TITLE: Proving that $E=mc^2$
QUESTION [9 upvotes]: What are the axioms of special relativity? Is there a book or paper that introduces the theory of special relativity in a rigorous manner, and proves that $E=mc^2$ after appropriate definitions?

REPLY [2 votes]: I'm not sure if this should be asked on a math board. 
@Ross Millikan answers "correctly"

The axioms are simple: 1) intertial reference frames are equivalent-physical laws are the same in all of them. 2) the speed of light is the same in all these frames. From those you derive the Lorentz transformation. This is done in most texts that cover special relativity.

But without me knowing the tedious details, already 95 years ago, Weyl wrote pedantically about the relation between physical axiomatizations and the Lorentz transformation. Apparently for uniquencess you must also set up axioms about some finiteness of the domains to be transformed to actually rule out fractional linear transformations, projective stuff, etc. etc. 
Speaking of tedious rigour, as an answer to the question, someone wrote a PhD thesis about setting up special relativity in first order logic here: First-Order Logic Investigation of Relativity Theory
with an Emphasis on Accelerated Observers.