Let be a well ordered set. If is nonempty, has a minimal element, which is its maximal lower bound or infimum (inf). If is bounded above, it also has a minimal upper bound or supremum (sup). If , we define the initial segment of to be

.

The elements of are called predecessors of .

**The Principle of Transfinite Induction.** Let be a well ordered set. If is a subset of such that whenever , then .

Proof. If , let . Then but $x\not\in A$.

**Proposition.** If is well ordered and , then $\cup_{x\in A}I_x$ is either an initial segment or itself.

Proof. Let $J=\cup_{x\in A}I_x$. If , let . If $\exists y\in J$ with , we would have for some and hence , contrary to construction. Hence $J\subset I_b$, and it is obvious that .

**Proposition.** If and are well ordered, then either is order isomorphic to , or is order isomorphic to an initial segment in , or is order isomorphic to an initial segment in .

Proof. Consider the set $\mathcal F$ of order isomorphisms whose domains are initial segments in or itself and whose ranges are initial segments in or itself. $\mathcal F$ is nonempty since the unique belongs to , and is partially ordered by inclusion (its members being regarded as subsets of ). An application of Zorn’s lemma shows that $\mathcal F$ has a maximal element , with (say) domain and range . If and , then and are again initial segments of and , and could be extended by setting , contradicting maximality. Hence either or (or both), and the result follows.

**Proposition. **There is an uncountable well ordered set such that is countable for each . If is another set with the same properties, then and are order isomorphic.

Proof. Uncountable well ordered sets exist by the well ordering principle; let be one. Either has the desired property or there is a minimal element such that is uncountable, in which case we can take . If is another such set, cannot be order isomorphic to an initial segment of or vice versa, because and are uncountable while their initial segments are countable, so and are order isomorphic.

**Remark. ** is called the set of countable ordinals.

**Proposition. **Every countable subset of has an upper bound.

Proof. If is countable, is countable and hence is not all of . Then such that , and is thus an upper bound for .

The set of positive integers may be identified with a subset of as follows. Set , and proceeding inductively, set . Then is an order isomorphism from to , where is the minimal element of such that is infinite.

It is sometimes convenient to add an extra element to to form a set and to extend the ordering on to by declaring that for all . is called the first uncountable ordinal.

“Gratitude is the key to happiness. When gratitude is practiced regularly and from the heart, it leads to a richer, fuller and more complete life… It is impossible to bring more abundance into your life if you are feeling ungrateful about what you already have. Why? Because the thoughts and feelings you emit as you feel ungrateful are negative emotions and they will attract more of those feelings and events into your life.” ~ Vishen Lakhiani

To be continued tomorrow 🙂

References

[1] Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, 2ed, page 9-10.

[2] Purpose Fairy’s 21-Day Happiness Challenge, http://www.jrmstart.com/wordpress/wp-content/uploads/2014/10/Free+eBook+-+PurposeFairys+21-Day+Happiness+Challenge.pdf.