Well-ordered | search
Click here to view Well-ordered news from 60+ newspapers.
Bookmark or Share- A well-order on a uncountable set - Mathematics Stack ExchangeYou cannot define a well-order on the Cantor set in ZF without AC (Axiom of Choice) because we can show without AC that there is a bijection from the Cantor set to the reals, and it has been shown (by using inner models of Forcing extensions) that it is consistent with ZF that the reals cannot be well-ordered.
- What does a well ordering of - Mathematics Stack ExchangeAs for the elements? Well, that is impossible to tell if the set is not canonically well-ordered, like the natural numbers. Consider the rationals, those are well-orderable (it is a countable set). What is the least rational in the well-ordering? What is its successor? We can't really tell. We can always choose a well-ordering that its first ...
- The Well-Ordering Theorem and Uncountably Infinite SetsAs Noah's comment and user224...'s answer shows, defining functions recursively on well-ordered sets is more subtle than this; you've assumed implicitly that every element other than the least element has a predecessor but this fails for every infinite well-ordered set other than those isomorphic to $\mathbb{N}$.
- How to prove a set is well-ordered - Mathematics Stack ExchangeA well ordered set is a totally ordered set in which every non-empty subset has a smallest element. The interval $[0, 1] \subseteq \Bbb R$ with the usual ordering has a smallest element but is not well ordered because (among other reasons) the subset $(0, 1)$ has no smallest element. $\endgroup$
- The difference between well order and total order?The simplest examples of well-orderings are the finite linear orders and $\mathbb{N}$ itself (the fact that $\mathbb{N}$ is well-ordered is the thing that makes proof by induction work!). However, there are bigger well-orderings; for example, "$\mathbb{N}+\mathbb{N}$," where "$+$" denotes the sum of linear orders (put the first "after" the second).
More
