On the nature of discrete space-time: The atomic theory of space-time and its effects on Pythagoras's theorem, time versus duration, inertial anomalies of astronomical bodies, and special relativity at the Planck scale

In this work, resolutions will be given for commonly stated problems associated with a model that assumes that space and time are discretized (i.e., atomized). This model is in contrast to the continuous space-time model that is used in all common physical theories and equations -- a model that assumes that spatial coordinates and time are continuous variables. The resolutions to the problems are arrived at, not by proposing any new theories or postulates, but by strictly adhering to: Ernst Mach's principle of non-absolute space, the tenets of logical positivism, quantum mechanics and general relativity. The problems associated with discrete space-time addressed in this paper include: Lorentz contraction (time dilation) of the ostensibly smallest spatial (temporal) interval, maintaining isotropy, violations of causality, and conservation of energy and momentum. Importantly, this work yields modifications to the standard formulae for time dilation and length contraction, with these modifications preserving the quantums of space and time and allowing for temporary travel at the speed of light. Also given are: a resolution to Weyl's tile argument, a modification of the 2500 year old Pythagoras's theorem, a reassessment of Henri Bergson's theory advocating a distinction between the time durations measured by scientists and an immutable "Time", and a discussion of whether Einstein's light-clocks are as ideal as most scientist believe. Also included is a demonstration of how discrete space imposes order upon John Wheeler's quantum foam such that the foam becomes a gravity crystal permeating all space and producing measurable inertial anomalies of astronomical bodies.