The last input column of the infinite truth table (one with an infinite number of rows and columns) is given by …010101010101… Its first column consists of two infinite strings…one all 0’s and the other all 1’s. Each subsequent column has twice as many strings of 0’s and 1’s concatenated such that they alternate. Every possible infinite binary string is found in some row of the table.
The columns are very “orderly”…their structure completely determined. An arbitrary row however will have the appearance of complete disorder, infinitely complex. The impression one gets from the table depends on the scale of observation. Taken as a whole there is complete determinism…complete order to the columns: the number of sets of adjacent 0’s and 1’s doubles from column to column all the way to …01010101010… The view across rows, however, is complete disorder with occasional islands of order: the bifurcation tree of chaos theory.
Then there is the final column…the column where one finds the definition of the logical function (operator). If the number of rows is 2n then the number of possible functions is 22^n….and they come in complementary pairs.
But one could turn the whole process around…given all of the 22^n long strings (the history of the universe) and their “semantic” referents…what they “mean”…how they “make sense”, the order in which they were derived/occurred/caused, one could show that the rows that produced them are not arbitrary in their construction, however “arbitrary” the connection between function and meaning may seem.
Consider the ordering of the infinite limit strings in the DD tree as one such ordering (there are many). Each string differs from its neighbor at exactly one bit location. The cross-reference, restoring the order, mapping a string back to the input string that codes its infinite path through the DD tree, would be a monumental task, impossible in fact, but not impossible in “theory”…not precluded by a “proof by contradiction”. Regardless the number of rows, the number of functions is countable. They cannot be counted in the usual order of the counting numbers…they follow the Sharkovskii ordering, the ordering that comes via the doubling of cycles, wherein we find an infinity of infinites.
The “uncountability” of Cantor’s diagonal is an artifact of attempting to subject the set of infinite binary strings to an order from which they do not derive. It completely ignores the role of complementation in the whole structure. That feature of the infinite truth table is present from the first input column to the last. The 0’s and 1’s are present in alternating strings starting with just two such strings in the first column…an infinite string of 0’s followed by an infinite string of 1’s.
Infinity “knows” no limit. The order of the universe is not subject to our idea of order. One will never find it by way of a unary operator alone…it is irreducibly, fundamentally, simultaneously unary (negation) and binary. With one and two we get three. With three we get “chaos”…a sensitive dependence on initial condition….there would be exactly one “seed” and one order, however impossible to find it out.
21st Century Paradox 