>>128881893. What does it actually mean to be equivalent, in a Cauchy sense? The standard definition is of course, given two sequences s_n, t_n of rationals, they are equivalent precisely when the sequence u_n = s_n - t_n -> 0. And the Cauchy definition of convergence (for reference) is:
u_n -> A provided that for all e > 0, there exists N, s.t. for n > N, |u_n - A| < e.
Already, there are some major issues here.
a) Since we are going with "choice" sequences, you can write s_n - t_n, but since there is no rule that allows me to recover the resulting sequence, you are essentially forcing me to compute the difference of the sequences pairwise. Unfortunately, since both of these sequences are infinite, this is impossible, and the only reason you say it is possible is because you are assuming (by fiat) the ability to do an infinite amount of work via ZFC (see below) which at least should be a point of contention and not standard dogma.
b) When you say "for all e > 0, there exists N", the baby examples in intro real analysis courses try to establish a functional relationship between N and e by manipulating absolute values and inequalities, but in general, such a thing is futile, since the sequences in question are arbitrary, so it is impossible to finitely establish such a relationship in the first place. So again, you are relying on doing an infinite amount of work to show two sequences equivalent to each other.
Which finally brings us to
4. The entire reason why all of this "works" in the first place, which is ZFC. Basically, the only reason you can magically assume the ability to do an infinite amount of work is simply because you assume you can do so by fiat (e.g. via the Axiom of Infinity). The standard argument to this is "well you need to start from somewhere, and ZFC and sets where you start from" and "you can always choose your own axioms instead of ZFC", but this is already cheating since you've changed the definition of axiom in the first place.
cont.
