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Not quite, you can't order all natural numbers alphabetically either but that doesn't mean you can't count them.

A better way to count based on names would be to first count all single letter names alphabetically, then all two letter names alphabetically, etc.

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Yes, you can. Order is just a relation, i.e. a set that contains pairs of elements. So for example, let X be the relation <. X is a subset of NxN, the cartesian product of N. We say that n<m if the pair (n,m) is in X. Moreover, we say that (n,n+1) is in X, and whenever (n,m) in X and (m,k) in X, then (n,k) in X. This constitues order on naturals.

True, but the GP claimed that if we can order alphabetically we can create a one to one correspondence with natural numbers.

Obviously we can order reals (using <), I interpreted the original post as referring not to your definition but to "order and count"

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