However, the definition usually used in practice in constructive mathematics is this:
In other words, given any function from a Kuratowski-finite set to , there exists an element of that is not in the image of . This is essentially a variation of Richard Dedekind's definition of a Dedekind-infinite set.
Note that you can make this definition work without previously assuming the existence of natural numbers, by using an infinity-free definition of Kuratowski-finite set.
Probably a lot to say about the relation between the various definitions of infinite set (the one above, the negations of the definitions of finite set, and others that might be studied). In the meantime, try the English Wikipedia.