Based on a discussion of the theory experiment connection it is proposed to tighten the connection by replacing the real and complex number basis of physical theories by sets Rn, Cn of length m finite binary string numbers. Here n=n(t) will be cosmological time dependent. The form of the numbers in Rn is based on the infinite hierarchy of 2n figure outputs from measurements of any physical quantity with an infinite range (distance energy etc.). A space and time based on these numbers is described. It corresponds to an infinite sequence of spherical scale sections Rn,e(e an integer). Each section has the same number of points but the size increases exponentially with increasing e. The sections converge towards an origin which is a space singularity. Iteration of a basic order preserving transformation, F< or its inverse shows exponential expansion or contraction of the space with the origin as a source or sink of space points. The suitability of Rn space as a framework for inflationary cosmology is based on a constant iteration rate for F< and time dependent scale factors e and N=n(t). The time dependences restrict all space initially to a region of scale sections Rno,e with no (small) and e≤eo (negative). Inflation which occurs naturally is stopped at a time when all points in the outermost Δ scale sections are expanding away from the origin at velocities >c. Then n increases to nI > no where nI is such that the outermost Δ scale sections of Rno space are contained in the scale section RnI,o of RnI space. This is needed if RnI,o space is to be similar to the usual R space. Hubble expansion and the redshift are accounted for by a continuing slow increase in n. Comparison with experimental data suggests that the rate of increase must be at least one n unit every 30-60 million years.