We investigate the complexity of three successive astronomical paradigms in the science of Physics, namely the Ptolemaic paradigm, the Copernican paradigm and the Keplerian paradigm and mention briefly some characteristic facts about the colossal Newtonian paradigm. This complexity can be understood according to five criteria, as proposed by Thomas Kuhn, the father of the epistemological notion of the paradigm, as well as the founder of an important epistemological school within the realm of

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... 20th century. We propose that there does not exists an overall formal criterion for deciding among these rival paradigms, that is of the existing astronomical paradigms at the age Johannes Kepler formulated its own breakthrough within the science of Astronomy. The further evolution of the science of Astronomy, as well as the advent of the telescope era for investigating the celestial phenomena surely decided for the Newtonian paradigm, which can be understood as the epitome of all past astronomical and cosmological paradigms, yet the advance of the scientific study of the celestial phenomena did not evolved within a linear fashion, on the other hand, it has undergone many changes, subject to the great historical turns, that is the eras of the mentioned astronomical paradigms, during their evolution and their abandonment from the scientific community of the astronomers, the scholars and the polymaths of their age, respectively. We propose that each of Thomas Kuhn criteria imposes its own "complexity measure" of these paradigms, while the overall complexity criterion has to be regarded as the accumulating, overwhelming, empirical evidence, for finally deciding the new way of evolution and the novel turn within the science of Astronomy, especially in the post-Keplerian and surely in the post-Newtonian era.