CARDINALITY theory is incompatible with FORMAL GRAMMARS, ENTROPY ANALYSIS, PROBABILITY & AMORTIZED COMPLEXITY

Let's go back to your argument and focus on the proposition whether a certain string
is likely on the LIST or not, without invoking cardinality.

> The probability of a list of infinite strings of length n
> Containing at least one string whose first r digits are all 0's.

> = 1 - probability of not containing any
> = 1 - (probability given row does not start with r 0's)^n
> = 1 - (1 - probability it DOES start with r 0's)^n
> = 1 - (1 - (1/10)^r)^n
> = 1 - ((10^r - 1)/10^r)^n

> Case 1
> n ->oo
> This goes to 1

> Case 2
> r ->oo
> This goes to 0

> Case 1 & 2
> This is indeterminate

> YOU assume that any given infinite-length-string can be found in any
> random plane. My argument refutes your assumption by showing that it
> is indeterminate.

Substitue 10^2r for the length of the list (number of rows)

The probability of a (list of infinite strings) of length 10^2r
Containing at least one string whose first r digits are all 0's.

= 1 - probability of not containing any
= 1 - (probability given row does not start with r 0's)^(10^2r)
= 1 - (1 - probability it DOES start with r 0's)^(10^2r)
= 1 - (1 - (1/10)^r)^(10^2r)
= 1 - ((10^r - 1)/10^r)^(10^2r)

Case 1
10^2r ->oo [r -> log(oo), r -> oo]
This goes to 1

Case 2
r ->oo
This goes to 1

Case 1 & 2
This goes to 1

> Yes. The number is certainly on the list. The list contains every
> possible string. Excellent.

_________________________

For instance, consider a normal random infinite plane of digits.

According to basic logic, this set (list) of reals should cover all sentences from the grammar

REAL GRAMMAR
--------------------
R = DR
D = [ 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 ]

Yet for every real generated there is a possible ANTI-REAL that could be generated.

e.g.
ROW 7 = 0.1029393848457..
ANTI-7 = 0.4444444454544..

According to basic entropy analysis, the rows of the random plane can be rearranged to form
a new permutation of the list to have any diagonal calculated from REAL GRAMMAR.

According to classical mathematics this is not possible, the random LIST is likely with probability P=1 to
accommodate any diagonal onto the random plane with a new permutation, but there are infinite possible ANTI-r
for each of infinite rows, that are contenders for the diagonal, that prevent this from happening.

> > Given a list containing a real number r, it is
> > impossible to set the diagonal (however defined) to anti-r
> > - William Hughes

Herc
--
Cantor's 2nd proof is exactly "Which box contains the box numbers which don't contain their own boxes?"
There isn't one! DUH! Hence higher infinities must exist! Exactly the same proof as anti-diagonals!

For the love of no god, give it a rest and take your fucking
medication you drooling psycho.

What is the O-time for these operations?

Danilo J Bonsignore

Danilo J Bonsignore

--

The shuffle algorithm to accomate a new diagonal.
best case linear, worst case non computable.

Typical 5 to 10 digit comparisons per row (length of list)

http://freewebs.com/namesort/linux.html

Herc