Notes and Observations

12.1 and 12.5 Theorem:

Let $\QTR{Large}{A\ }$and $\QTR{Large}{B}$ be Well Ordered then

  1. $\QTR{Large}{A}$ is order isomorphic to a segment of $\QTR{Large}{B}$.

    or

  2. $\QTR{Large}{B\ }$is order isomorphic to a segment of $\QTR{Large}{A}$.

    (For some MATH,MATHis order isomorphic to MATH.)

    or

  3. $\QTR{Large}{A}$ and$\QTR{Large}{\ B\ }$are order isomorphic.

    (If MATH n then $\QTR{Large}{A}$ is order isomorphic to $\QTR{Large}{n}$.)

    ($\QTR{Large}{A\ }$is order isomorphic to MATH.)

----------------------------------------------------------------------

Another Example.

MATH - The Well Ordered Set consisting of the Natural Numbers followed by a copy of the Natural Numbers followed by $\QTR{Large}{\{}$1,2,3,4,5,6,7$\QTR{Large}{\}}$

Let $\QTR{Large}{A\ }$be Well Ordered then

  1. $\QTR{Large}{A}$ is order isomorphic to a segment of MATH,or MATH itself.

    or

  2. For some MATH,MATHis order isomorphic to MATH.)

The Axiom of Choice does not play a role in this.

----------------------------------------------------------------------------------

Another Example.

MATH- The ordered Set consisting of the Natural Numbers followed by the Integers.

Consider the following two self-order isomorphisms.

MATH

and

MATH

where

MATH and MATHn$\QTR{Large}{)=}$n+1 for n MATH

Note that

MATH but MATH.

Consider the structure of the Well Ordered subsets.