Note I'm describing everything from Wildberger's point of view below (or at least what I think he believes).
>>12466642Not necessarily. As an example, you can construct sqrt(2) algebraically, even though analytically it requires an infinite amount of work to enumerate. Some sequences are fine as well, but an "infinite sequence" is nonsensical. As an example, [n> = 1, 2, 3, ... is an ongoing sequence (his terminology) and we can continue producing as many terms as we want and Nat is a type (in a CS sense) that represents the natural numbers. This is okay; however, [1, 2, 3, ...] (considering the list of numbers from that infinite sequence) or Nat = N (the set of all natural numbers), doesn't make sense because you haven't done all the work to define those objects. You've given a formula or a description that one can use to get as many terms as they want, but saying [1, 2, 3, ...] is the entire infinite sequence and manipulating that object (e.g. putting it in a set) is not okay because you can never actually do the leg work to consider the sequence in its entirety as an object (let alone manipulate it).
>>12466789The modern mathematicians view of functions is essentially that of choice. As in, a general sequence a_n does not necessarily need to specified by any concrete rule (each term can be chosen randomly).
Looking at the definition of limit: for all e > 0, there exists n > N, s.t. |a_n - L| < e, we see we need to pair each e (a real number) with some N, which in general is an infinite amount of work. Now the baby examples they give in your intro analysis class allow you to relate N and e in somennice way since a_n is given by a relatively simple mathematical formula, but this is a special case and need not necessarily be true if we take the choice point of view. As a result, even the standard definition of a limit would require an infinite amount of work to check since the rules that can generate a general sequence are arbitrary.
