Extrapolating linear ratios

Cantor's diagonal argument shows that the anti-diagonal number AD
cannot exist at the n-th place of a diagonalized sequence. From this
fact it is concluded that the AD cannot be a member of the complete
infinite sequence. But this conclusion is invalid unless the ratio of
0 ADs and 1 numbers

0/1 = 0

at the n-th place of the sequence can be extrapolated to yield 0 ADs
for the complete sequence. This conclusion is not self-evident,
because in mathematics we cannot conclude in general from 0*n = 0 that
0*oo = 0 is correct.

Without this possible extrapolation, however, Cantor's proof is
invalid. But if this extrapolation holds, then it shows also that the
number of elements of the set of positive even numbers cannot be
larger than every number of the set.

Let n be the cardinal number of the set. The ratio of positive even
numbers > n and positive even numbers =< n cannot be less than 1. The
minimum of this ratio

|{2(n+1), 2(n+2), ..., 4n}| / |{2, 4, 6, ..., 2n}| = 1

is taken for initial segments of the form {2, 4, 6, ..., 4n}. For all
other finite sets of positive even numbers this ratio is larger. By
extrapolating as above we see that every infinite set of positive even
numbers contains numbers that are larger than the cardinal number of
the set.

This can also be seen by a geometric argument: Counting elements means
to add a step width of one unit per count on the real axis. But every
even number consumes two units on the real axis. Therefore it is
impossible to compress any set of even numbers to a diameter that is
smaller than the number of elements.

Therefore, every set of positive even numbers contains numbers larger
than its cardinal number.

Regards, WM

.

Relevant Pages

  • Re: Extrapolating linear ratios
    ... cannot exist at the n-th place of a diagonalized sequence. ... Without this possible extrapolation, however, Cantor's proof is ... The ratio of positive even ... "Understanding Godel isn't about following his formal proof. ...
    (sci.logic)
  • Re: Six Simple And Reasonable Questions ...
    ... "St. Thomas the Doubter"). ... This creates a ratio for a 1,000aa system of about ... total sequence space minus the sequences that have cytochrome c ... "sequence specificity" and taking the 20th root of a number is simply ...
    (talk.origins)
  • Re: Kenneth Millers Interview with NOVA
    ... first we have to calculate the likely gap size. ... Rather it is the ratio of the number of sequences that have a specific ... by recognizing that more than one sequence (so ... minimum likely gap distance is therefore always the minimum possible ...
    (talk.origins)
  • Re: Direct Experimental Evidence for Non-Beneficial Gaps
    ... So we are all agreed that a random sequence of amino acids is very ... trying to disprove evolution using a static model. ... what is your suggested ratio for CytoC functionality in ... between the size and sequence flexibility of a protein and Dursten's ...
    (talk.origins)
  • Re: Howard Hersheys Challenge of Sean Pitmans Assumptions
    ... biasing and confounding the calculation of "average gap size". ... also makes the use of total sequence space as the denominator absurd ... and makes your claim that the ratio you present is the ratio of ... the "average gap size" for proteins of 100aa in length. ...
    (talk.origins)
Loading