Why would we identify a sequence of pulses as integers if, for example, the passage of time is perceived differently by aliens? Or if mathematics within alien life is based on atomic units of group like set logic, and so the sequence of pulses itself is seen as "1".

The ancient Egyptians expressed axioms by a series of concrete examples. Stating them as actual axioms or theorems with proofs (by Greeks, Indians and Chinese, among others) was an improvement.

How can we communicate actual axioms to extra-terrestrials?

Presumably we can create a kind of Rosetta Stone with different representations of logical and algebraic expressions, and maybe they can decode that if they can figure out how to decode whatever "broadcast" representation of this we come up with. And that assumes that they recognize such a broadcast as something from intelligent life.