Talk:Tarski's axiomatization of the reals
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Quoting from the article:
The Axioms of:
- Order imply that R is a dense set and a total order;
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- 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;
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- 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)
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- 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)
[edit] Axiom 5
Axiom 5, as stated, does not imply that R is a divisible group. TianDe 21:37, 30 March 2007 (UTC)
- Indeed, one needs more or less all the axioms together to show divisibility. Namely, the axioms imply that R is a dense Dedekind complete ordered Abelian group, and any such is easily seen to be divisible. -- EJ 10:48, 2 April 2007 (UTC)
