I'm learning set theory by using Halmos's book. In the section about order, the author states that "The set of all natural numbers (with its customary ordering by magnitude) is an example of a partially ordered set with a first element (which is 0) but no last".
As I understand, we can compare every 2 natural numbers, so the set of all natural numbers must be totally ordered. Could you please explain me why it should be the case as stated by Halmos ?
Thank you very much for your help!