If you're having math problems, I feel bad for you, son...

[tama]

29. Deep shit ;)

Two excerpts from TGD MATHEMATICAL ASPECTS OF CONSCIOUSNESS THEORY (http://tgd.wippiespace.com/public_html/mathconsc/mathconsc.html), about post-Cantorian notions of infinity:

2.9
Comparison with the approach of Cantor
The main difference between the approach of Cantor and the proposed approach is that Cantor uses only the basic arithmetic concepts such as sum and multiplication and th e concept of successordefining ordering of both finite and infinite ordinals. Cantor’s approach is also purely set theoretic. The problems of purely set theoretic approach are related to the question what the statement ’Set is Many allowing to regard itself as One’ really means and to the fact that there is no obvious connection
with physics.
The proposed approach is based on the introduction of the concept of prime as a basic concept whereas partial ordering is based on the use of ratios: using these one can recursively define partial ordering and get precise quantitative information based on finite reals. The ordering is only partial and there is infinite number of ratios of infinite integers giving rise to same real unit which in turn leads to the idea about number theoretic anatomy of real point.
The ’Set is Many allowing to regard itself as One’ is defined as quantum physicist would define it: many particle states become single particle states in the second quantization describing the counterpart for the construction of the set of subsets of a given set. One could also say that integer as such corresponds to set as ’One’ and its decomposition to a product of primes corresponds to the set as ’Many’. The concept of prime, the ultimate ’One’, has as its physical counterpart the concept of
elementary particle understood in very general sense. The new element is the physical interpretation: the sum of two numbers whose ratio is zero correspond to completely physical finite-subsystem-infinite
complement division and the iterated construction of the set of subsets of a set at given level is basically p-adic evolution understood in the most general possible sense and realized as a repeated second quantization. What is attractive is that this repeated second quantization can be regarded also as a model of abstraction process and actually the process of abstraction itself.

(...)

6.2
Leaving the world of finite reals and ending up to the ancient Greece
If strong number theoretic vision is accepted, all physical predictions of quantum TGD would be numbers in finite algebraic extensions of rationals at the first level of hierarchy. Just the numbers which
ancient Greeks were able to construct by the technical means at use! This seems rather paradoxical but conforms also with the hypothesis that the dicrete algebraic intersections of real and p-adic 2-surfaces
provide the fundamental cognitive representations.
The proposed construction for infinite primes gives a precise division of infinite primes to classes: the ratios of primes in given class span a subset of rational numbers. These classes give much more refined classification of infinities than infinite ordinals or alephs. They would correspond to separate phases in the evolution of consciousness identified as a sequence of quantum jumps defining sequence of primes → p1 → p2 ...... Infinite primes could mean a transition from space-time level to the level of function spaces. WCW is example of a space which can be parameterized by a space of functions locally.
The minimal assumption is that infinite primes reflect their presence only in the possibility to multiply the coordinates of imbedding space points by real units formed as ratios of infinite integers. The correspondence between polynomials and infinite primes gives hopes of mapping at least the reduced WCW consisting of the the maxima of K ähler function to the anatomy of space-time point. Also WCW spinors and perhaps also the the modes of configuration space spinor fields would allow this kind of map. One can consider also the possibility that infinite integers and rationals give rise to a hierarchy of imbedding spaces such that given level represents infinitesimals from the point of view of higher
levels in hierarchy. Even ’simultaneous’ time evolutions of conscious experiences at different aleph levels with completely different time scales (to put it mildly) are possible since the time values around which the contents of conscious experience are possibly located, are determined by the quantum jump: also multi-snapshots containing snapshots also from different aleph levels are possible. Un-integrated conscious experiences with all values of p could be contained in given quantum jump: this would give rise to a hierarchy of conscious beings: the habitants above given level could be called Gods with full reason: those above us would probably call us just ’epsilons’ if ready to admit that we exist at all
except in non-rigorous formulations of elementary calculus!