Talk:Well-ordering principle

From Wikipedia, the free encyclopedia

[edit] Z vs N

In the algebraic definition, I think it's usual to talk about all of Z rather than just N, because it is Z which is an integral domain, and Z which amalgamates the relevant algebraic properties. The natural numbers themselves are well-ordered in a set-theoretic (and order-theoretic) sense, of course; but the property of interest is algebraic here, no?

Will this be understood as I've left it? --VKokielov 03:42, 11 September 2006 (UTC)

If I understand you correctly, you (or perhaps Birkhoff-MacLane, which I doubt) define the phrase

R is a well-ordered integral domain

as an abbreviation for

R contains a well-ordered subset, called the natural numbers, of which any subset always contains a least element.

This does not make sense to me. First, "always contains a least element" just repeats "well-ordered". Second, I assume that by "called the natural numbers" you mean "isomorphic to the natural numbers". Third: the rational numbers also have this property (namely, that they contain the well-ordered subset of the natural numbers).

You may mean the following:

The ring of integers is characterized among the integral domains by the fact that it can be linearly ordered in such a way that

1. the order agrees with the operations (i.e., addition and multiplication are monotone)
2. the positive elements are well-ordered.

Aleph4 11:38, 11 September 2006 (UTC)

Right -- so I've caught myself and fixed it. --VKokielov 22:57, 11 September 2006 (UTC)