limitation to the halting proof
    ... there are also rules for manipulating INFINITE ... one only has so many atoms, regardless of representation. ... DEFINE what YOU mean by "countable reals". ... the UTM takes TWO n's as arguments. ...
    (sci.logic)
  • Re: Cantors diagonal proof wrong?
    ... It's an infinite set, there's always one more. ... One representation of the binary expansions to represent a number is ... Consider the expansion as representation in a finite "base", ... gaps, the reals are not complete without the line, taking into account ...
    (sci.math)
Quantcast