> by the way, most infinite orders are isomorphic to the natural numbers as well, with the exception of the ones for which Cantor’s diagonal argument applies T

> by the way, most infinite orders are isomorphic to the natural numbers as well, with the exception of the ones for which Cantor’s diagonal argument applies There are at least two things wrong with this statement. First, "the ones for which Cantor’s diagonal argument applies" is a bit vague. I assume it's supposed to be a reference to uncountable sets, but as written, it's (probably) referring to the general version of the argument that shows that the powerset of a set is strictly larger than the set itself. Thus, a set "for which Cantor's diagonal argument" applies is a powerset. But not all uncountable sets (or even all uncountable total orders) are powersets. For example, any strong limit cardinal[0] can't be a powerset. Obviously, no uncountable total order can be isomorphic to a subset of the natural numbers. You can then well-order that strong limit cardinal to get a total order which isn't a powerset and isn't isomorphic to a subset of the natural numbers. Second, even countable total orders are much more varied than subsets of the natural numbers. For example, the integers form a total order which can't be isomorphic to a subset of the naturals. The integers are unbounded below, but any subset of the naturals is bounded below (by 0, e.g.). As another counterexample, the set of rational numbers in [0, 1] forms a total order which is dense: between any distinct elements of the total order, there's another distinct element between them. You can get a nice theorem along these lines, though. Every countable total order is isomorphic to a subset of the rational numbers. [1] Of course, the word "most" here is ambiguous, but seeing as there are uncountably many non-isomorphic, countable, total orders (for example, the number of countable ordinals is uncountable), but only countably many non-isomorphic subsets of the naturals, I think it's inappropriate. [0] https://en.wikipedia.org/wiki/Limit_cardinal https://en.wikipedia.org/wiki/Limit_cardinal [1] https://www.whitman.edu/mathematics/higher_math_online/section05.06.html https://www.whitman.edu/mathematics/higher_math_online/secti...