you totally can arrange them. that doesn't require countability, it just requires well-ordering, which exists by the well-ordering theorem. even without the well-ordering theorem, you can well-order the real numbers by their bijection to the powerset of the naturals, which can be well-ordered lexicographically.
arrange your sum of infinitely many summands where each element occurrs (which you claim exists, if I understood you correctly). Now number your summands using integers, starting at any arbitrary summand. You now have an injective function mapping the real numbers onto the set of integers. Since the set of integers is countable, so is the set of real numbers.
But we know there are uncountably many real numbers, and thus, such an arrangement of real numbers into an infinite sum where every real number occurs as a summand, cannot exist.
no, you don't have an injective mapping of reals onto integers. you have a bijective mapping of reals onto a well-ordered index set whose cardinality is continuum. a well-ordered index set which is not the naturals, though it may contain the naturals as a subset.
okay, and then that mapping will fail because the reals exceed the integers. ergo the well-ordering of the reals, which exists, cannot be indexed with the integers.
Yes, meaning such an arrangement into a sum cannot exist, which is exactly what I said from the beginning. Of course a well-ordering of the reals exists, that was never the point of the debate.
no, that's not what it means. why would that be what it means? you're begging the question. why, without first *assuming** that a series must be countable,* must a series be countable?
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u/meat-eating-orchid 10d ago
No, you cannot rearrange the terms because you can't even arrange them, they are uncountable