User blog comment:LittlePeng9/Long hierarchies of functions (on my other blog)/@comment-11227630-20171031155258/@comment-80.98.179.160-20171107174619

And $$|\omega_1+1|\ne|\omega_1|$$.

Proof:

Say you have $$[0,1[$$ as your set (named $$A$$), and $$B$$, that is $$A\cup\{2\}$$.

$$B$$'s length is 1 more than $$A$$'s.