Does SR explicitly assume $c$ to be finite?
If so, by what statement in Einstein's original paper is this implied?
If not, what to make of equations containing $c$? (e.g., $E = mc^2$)
Formulating the question in a different way: without knowledge of any physics other than inertial frames, do Einstein's two postulates uniquely identify the transformation between inertial frames as Lorentzian (with a finite limiting speed)? Or would it also allow the Galilean transformation?
I'm trying to figure out what is part of the axiomatic system of special relativity and what is not. I'm assuming the finiteness of the speed of light is part of it, but I'm not sure which statements of the theory make that explicit.
The invariance of the speed of light is explicit in the postulate and conversely it's clear the numerical value of $c$ is left to be determined empirically, as it has no impact on the essence of the theory.
EDIT: Based on the many comments I would like to clarify that my question is not if taking $c \to ∞$ leads to classical mechanics, which is obvious. My question is whether the finiteness of $c$ is part of the core of special relativity or left for empirical determination. In the latter case, the theory (alone) would allow both galilean and lorentzian transformations. In the former case, the postulates select the lorentz transformation uniquely, without the need for further empirical facts. This, I presume, is how special relativity is mostly interpreted, but I'm wondering where the finiteness of the speed of light is stated in Einstein's original paper.