Recently, [3], it was shown that Special Relativity is in fact based just about on one single physical axiom which is that of Reciprocity. Originally, Einstein, [1], established Special Relativity on two physical axioms, namely, the Galilean Relativity and the Constancy of the Speed of Light in inertial reference frames. Soon after, [2,4,5], it was shown that the Galilean Relativity alone, together with some implicit mathematical type conditions, is sufficient for Special Relativity. The references in [7,3] can give an idea about the persistence over the years, even if not the popularity, of the issue of minimal axiomatic foundation of Special Relativity. Here it is important to note that, implicitly, three more assumptions have been used on space-time coordinate transformations, namely, the homogeneity of space-time, the isotropy of space, and certain mathematical condition of smoothness type on the coordinate transformations. In [3], a weaker boundedness type condition on space-time coordinate transformations is used instead of the usual mathematical smoothness type conditions. In this paper it is shown that the respective boundedness condition is related to the Principle of Local Transformation Increment Ratio Limitation, or in short, PLTIRL, a principle introduced here, and one which has an obvious physical meaning. It is also shown that PLTIRL is {\it not} a stronger assumption than that of the mentioned boundedness in [3], and yet it can also deliver the Lorentz Transformations. Of interest is the fact that, by formulating PLTIRL as a physical axiom, the possibility is opened up for the acceptance, or on the contrary, rejection of this physical axiom PLTIRL, thus leading to {\it two} possible theories of Special Relativity. And to add further likelihood to such a possibility, the rejection of PLTIRL leads easily to effects which involve unlimited time and/or space intervals, thus are not accessible to usual experimentation for the verification of their validity, or otherwise. A conclusion is that a more careful consideration of the assumptions underlying Special Relativity is worth pursuing. In this regard, a corresponding trend has lately been observable in Quantum Mechanics and General Relativity. In the former, the respective analysis is more involved than has so far been the case for Special Relativity. As for the latter, the technical and conceptual difficulties are considerable. Regarding Quantum Field Theory, the situation is, so far, unique in Physics since, to start with, there is not even one single known rigorous and comprehensive enough mathematical model. This paper is a new version of [20].
