# List

A list is an or­dered col­lec­tion of ob­jects, such as 1, 2, 3 or “blue”, 0, “shoe”. Lists are gen­er­ally de­noted by square brack­ets, with el­e­ments sep­a­rated by com­mas. Lists al­low rep­e­ti­tion (un­like sets) and are or­dered (un­like bags).

Some­times lists are typed, in which case they con­tain only one type of el­e­ment. For ex­am­ple, 1, 2, 3 is a well-typed list of num­bers, whereas the list “blue”, 0, “shoe” is not well-typed (be­cause it con­tains both num­bers and strings). The set of all lists of type $$X$$ is usu­ally de­noted $$[X]$$ or $$X^{\le \omega}$$, where $$\omega$$ is the small­est in­finite limit or­di­nal. The set of all finite lists of type $$X$$ is usu­ally de­noted $$X^{< \omega}$$, and the set of all in­finite lists of type $$X$$ is usu­ally de­noted $$X^{\omega}$$.