Let be well ordered. If , then has a minimal element, which is its maximal lower bound or infimum(); conversely, bounded above has a minimal upper bound supremum().
If we define the initial segment of to be
and all are called predecessors of .
Theorem 0.4.1 (The Principle of Transfinite Induction). Let be a well ordered set and such that whenever . Then .
Proof. If , let . Then but . (Contrapositive.) ∎
Proposition 0.4.2. If is well ordered and , then is either an initial segment or itself.
Proof. Let , and let . If there were with , we would have for some , hence , a contradiction. Therefore and , so .∎
Proposition 0.4.3. If are well ordered, then either is order isomorphic to , or is order isomorphic to an initial segment of , or is order isomorphic to an initial segment of .
Proposition 0.4.4. There is an uncountable well ordered set such that is countable for each . If is another set with these same properties, is order isomorphic to .
Proposition 0.4.5. Every countable subset of has an upper bound.
There is an order isomorphism , where , given by and . So and is an order isomorphism, where is the minimal element of such that is infinite.
, the extension of obtained by declaring for all . is called the first uncountable ordinal.