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What is infinite except that which is NOT finite? To ignore the role of the complement (ignore its absolutely foundational role…the only absolute: there is no absolute) is to ignore everything related to it…including the infinite. Therein is the contradiction in the whole of “Cantor’s Garden”. It ignores the “orthogonal” point of view for whichContinue reading “How fundamental is complementation?”

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. EveryContinue reading “The Infinite Truth Table”

A measurement amounts to a determination that a handoff of information is possible…if so, the handoff happens…without fail…as in the behavior of a particle in a potential …the trajectory is that of least action. The limiting features that determine possibilities for handoff are clock frequency and relative bit size ….scale (gear mesh model…pitch of theContinue reading “The handoff”

There are label on vertices and labels on branches The labels on the branches (either 0 or 1) yield a vertex label which is the record of the path through a series of labeled branches that leads to that vertex. Standard Binary tree the branch labels follow a two bit pattern …0-1 | 0-1 |Continue reading “Tree structure refresher”

The discrete derivative tree yields a reordering of labels on vertices in each row of the standard Binary Tree such that the elements of each row are strings that make a cycle of a single bit change between neighbors. Since the number of elements doubles from row n to row n+1 there is a ratioContinue reading “Period Doubling Tree”

The question of “absoluteness” in mathematics has the same flavor as the old question that so troubled the monarchies of Europe…it’s political, social, economic and religious. Any analysis, especially a “philosophical” one, that disregards that and pretends to be “analytical” is a farce. I’m with the republicans.

If the purported constructions fail, and they do, can we just set up an axiomatic structure that gives us all of the structure in the current theory of the real numbers? No. The theory of complete ordered fields is only decidable with the operations addition and multiplication. As soon as exponents and trigonometric functions areContinue reading “Can we “axiomatize” our way to R?”

Question… In the current theory of the real numbers does there exist a pair of distinct rational numbers between which there is NOT an infinite number of irrationals? No. How then does a “cut” of Q (the rationals) determine a unique irrational number? Usual answer: The sqrt2 “cuts” the rationals into two sets: those whoseContinue reading “The Cauchy Criterion vs Dedekind “gaps””

Both facts and counterfactuals can be modeled as real numbers. The state of every structure in the world corresponds to some real number…counterfactuals are their complements. There is no “shortage” of real numbers. A real number is a row in an infinite truth table the final column of which chooses between fact and counterfactual… thereContinue reading “A “null result” can still constitute a measurement”

How does one define the usual logical operators (think back to your first course in symbolic logic)?…by way of tables. Let’s think of the binary representation of (as a coded expression of the continued fraction form of…) an irrational number (…as provided by the cross-reference capability of the Stern-Brocot tree) as a row in anContinue reading “Binary Trees as definitions of logical operators”