The awesome aspect of the 3n+1 conjecture is that it involves the infinite set of positive integers. Infinity is an awesome concept, and seems to be difficult to deal with in convincing ways. Since this work has had to face the difficulties of infinity, and since various possible approaches have arisen through a period of time, it seems worthwhile to draw all these possibilities together on this single page, either by reference or by a brief discussion herein.
We have discussed the importance of distinguishing intensive and extensive metrics and pointed out that the use of the density of integers in infinite sets avoids the dangers of adding and subtracting their cardinalities. The whole process of summation of the densities of the integer sets in l.d.a.s to determine that the abstract predecessor tree does contain all the positive integers is critically dependent on the validity of the intensive metric, density.
We have devoted two pages to a sieving methodology based on the observation of the integers included in l.d.a.s, and developed into an a priori sieve based on the contents of all the l.d.a.s in the abstract predecessor tree. Since it is in the nature of sieves to exhaustively cover the universe under investigation, the apparent success of this sieve lends support to the notion that the abstract tree does contain all the positive odd integers.
We have produced an algorithm which places any arbitrary positive integer into its location in the abstract predecessor tree. This algorithm depends only on the simple arithmetic properties of multiplication (by 3 or 24), division (by 2 or 3 or 4 or 8), addition (of 1), and subtraction (of 1). Such operations clearly apply to very large numbers with the same force that they do to familiar numbers. It immediately follows that the abstract tree does contain the complete set of odd positive integers. In effect, we have shown how to map the positive integers into the contents of the abstract tree. This mapping is the very basis for proof of set equivalence and completely avoids problems with the cardinality of the sets involved.
Another approach to proving infinite content of a set of positive integers might be to develop a scheme which arrays them in so systematic a way that it is persuasive that all are included. [Any such would be the basis of an inductive proof.] Such a scheme arose during characterization of the contents of the extensions in the binary predecessor tree (containing only odd integers) and in the right descents in a binary predecessor tree containing both odd and even positive integers. Effectively, use of a polar plot of log4 of the odd integers 1 for the former and of log2 of the even integers for the latter makes it clear how each successive order of magnitude (base 4 or base 2) fills in the radial space in ever greater detail which continues without limit. Since these polar plots are suggested by features of either style of the binary predecessor tree (not the abstract tree this time), they suggest in a more direct way (i.e. avoiding any necessity to map from the abstract tree to the binary tree in detail) that the binary tree does indeed contain all the required positive integers.
All these approaches to the problem of infinity in the Collatz 3n+1 conjecture point in the same direction: The Collatz predecessor tree does contain all the positive integers. If this were proven, the Collatz conjecture would be proven.
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