FOM: non-standard models

[Walter Felscher]

On Sun, 30 Nov 1997, Neil Tennant wrote:

> I would like to know the earliest documented observation that there are
> non-standard models for the theory of arithmetic. I am interested not so
> much in date of first publication, as date of first realization of this fact.

Skolem in "Ueber einige Grundlagenfragen der Mathematik" [Skrifter 
Videnskapsakademiet i Oslo, I no. 4 (1929) 1-49] conjectured them when 
writing about characterizations of the number series

   Man bildet sich nun ein, diese Charakterisierung koenne absolut 
   gemacht werden, naemlich dadurch, dass man die Gueltigkeit der 
   vollstaendigen Induktion fordert. In dieser Forderung tritt jedoch der 
   Begriff der "Aussagenfunktion", oder wenn man will "Menge" auf, und in 
   einer folgerichtigen formalistischen Mathematik muss dieser Begriff 
   wieder durch die Forderung gewisser Schlussregeln charakterisiert werden. 

Hao Wang on p.41 of "A survey of Skolem's work in logic" [Thoralf 
Skolem: Selected Works in Logic, ed.J.E.Fenstad, Oslo 1970, pp.17-52] 
comments that passage writing

   It is for the first time suggested in print (for an unpublished 
   anticipation, compare Dedekind's letter, JSL 22 (1957) 150) that given 
   any set of theorems on natural numbers, we can find a nonstandard model 
   in which these theorems are true.

In "Ueber die Unmoeglichkeit einer Charakterisierung der Zahlenreihe 
mittels eines endlichen Axiomensystems" [Norsk Matematisk Forening, 
Skrifter Ser. 2 , no.1-12 (1933) 73-82 ] Skolem then constructs his 
famous nonstandard model N* . 

W.F.