Talk:Tarski's axiomatization of the reals

From Wikipedia, the free encyclopedia

Quoting from the article:

The Axioms of:

• Order imply that R is a dense set and a total order;

The axioms of order imply no such thing. The empty relation satisfies all of them. Or, for a less trivial example, the product order on R×R satisfies the order axioms, but is not total.

• Addition imply that R is an Abelian group under addition;

The axioms of addition imply no such thing. Any associative right quasigroup, with < defined as the empty relation, satisfies the axioms. It does not have to be commutative, it does not even have to be a group. -- EJ 20:33, 6 August 2006 (UTC)

Checking with the original revealed that the problem is in fact deeper, as the real Tarski's axiom 4 is not just associativity, it has commutativity built-in, so to speak. Fixing that. -- EJ 17:59, 9 August 2006 (UTC)