-Many paradoxes involving infinity arise from intermixing intensive
and extensive metrics. (*--*)
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-Certainly an extensive metric of these infinite sets gets no grip on the
problem of the Collatz conjecture.
-Presumably, use of an intensive metric (i.e. density, a rational)
will do better.
- Two or three
*such* paradoxes were encountered in this work and resolved by applying
density as a measure.
-Much work revealing problems in infinite sets (e.g.
dust, a set of zero measure but
infinite content) is seated within the continuum of the number line.
-But the Collatz conjecture involves only integers, thus
avoiding this trap involving "measure".
-The assertion that the predecessor tree contains all the odd
integers on the basis of infinite summations of their densities (which gives 1/2)
is grounded in the density measure.
-Summation of the densities of the powers-of-2 multiples of all the odds
in the predecessor tree gives 1/2 also.