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

$I_x=\{y\in X: y.

The elements of $I_x$ are called predecessors of $x$.

The Principle of Transfinite Induction. Let $X$ be a well ordered set. If $A$ is a subset of $X$ such that $x\in A$ whenever $I_x\subset A$, then $A=X$.

Proof. If $X\not=A$, let $x=inf(X\setminus A)$. Then $I_x\subset A$ but $x\not\in A$. $\Box$

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

Proof. Let $J=\cup_{x\in A}I_x$. If $J\not=X$, let $b=inf(X\setminus J)$. If $\exists y\in J$ with $y>b$, we would have $y\in I_x$ for some $x\in A$ and hence $b\in I_x$, contrary to construction. Hence $J\subset I_b$, and it is obvious that $I_b\subset J$$\Box$

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

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

Proposition. There is an uncountable well ordered set $\Omega$ such that $I_x$ is countable for each $x\in \Omega$. If $\Omega'$ is another set with the same properties, then $\Omega$ and $\Omega'$ are order isomorphic.

Proof. Uncountable well ordered sets exist by the well ordering principle; let $X$ be one. Either $X$ has the desired property or there is a minimal element $x_0$ such that $I_{x_0}$ is uncountable, in which case we can take $\Omega=I_{x_0}$. If $\Omega'$ is another such set, $\Omega'$ cannot be order isomorphic to an initial segment of $\Omega$ or vice versa, because $\Omega$ and $\Omega'$ are uncountable while their initial segments are countable, so $\Omega$ and $\Omega'$ are order isomorphic. $\Box$

Remark. $\Omega$ is called the set of countable ordinals.

Proposition. Every countable subset of $\Omega$ has an upper bound.

Proof. If $A\subset \Omega$ is countable, $\cup_{x\in A}I_x$ is countable and hence is not all of $\Omega$. Then $\exists y\in\Omega$ such that $\cup_{x\in A}I_x=I_y$, and $y$ is thus an upper bound for $A$. $\Box$

The set $\mathbb N$ of positive integers may be identified with a subset of $\Omega$ as follows. Set $f(1)=\inf\Omega$, and proceeding inductively, set $f(n)=\inf(\Omega\setminus\{f(1), ..., f(n-1)\})$. Then $f$ is an order isomorphism from $\mathbb N$ to $I_\omega$, where $\omega$ is the minimal element of $\Omega$ such that $I_\omega$ is infinite.

It is sometimes convenient to add an extra element $\omega_1$ to $\Omega$ to form a set $\Omega*=\Omega\cup\{\omega_1\}$ and to extend the ordering on $\Omega$ to $\Omega*$ by declaring that $x<\omega_1$ for all $x\in\Omega$. $\omega_1$ 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.