FOM: Thomae and non-archimedian domains

[Walter Felscher]

Bill Tait, on Nov.12th, wrote in connection with Goedel and
infinitesimals that

   Cantor is quoted by Dauben as saying that Johannes Thomae
   (who had an office down the hall from Frege) was the
   first to ``infect mathematics with the Cholera-Bacillus
   of infinitesimals''.

While I do not know what Cantor actually may have referred
to, Thomae (1840-1921 , since 1872 Professor in Halle, since
1879 in Jena) does have a documentable connection, not with
infinitemals, but with non-archimedian extensions of the
reals: in his

   Abrisz einer Theorie der complexen Funktionen, Halle 1870

and his

   Elementare Theorie der analytischen Functionen einer
   complexen Ver„nderlichen.,  2te Aufl., Halle 1898

he represented the orders of growth of real functions by
lexicographically ordered semigroups of sequences of
integers. It then was Paul du Bois-Reymond 1882 who
expressed the idea that the totality of orders of growth
(which he called the infinitaere Pantachie) should be viewed
as an expansion of the continuum; for a more modern
presentation of his work cf. G.H.Hardy, Orders of Infinity,
Cambridge 1924.

It may not be superfluous to point out that this achievement
of Thomae's is NOT connected with his opinions on the
foundation of numbers, analyzed so masterfully  in Frege's
"Ueber die Zahlen des Herr Thomae".

W.F.