Re: infinity


sci.math_030803:
This is different than the "limit of list" (3) argument, which supposes
that as the sequence of the list progresses that there would always be
a number getting nearer the antidiagonal, thus that any antidiagonal
would eventually be generated, it is after all an infinite list and
never terminates.

sci.math_030803:
Thus, from that two different infinite expansions in the same base can
have the same value, infinite lists are easily constructible in design
that contain as an element the number equal in value to one of the
list's diagonal numbers."

sci.math_030803:
Virgil states again "real number not on list" (18), and goes on to say
"diagonal argument says diagonal argument" (17). I say "listed element
is counterexample" (10) and "binary is sufficient" (6). I go on to say
"infinite set is infinite" (20) and "list of reals is list of reals"
(21) about "limit of list" (3) and "real number not on list" (18).

sci.math_030803:
Manukah calls me wrong and then says "ordering of list to derive
element on list" (22) is contrived. I say "list allows reordering"
(23) and "listed element is counterexample" (10) and also "limit of
list" (3)/"infinite set is infinite" (20).

sci.math_030803:
Virgil puts forward again "no point in constructing antidiagonal on the
list" (9) and "antisemitridiagonal" (4). I repeat "list of rationals is
similar to list of reals" (25) and "eroding explanation" (12). I also
put forth some "antidiagonal not in set" (26), and then again "infinite
set is infinite" (20).

sci.math_030803:
I say "rationals can map to powerset of integers" (29) and Steve Leibel
asks how and I say "infinite sets are equivalent." (30) David Kastrup
says "rationals can not map to powerset of integers" (31), I say
"powerset of integers is not powerset of rationals" (32).

sci.math_030803:
Because of that erosion of the argument, its proponents amble down the
slope from its relative footing, as most people would not immediately
discern the flaw in the plain antidiagonal argument, into finding some
new rule. Thus the semi-tri-diagonal argument, with using two of the
three elements of the tri-diagonal of the infinite matrix of the
sequences, is said to constitute some better rule, and the proponents
rest.

sci.math_030804:
The function gamma(z) is also given in Euler's infinite product form:

sci.math_030804:
Anyways that's the easy case, we can say with certitude that there is
only one sequence with all zeros, and only one sequence with all ones.
For having one or more zeros with one or more ones there are infinitely
many.

sci.math_030804:
All of these numbers are rationals that have denominators with powers
of two, we haven't gotten to any with infinite ones and infinite zeros
except as we have been discussing them and about half of them.

sci.math_030805:
What would these probabilities tell us? Well, it might be possible to
easily derive what their limits are as they are evaluated for large n
as multiplied by other expressions and then to prepare a whole bunch of
forms with various fractions so that if/when the (2 sqrt2) / sqrt(n Pi)
case for even numbers of ones and zeros is proven that all the other
identities could be proven at once, giving an infinite family of
expressions for n! as n->oo in terms of 2^n and ((qn)+-x)! and
(((1-q)n)-+x)! for rational q.

sci.math_030806_b:
Another thread today on
sci.math is discussing "probabilities that equal zero". Pick a number
between zero and one, I'll probably bet it's one half, with probability
at least infinitesimally greater than one half. After all, the average
value of all of the reals between zero and one is one half. Of any
characteristic of the amount of ones and zeros of any infinite string
of coin tosses, it is more probable that it has nearly 1/2 ones and 1/2
zeros than any other fraction. Then again, on "pick a number between
zero and n for unknown n", I pick four.

.

Relevant Pages

  • Re: abundance of irrationals!)
    ... No such "sum" exists except as, and only as, the limit of the infinite ... If an infinite sequence does not have to "achieve" its limit, ... > It is only for lists not initially known not to be surjective that ...
    (sci.math)
  • Re: In a Bad Mood
    ... >> the same digit can occur). ... > There is no unique sequence of digits to anti_diag. ... does the infinite list map to the entire number line? ... but the LISTS INFINITE FORM is required to ...
    (sci.logic)
  • Re: Request for Review of ZF Inconsistency Proof
    ... denote the infinite set of denumerable binary strings with alphabet ... sequence of 0s and 1s (or alternatively, ... "tail of zeros"), via reverse binary notation. ...
    (sci.logic)
  • Re: limitation to induction on finite bounds
    ... >told the infinite form of the set you break down. ... given your above non-standard "fairy maths" rule about infinite lists of ... sequences, here's another go at showing that the infinite sequence .333... ... let every member of the infinite list be an *infinite* sequence: ...
    (sci.logic)
  • Re: limitation to induction on finite bounds
    ... >told the infinite form of the set you break down. ... given your above non-standard "fairy maths" rule about infinite lists of ... sequences, here's another go at showing that the infinite sequence .333... ... let every member of the infinite list be an *infinite* sequence: ...
    (sci.math)