User blog comment:Ikosarakt1/Uncountable function cannot exists/@comment-1605058-20130707061148/@comment-5150073-20130708181433

Argument n in the function f(n) works with pretty similar way with n-th term of the fund. sequence to its corresponding order type. But if we're supposing that f(n) has order type omega_1, then we can construct a sequence to it, which, as we know, is impossible. So, we got fooled, and the order type for f(n) isn't omega_1, rather: some another (possibly large) countable ordinal.