Count and Measure

• There is only one certainty: There is no such thing as a perfectly impermeable boundary (all are conditional, porous)

• Sharkovskii’s ordering of N as an example of a porous boundary: an infinity of infinities

• The BITXOR/discrete derivative completely changes the order type of the set of infinite binary strings

• A favorite argument by those who defend Cantor’s diagonalization is to insist that the order types (N and ‘R’, infinite binary strings) don’t match …no bijection. They never once consider that the set of binary strings can be reordered.

• When the set of binary strings is considered as the set of DD’s, what is not obvious is that the ‘initial value’ is missing…complementary strings have the same DD…hence complementary pairs cannot be subject to ‘choice’ (Fraenkel 1927), chosen ‘separately’…the information that would make that possible is ‘missing’, out of reach, unavailable, hidden,…

• Instead of pairs of complements of whole strings symmetrically arrayed in the interval [0,1] (which, counter intuitively, results in complete asymmetry…the binary tree is the Möbius strip: 00->01->10->11->00) we get complementation of exactly one bit in exactly one location in the string (resulting in complete symmetry…of two kinds: at 1/2 and 1/3…the ‘symmetry’ at 1/3 ‘ascends the tree’ at 1:2 proportions…after the initial value, of course)

• In the usual binary order (of the strings representing elements in [0,1]) the only complements exactly next to each other are those at the powers of 1/2…all others are arrayed displaced across 1/2

• DD results in N copies of itself where the only thing ‘swapped’ (reflected/complemented) on either side of 1/2 (in the new, reduced by 1/2 version…look at the left and right edges of the pattern in the picture) is the initial value instead of the whole string being reflected/complemented across 1/2

• In a very real sense we get ‘access’ to the other ‘end’ of the string …since those ‘ends’ (when arrayed together with all the others in the S-B/binary tree) is nothing more than a horizontal string of 010101… the tree has been inverted…we see its ‘roots’

• The horizontal string of all 0’s has no complement…neither does the set of all sets not members of themselves …the ‘empty set’ does not, cannot, exist…(there is no ‘outside’)…it is the quintessential set that is not a member of itself.

• Look at the structure of the Von Neumann universe…built up from the empty set….what do you see: the set/universe of all sets not members of themselves …oh sure, it’s not a set 😂…to quote Wittgenstein, “it’s a joke”…“Mathematics is ridden through and through with the pernicious idioms of set theory”….“the way people speak of a line as composed of points”….“a line is a law and isn’t composed of anything at all”

• The whole structure of the set of binary strings changes

• BITXOR of 101010… as an example of cyclic structure…of ‘the doubling route to chaos’: after an infinite number of steps we arrive back at an infinite string of 101010…, then an infinite string of 1’s, and finally back to 10000…

• While that cycle certainly does not contain all possible strings, it does show how an initial ‘horizontal’ cycle leads to a ‘vertical’ cycle

• A ‘random’ string is a cycle of frequency 1 and infinite wavelength, 101010, is the shortest possible non-trivial cycle

• Establish that there exists a way of representing the rational numbers that does not involve radix notation of any particular basis (particularly base 2), rather it involves nothing more than counting and repeated division…division of the unit by a string of not necessarily identical divisors: the binary sequencing of the Stern-Brocot tree

• Note that base 2, is based on repeated division by 2 and note that sqrt(2) …(everybody’s favorite irrational number…it’s the only one ever mentioned when presenting Dedekind cuts)…has the repeated division [1;2,2,2,2,…]

• What we need is a way of showing the relationship between the local structure and the global… between count and length. We need a link. GM provides such a link: it provides an ‘internal’ ratio that is also the ‘external’ ratio…1:2 as 3:6…in binary its code is 101010…2/3…as a Gray code it is 1111…and that string of ones cuts the number of code words at what approaches the 2/3’s mark (as shown by the Lichtenberg sequence) as their length approaches infinity …a link between subdivision and count

• The connection to ‘four pieces’ is that the Gray coding is a result of the reversal of every other 01 pair in a binary subdivision…you need four to express/accomplish it, two won’t do…hence the contrast with the form of the binary ‘truth table’….instead of an Aristotelian truth table we get one with a hole in it: 1011 = 2 in base tau

• “How do you propose to count them (the set of infinite binary strings)?”…answer: “I don’t propose to count them (since I really don’t have access to them to do so… they are infinite after all, and I am not), rather I will have them count themselves”…internal proportion the same as the external (projectively…the circle at infinity…the cross ratio)

• Contrast the two ways of representing the rationals in [0,1]: S-B tree vs radix notation

• Show that the only element in S-B mode of representation that involves a single ‘basis’ is the Golden Mean…all others are mixed divisors/indices in repeated division…it is the only candidate for a ‘canonical’ basis that can commensurate ‘count’ and ‘measure’…it is the only one that involves an ‘even’ division….division of both number and measure into parts that are exponentially related.

• Show that that division of the set of all binary sequences is in essence division into three exponentially related parts…that two consecutive exponents results in a division into three ‘equal’ parts…those parts ‘telescope outward’ …vertically, not horizontally….thus relating two different divisions: simultaneously into two ‘parts’: x^1 and x^2 and three parts: [00000…, 011111…, 111111…, 1] and show the relation to [00000…, 010101…, 101010…, 1]

• Show that the ratio of the number of sequences to the left of the string of 1’s in the Gray code version of a set of binary sequences vs the number in the whole set approaches 2/3 as n -> oo …why can we expect the order type to ‘allow’ a count?…the order type has completely changed

• Discuss the Lichtenberg sequence, ‘ruler sequence’, the Towers of Hanoi, and the Sierpinski triangle (original construction)

• Show that therefore the S-B tree presents a subdivision of the set of binary strings of a different kind: one founded on counting and division rather than division alone.

• Show that ’uncountable’ (as usually construed) implies there exists a line segment (in [0,1]) that cannot be divided into four pieces….a refinement of the argument of Vitali (who assumed the usual ideas regarding the continuum represented as the set of infinite binary sequences) that does not depend on the Axiom of Choice

• Dividing a segment into four pieces provides a means of producing a Gray coding of the rational numbers…a countably infinite Gray code….that ordering extends to the whole set of binary strings rendering it countable

• If a number is expressible as a binary string, then its discrete derivative is an element in a Gray code….If a number is not an element in a Gray code, then its anti-discrete derivative is not expressible as a binary string.

• The continuum cannot be faithfully modeled as a set of binary strings…the map is not the territory.

• The continuum is non-Archimedean

• Divisible into four pieces <-> countable …uncountable <-> not divisible into four pieces

• The set of all binary sequences can be usefully modeled as a countable infinity of countable infinities….w^w

• Show that the representation of the continuum as the set of ‘infinite binary sequences’ is justified by nothing more that ‘utility’

• Consider the fact that ‘utility’ is itself conditional: it depends on a culture’s ‘system’ of values.

Addendum

• I propose a more honest model of the continuum…one with holes where in fact we have no idea what is ‘there’

• 101010… is the generator of a group whose binary operators are both arithmetic (addition and multiplication) and ‘geometric’ (reflection and ‘concatenation’)…and whose unary operators are negation and BITXOR (which itself is simultaneously a kind of unary and binary operator)

• Paradox…a contradiction between ’meta’ levels…between local and global views. The essence of the turn-of-the-century controversy over Cantor’s ideas, ’proof’ and set theory in general…as recounted by Gaifman in ‘Naming’

• Related to the mathematics of this ‘view’: How difficult it is to literally ‘keep it straight’ (in your head and on the road) controlling the direction of the back end of a horse trailer when backing up while looking into your mirrors

• Show the relationship to Geometric Algebra (2×2 matrices), infinitesimals, dual numbers, ‘Dyhedrals’ (2×2 matrices), the Möbius strip,…

• ?? If we use r = GM as the base of a kind of dual number system (as in a = a + r…namely Q shifted by r) then for all such ‘values’, a, b, a-b is a rational number…in other words, we are operating within a ‘single equivalence class of R/Q+’…essential element in Vitali’s proof…remembering that topologically R/Q+ is a single point??

• The contrast of our four-piece construction to Vitali’s begins with the idea that the cosets r +Q are not actually disjoint as assumed (given the representation of R as infinite binary strings…that representation is exhausted by Q) ….that there is not enough ‘power’ in radix notation to separate R/Q+ into distinct and disjoint cosets (absent the AoC? …in a constructivist universe?)