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If you start with Peano axioms for the natural numbers, then $0$ is part of the language, but $1$ is not. We use $1$ as a shorthand for the term $s0$.

Now we can use the axiom $\forall x(sx

eq0)$, and infer that in particular for $x=0$ it is true that $s0

eq0$. Congratulations, we proved that $0

eq1$ axiomatically.

You can choose different contexts, like set theory, field theory, ring theory or other contexts in which we can interpret $0$ and $1$. You can also find contexts in which $0=1$ is a provable statement. For example the theory whose single axiom states $0=1$. True this theory describes very little of what we expect from the natural numbers, or the symbols $0,1$ to mean. But it is a mathematically valid thing to do.