-----Messaggio originale----- Da: Richard Grandy <firstname.lastname@example.org> A: email@example.com Data: venerdi 23 aprile 1999 1.32 Oggetto: Re: [HM] Question regarding Dedekind and numbers
> > I don't know the answers to your first two questions, but I have a > question about the third. Any first order set of axioms for the reals will > have both standard uncountable models and by the Lowenheim-Skolem theorem > non-standard countable models, it is is not true that any two models are > isomorphic. Perhaps you intended to ask something slightly different? >
Right. First order axiomatizations are always not categorical for axioms satisfied in infinite models.
However, if my logic background is not too weak, I should add that there are formalizations which can not be embedded in the first order PC. For example David Hilbert in his "Grundlagen der Geometrie" (1899) in the fifth group of axioms introduces the "completeness axiom" (later substituted by the "linear completeness axiom" by Bernays), which says 'very approximately' that the accepted model is the least in which all the previous axioms can be satisfied: a very strange axiom indeed! Surely out of the first order PC. In this axiomatic system geometry is proven cathegorical and isomorphic with R^3.
Real numbers are usually axiomatized in modern algebra as an "archimedean complete ordered field". I think that Archimedes' axiom can be written employing natural numbers in a first order axiomatic system, ordering and field axioms are first order with equality, but again it seems to me that completeness is something that can not be first order, either in its Cauchy or in its Dedekind or in its Cantor form, because therein you must always quantify over infinite sets of rational numbers.
Probably Dirk's question did not look for this logical aspects (which would deserve a more technical answer by an expert), but just for the historical reference to a proof of "categoricity" in a pre-hilbertian style (as Dedekind's proof of the "categoricity" of his formalization of arithmetic). In Dedekind's "Stetigkeit und irrationale Zahlen" (1872) I did not find such a proof, even if the genetic construction of the real numbers could allow at least a prove analogous to Hilbert's.