>>13990021>>13989325Everyone in this argument is stupid, because the two sides are working under different axiom systems so of course you are going to get different results; the other poster rejects the axiom of infinity and you don't.
Do you see that little sideways 8 above your capital sigma? That is called an "infinity" sign. To calculate "sqrt(2)", you need to perform an infinite number of operations.
And no, being able to calculate "sqrt(2)" to an arbitrary number of places doesn't really matter. At the end of the day, your "real numbers" boil down to (most likely) equivalence classes of Cauchy sequences (which are infinite sets) or Dedekind cuts (which are infinite sets, which, in the general case, require an infinite number of conditions to describe).
Therefore, any definition of real numbers necessarily requires an infinite set or an infinite number of operations in the underlying theory to work.
What you posted is not "sqrt(2)"; rather, it is an algorithm to compute the "sqrt(2)". If you were asked to exhibit a "real number" according to it's definition (e.g. a Dedekind cut), you would necessarily have to describe an infinite set. And since you don't have a prior theory of algorithms, if I come by and give you a random algorithm, you have no idea whether it is equivalent to "sqrt(2)".
This is why mathematicians don't work with algorithms, they just pretend to be able to do an infinite number of operations via the axiom of infinity, so the underlying "can tell" if two real numbers are equal, even if we physically can't.
So the only argument you have is that "infinite set theory" is more "expressive", to which Wildberger would say that is nonsensical, since letting your theory assume the ability to do an infinite number of things is an asinine thing to assume in the first place.
This argument will never be resolved because the choice of axioms is completely subjective. So can all of you just shut up and move on with your lives please?
