>>108 追加

英語版
https://en.wikipedia.org/wiki/Well-order
Well-order

Examples and counterexamples
Natural numbers
The standard ordering ≦ of the natural numbers is a well ordering and has the additional property that every non-zero natural number has a unique predecessor.
Another well ordering of the natural numbers is given by defining that all even numbers are less than all odd numbers, and the usual ordering applies within the evens and the odds:
0 2 4 6 8 ... 1 3 5 7 9 ...
This is a well-ordered set of order type ω + ω. Every element has a successor (there is no largest element). Two elements lack a predecessor: 0 and 1.

Reals
The standard ordering ≦ of any real interval is not a well ordering, since, for example, the open interval (0, 1) ⊆ [0,1] does not contain a least element. From the ZFC axioms of set theory (including the axiom of choice) one can show that there is a well order of the reals. Also Wac?aw Sierpi?ski proved that ZF + GCH (the generalized continuum hypothesis) imply the axiom of choice and hence a well order of the reals. Nonetheless, it is possible to show that the ZFC+GCH axioms alone are not sufficient to prove the existence of a definable (by a formula) well order of the reals.[1] However it is consistent with ZFC that a definable well ordering of the reals exists?for example, it is consistent with ZFC that V=L, and it follows from ZFC+V=L that a particular formula well orders the reals, or indeed any set.

つづく