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?
Richard Grandy
who grew up in Pittsburgh when it wasn't so sunny
>Dear listmembers,
>
>I am a graduate student of philosophy and I'm doing my current
>research on an interpretation of Richard Dedekind's works on the
>foundations of mathematics as 'axiomatic' and 'structuralistic'.
>
>In the course of this reseach I hit upon some questions that I hope
>some of you might be able to answer:
>
>1. Has anybody information about the whereabouts of the letter from
> Dedekind to Lachmann mentioned by Emmy Noether in "Gesammelte
> mathematische Werke", vol.3, 1932: "So liegt ein Brief des
> Zweiundachtigjaehrigen vor - Antwort an einen damaligen Studenten,
> Lachmann, jetzt im Besitz von G.Hamel - wo er die 'Erweiterung des
> Reichen N der natuerlichen Zahlen zu dem Reiche G der ganzen
> rationalen Zahlen' ... andeutet." (p. 490)?
> (I've already contacted Prof.Scharlau and the Dedekind-Nachlass in
> Goettingen on this matter without success.)
>
>2. Who (and when) was the first to introduce the integers and rational
> numbers as equivalence classes of pairs of natural numbers/integers?
>
>3. When was the first proof published, and by whom, that establishes
> the isomorphism between any two models for the axioms of the real
> numbers?
>
>Any help is kindly appreciated,
>from sunny Pittsburgh,
>
> Dirk Schlimm
>
>
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> O O
>
> \___/