You cannot observe infinity operationally. Take 0 and add 1 repeatedly. For what n does n+1 become infinite? Never. Since you can't construct infinity you can only believe in it like God.

Hence the jargon "completed infinity". Semantically --- but not in the symbols themselves 0+1+1..." one can pass from finite to infinite by arguing since every n has a successor define Z to be the set of all successors "completing into infinity".

Not having infinity is the real reason a+b=b+a can't be proved in ultra finitism. Induction which depends on the idea of completed infinity is what is otherwise is used.