No, for a quite fundamental reason: numbers are a thing we made up, and we made them to work a certain way, and they don't work that way. Part of the definition of an integer is that every integer has a successor, which is also an integer. And—presuming we have an analytical understanding of a function—a function of the integers won't suddenly "malfunction" if fed a sufficiently-large one. A function isn't a computer.

I mean, you could have your own axiomatic set-theory where mathematical induction doesn't work, but I'm not sure how to construct it without it being a very strange set-theory indeed.