Infinity

Infinity is not a ‘thing’…it’s a relation.

Vector Space

• The square roots that come out of the tau(p,q) spaces serve as the eigenvectors of an infinite dimensional vector space. That space is founded on a 2D integer lattice, (p,q).

• The lattice gives rise to a countable collection of vectors defined via the square root

• The square roots form a basis for the collection of vectors that show up as ‘Real numbers’ when viewed via radix notation

• The collection of binary strings we see as the result of DD does not represent the same thing as that given by subdivision (by 2’s), even though it uses the same binary coding….it is not a rearrangement of that subdivision…it refers to an entirely different space

• The S-B tree provides a link…a means for translating between the ‘languages’ via continued fraction forms of values…in which form all strings of repeating patterns correspond to square root functions….those functions form a countable eigenvector basis for a vector space.

• The square roots involved are generated by the logarithmic spiral. 1,1->sqrt(2) || sqrt(2),1-> sqrt(3) || … and on and on out to the circle at infinity where the ‘1’ becomes the infinitesimal step along the circle

• The logarithmic spiral generated by tau is a model of that spiral that is bounded in projective space by the line at infinity…it yields the cross ratio with the proportions 1:2|3:6…not a coincidence that it comes out of the 2,1->sqrt(5) step….it links the division of the elements of the S-B tree into powers of 2 and powers of 3…and leads to Sharkovskii’s observation….

• The 2 and 3 of Sharkovskii’s order are linked back in a kind of Möbius strip yielding an infinite regress…an infinity of infinities of infinities…a cycle nonetheless…2^(2^ n)… the number of possible ‘truth table’ functions

KEY IDEA

• KEY IDEA: the reordered strings are not representing the same thing…thus the fact that one is countable and the other not is not a contradiction…even though both are the set of infinite binary strings and one is ‘generated’ from the other …only a paradox…it describes exactly what Skolem showed in the L-S Paradox: if a countable first-order theory has an infinite model, then for every infinite cardinal number κ it has a model of size κ, and that no first-order theory with an infinite model can have a unique model up to isomorphism. As a consequence, first-order theories are unable to control the cardinality of their infinite models. (Wiki on L-S Paradox)

• …that is, what we think the language is describing is not uniquely determined…it may be interpreted many ways

• The difference is the domain of the Language

• We go from a two dimensional domain (base 2) to an infinite dimensional domain (square roots of n in N)…a ‘meta’ level

• The strings of the DD version are a representation of the countable eigenvector basis for the set of binary strings we see in base 2…practically speaking they form a basis for Q (which we see in the S-B tree)…and R is a vector space over Q

• The Minkowski ? function gives us a link in the form of a fractal function…it reduces the infinity of infinities of infinites by one ‘layer’…3 -> 2…. 2^(2^n) -> 2^n …that relation is enough to collapse …((((2^(2^(2^(2^(…2^(2^n)))))) …notice the infinity at the middle and end…really a circle of countable steps: N sandwiched in between…N is the foundation of an infinite ‘bifurcation’

• Given Sharkovskii’s ordering we can support the ITV…the implications are bi -conditional …trouble is, that ordering is ‘punctured’ by an infinity of infinities…thus it offers only a conditional (lower level) ‘completeness’…the universe really is full of ‘holes’…no boundary is perfectly impermeable

• The logarithmic spiral provides a means of determining every possible ‘direction’ in projective space since we have ‘access’ there to a step of ‘1’ on the line/circle at infinity….we have access to the infinitesimal….that infinitesimal is the a and b of the 2×2 matrix representation of a space in geometric algebra… to hyperbolic space via the Dihedrons

The ‘REAL WORLD’

• There are not ‘many worlds’ there are only ‘two’

• They are the worlds we see in the Elitzer-Vaidman bomb thought experiment: the one world where we find our two ‘real’ photons and the other one where there are two ‘virtual’ photons…those two are ‘real enough’ to provide information….we get an interaction-free measurement

• The boundary between the two ‘worlds’ is permeable…photons can go and come in pairs…can ‘trade worlds’