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)

Contents 1 Axiom 5 2 Axiom 4 3 Format 4 Field structure 5 How Tarski (1994) is possible

[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)

[edit] Axiom 4

Axiom four looks weird. Is it correct? It looks like it could either be a typoed statement of associativity--my first guess--but perhaps something more complex which I'm not immediately getting. Can someone who knows better/has access to the papers on this confirm the correctness of that form? Thanks. --Endersdouble 03:03, 24 May 2007 (UTC)

The axiom is correct. It is not weird, it is clever. It sort of combines associativity and commutativity into one axiom (though by itself it does not imply either, you need at least axiom 5 as well). This kind of concise axiom optimization is in fact the whole point of Tarski's axiomatization. -- EJ 08:32, 24 May 2007 (UTC)

That was one thought of mine--looked like some sort of combination. Didn't know enough about the formalism to be sure, however. Thanks for assuaging me. (Should something be added to the page to explain this?) -- Endersdouble 16:57, 24 May 2007 (UTC)

I have put a hint to that effect in the lead section. Do you think it's better now? -- EJ 12:12, 1 June 2007 (UTC)

Yeah, that's good. Thanks. Endersdouble 21:50, 1 June 2007 (UTC)

[edit] Format

The axioms are currently formatted using bold fonts like this:

'''Axiom 1'''. "<" is an [[Asymmetric relation|asymmetric]] relation.

'''Axiom 2'''. If ''x'' < ''z'', there exists a ''y'' such that ''x'' < ''y'' and ''y'' < ''z''. In other words, "<" is [[dense order|dense]] in '''R'''.

I would prefer to format them as definition lists, like this:

;Axiom 1.: "<" is an [[Asymmetric relation|asymmetric]] relation.
;Axiom 2.: If ''x'' < ''z'', there exists a ''y'' such that ''x'' < ''y'' and ''y'' < ''z''. In other words, "<" is [[dense order|dense]] in '''R'''.

This Wikitext gets translated into definition lists <dl><dt>…</dt><dd>…</dd></dl> which I think is more of a semantic markup and presumably would be easier to parse for screen readers and the like. — Tobias Bergemann 12:18, 1 June 2007 (UTC)

It sounds like a good idea. -- EJ 13:00, 1 June 2007 (UTC)

[edit] Field structure

I have finally taken a look on the three papers mentioned in the last paragraph of our article, and it has confirmed my deep suspicion that this "almost homomorphism" business is completely off-topic. What they do is to give an alternative construction of the field of real numbers from scratch (or rather, from integer sequences), whereas what Tarski needs to do is to show that any structure satisfying his axioms can be uniquely expanded by a multiplication operation which makes it an ordered field, which is a quite different problem. (Well, in fact, one could construct "Tarski multiplication" in a rather indirect way by taking a separately contructed field of real numbers, and showing that all pointed densely completely ordered Abelian groups are isomorphic. But then it is irrelevant how the other reals were actually constructed.) IMO the almost homomorphism stuff does not belong here but to the construction of real numbers article, and, surprise surprise, it is already there. (Another issue is that the construction appears to be due to Stephen Schanuel, not the three people mentioned.)

I thus intend to remove the second paragraph in the "How these axioms imply a field" section of the article, but I ask here first, in case somebody knows better. -- EJ 14:00, 6 June 2007 (UTC)

The almost-homomorphism stuff is actually related, if somewhat tangentally. The trick is that the bootstrapping of multiplication in Tarski's system uses the Eudoxus definition of magnitude, which is also the inspiration for the A-H construction. Therefore it is correct, if somewhat misleading, to say that the A-H reals are a "more elegant" version of Tarski's axioms -- Scott 17:59, 7 July 2007 (UTC)

[edit] How Tarski (1994) is possible

The sole reference for this entry is Tarski's text, published in 1994. Someone has asked how that could be possible, given that Tarski died in 1983. Answer: 1994 is the publication year of the Dover reprint of a book Tarski published in German in 1936 and in English in 1941 and 1946.132.181.160.42 04:56, 14 September 2007 (UTC)