A Theory of NumberFor denumerability N of elements n, the probability P(a(f)) = a/n = 0 for finite a. The problems also pertain at P(sub-N) = n/(n + n) = 1. The issue is addition, where n + n = n, etc. Part of the paradox is representational. High symmetry allows radial proximity, so that @ = 2 is a circle, etc. The representation forces a limit function of increasing density, with @2 expressible within any trigonometric arc, mod@. Transfinite numbers define orders of mapping. To conceptualize choice from sets @(N), points are assigned infinite sets, numerical universes multiplying to the touch, pointwise. The point itself cannot be chosen, since a finite choice cannot exhaust an infinite set. P(a(f)), for finite a and infinite sample space, is satisfactorily 0. Each number a is renamable as @(N)-{a}: the universe less itself (a characteristic way of negatively reconstructing a mathematical object) . We represent numbers as holes in an otherwise continuous mapping, as point discontinuities of @ fields. What are the properties of such a mapping? Most telling is its probability theory, above: P(a) = 0, for all a, due to the fractallic density of the field. The object remains spatially opaque--the boundary condition of objecthood. Note: The opacity is perspective, by which numbers are characterized. The form of the distribution is combinatorial, so that the point appears independent of its surroundings, projecting identity. The universe @(N) maps to every point, i.e. each element is deletable, by which it is known.