User blog comment:Edwin Shade/Understanding The Infinite/@comment-80.98.179.160-20171113173859/@comment-32876686-20171113190048

When we are dealing with finite numbers, $$a+b=b+a$$, but in ordinal arithmetic, there is no axiom that preserves commutativity, so $$a+b$$ does not always equal $$b+a$$.

In mathematics you can create systems that follow different rules, perhaps ones that even defy intuition, but it is still correct as long as your system of rules has no internal contradictions. So a given system need not follow the axioms of another.

By following the rules of ordinal arithmetic $$\omega\neq (1+\omega)-1$$ because $$\omega-1=\omega$$, also by the rules of ordinal arithmetic.