The minimal uncountable well-ordered set

[topsquark]

I once asked about this on MHF and didn't really get anywhere with it. (I thought things made sense and eventually ended up just as confused as before.)

Does anyone have an example of the minimal uncountable well-ordered set, where every section is countable? I'm still at the point in my self taught Mathematical skills that I need examples in order to understand the topic. Sad, but true.

-Dan

 

[Jhoneine]

A good example of a minimal uncountable well-ordered set is the set of real numbers. This set is uncountable because it is infinitely large and it is well-ordered because every element can be compared to every other element and there is an overall ordering of the elements from smallest to largest. Moreover, the set is minimal because for any two elements in the set, there is no subset of the set that contains only those two elements.To see how this set has countable sections, we can consider the intervals between rational numbers. Since the rational numbers are countable, the set of real numbers can be divided into countably many distinct intervals. For example, the interval between 1/2 and 1 can be divided into the following countable sections: [1/2, 7/10), [7/10, 8/10), [8/10, 9/10), [9/10, 1]. This shows that every section of the set of real numbers is countable.