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 (infA). If A is bounded above, it also has a minimal upper bound or supremum (supA). If x\in X, we define the initial segment of x to be

I_x=\{y\in X: y<x\}.

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.