Metrics of Infinity

I am using the word "metric" here even though my chemical background used "measure".   This is in recognition of the technical mathematical term "measure" which plays a role in integrating certain discontinuous functions.   I clearly do not understand this term,   but also do not wish readers to be confused by any misuse of it,  so use a different word.

Chemists  (at least those teaching freshman classes in my day)  are careful to distinguish two broad classes of metrics:   extensive and intensive.  Extensive metrics indicate the size of a sample in some way and include weight,  length,  volume,  and moles or equivalents.  Intensive metrics involve the ratio of two extensive metrics.  Intensive metrics include concentration  (e.g. moles per volume),  specific gravity  (weight per volume),  pressure  (force  (weight)  per square metric),  partial pressure (pressure of one gas in a mixture of gases as a fraction of the total pressure),  and many more abstruse metrics  (who remembers partial molal volumes?).

This distinction between extensive and intensive metrics helps avoid confusion.   It allows a chemist to distinguish between the quantities of chlorine and bromine available  (as anions)  in sea water even though for practical purposes the sea contains an infinite amount of both.  Perhaps it is this confusion of metrics which led to the historical aversion by mathematicians to the subject of infinity, and it certainly has led to many of the paradoxes involving infinity.

Mathematicians seem wedded to an extensive metric of infinity, using cardinality and/or countability via one-to-one mapping to measure infinity.  Thus they can conclude that all the positive integers,  all the positive even integers, all the positive odd integers,  and,  indeed,  all the positive integers congruent to any integer modulo any larger integer have the same degree of infinity.   This viewpoint motivates the statement that my view of the abstract Collatz predecessor tree involves an infinite set of infinite sets.  (The set of  l.d.a.s  grows without limit as the abstract tree is pursued to greater and greater depths and every element in every l.d.a.   shows itself to be an infinite set as its n goes to infinity.)   This certainly seems to make it a serious challenge to prove the conjecture.

Many of the paradoxes cited to demonstrate the peculiar properties of infinite sets derive directly from contrasting the use of an extensive metric with that of the intensive metric.  The latter seems to be more intuitive in many instances. Indeed,  the first paradox encountered in this work exactly mimics some classical paradoxes.  These are clear examples of the result of confusing two different kinds of metrics on infinite sets.

If we stick to an intensive metric to develop the key property of the infinite set of positive integers we will avoid this confusion.  We will employ the notion of the density of the integers appearing in a set as compared to the maximally dense integers.  This is calculated as the ratio of those from the set in some interval divided by the number of all odd integers in that same interval.   Just as in the Principle of Cavalieri  (cited as "quite rigorous" for its purpose)  the use of a ratio to get a result from what is otherwise an intractably infinite situation,  the use of densities in this work will prove invaluable.

Just as we can add partial pressures in Chemistry to show we have made a complete gas analysis,  we can add the densities of sets of odd integers encountered among the l.d.a.s to show the presence of all the positive odd integers in the abstract predecessor tree. .

Clearly the density of the odd and the even integers among all the integers are both exactly 1/2 so that the total density of the integers is 1.  Similarly, the sum of the densities of 0[3],  1[3],  and 2[3],  each exactly 1/3,  is 1.  Were we to sum the densities of two sets of integers.  e.g. 0[2] (density 1/2) and 1[4] (density 1/4),  we would learn that 3/4 of the integers were accounted for between them and that another 1/4 were missing.   This simple addition of fractional densities is dependent on the sets whose densities being added are mutually disjoint.  This is true for all the elements of all the l.d.a.s.

We need to sum the densities of the odd positive integers resulting from a large number of sets characterized as dn+c,  i.e. integers in {c[d]}.  We will start the ranges over which the densities are measured at 1,  in consideration of 1 being its own predecessor at the root of the predecessor graph.  The integer 1 is counted only once in that one of the l.d.a.s headed by the set of extensions  {1, 5, 21, 85, ...}.  

An arbitrary selection of the interval in which the densities are calculated might lead to inaccuracies because at the end of an arbitrary interval,  some sets' entries will have appeared and others' will not have appeared.   However,  at the end of each interval of length d,  at 0[d], the densities of the odd integers are accurately 2/d.   The set densities produced from any two d's will be accurately determined at 0[d*d'].   The sets arising from a large number of d's will be accurately determined at 0[(d*d'*d"*d'"...].   By considering integers out to the productof the d's,  we can ensure that the summed densities will not suffer from inaccuracy.

It developed that a table of cardinalities of the appearance of sets with a given  (a,b)  value in sets represented as c[2^a*3^b]  had properties which enabled a closed infinite summation resulting in exactly the total density required for all the odd integers.  Thus, using densities as the metric, we have managed to calculate the important property exhibited by the infinite set of infinite sets.

Another of Infinity's Tricks

Cantor's famous work involves iteratively taking away the central third of the interval between zero and one and then the central third of the two outer thirds,  and so on.  He then sums 1/3 + 2/9 + 4/27 + 8/81 + ...  (which sums to 1).  This is reminiscent of one of my summations, i.e.  that of the arrival of  l.d.a.s at leaf nodes in the abstract predecessor tree.   I claim that this indicates that all integers do appear in the abstract tree.  But Cantor was able to show that numbers remain in his interval,  in apparent contradiction to common intuition and to my hopeful claim.

The difference is in our basis sets.   Cantor's includes the real numbers as well as the rational,  whereas mine is restricted to the positive odd integers.   So when Cantor takes out an interval,  he leaves "dust",  but when I take out 1/3 of the integers,  only integers are left,  and there are no real numbers ever involved to survive as "dust".   Therefore,  I do not believe this example of an infinite summation giving unity without including all the items in the set is applicable to my work.

Similar observations apply with respect to the mathematical term "measure" which applies to sets of numbers which are mostly continuous, but contain some discontinuities.   The discussion involves taking limits on both sides of the discontinuities, an activity which clearly does not apply to sets of integers.   We must not be dismayed,   therefore,   when a mathematician tells us he knows of sets whose measure is zero yet contain an infinite number of points,   implying that when I find that the sum of the density of integers represented in the l.d.a.s is 1,   that the residual set   (metric zero,   of course)   may still be infinite.   That whole argument overlooks the fact that the Collatz process visits only integers.