> In another message, the difference between the two editions of
> Landau's Grundlagen der Analysis was mentioned. When I was a freshman
> in Jerusalem, and read that book (in the English translation) I heard
> that Landau discovered the error in the 1st ed. while in Jerusalem.
According to the December 1929 Preface for the Teacher, in which the
'error' and its correction are discussed, this edition (published in
Leipzig in 1930) containing the correction of the error, was the first. The
error was not discovered in some (non-existent) earlier edition but in a
manuscript notebook, consisting of lecture notes. The error was discovered
by Landau's assistant Grandjot, who used Landau's notebook as basis for his
lectures. (Landau adds that by the end of 1929 Grandjot was professor at
the U of Santiago.) In fact, it was the discovery of the error, and its
rectification--using an idea suggested by K\'almar [presumably the logician
L\'aszl\'o K\'almar]--that persuaded Landau to publish the notebook as a
book.
The dates seem to be consistent with Avi's conjecture that the error was
discovered while Landau was in Jerusalem.
BTW, the nature of the error was the following. Landau's system, like
Peano's, is second-order. In this system + and * are not primitive notions
but defined ones. Accordingly, the basic equations characterizing these
operations,
x + 1 = x', x * 1 = x,
x + y' = (x + y)', x * y' = x * y + x,
are not axioms (as they are in the first-order poor-man's version of
Peano's system) but definitions. The error in the notebook was the absence
of proofs that there are unique functions + and * satisfying the above
difining conditions; in other words, that the (implicit) definitions are
legitimate.
In the book, these proof appear as Theorem 4 (which is at the same time
Definiton 1) and Theorem 28 (which is also Definition 6).
On another matter: some members of this list have wondered about Landau's
remark in the Preface that one cannot prove the consistency of the Peano
axioms. How could he claim this in 1929, before the publication of
G\"odel's second incompleteness theorem? Indeed, an abstract announcing the
incompleteness theorems was first made public in October 1930, almost a
year after the date of Landau's Preface. But surely some people, mainly in
Vienna, must have known about these results before they were made public.
Could Landau have heard about them through the grapevine before writing his
Preface?
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Moshe' Machover | E-MAIL: moshe.machover@kcl.ac.uk %%
%% Department of Philosophy | FAX (office)*: +44 171 873 2270 %%
%% King's College, London | PHONE (home)*: +44 181 969 5356 %%
%% Strand | %%
%% London WC2R 2LS | * If calling from UK, replace %%
%% England | +44 by 0 %%
%% http://www.kcl.ac.uk/kis/schools/hums/philosophy/staff/moshem.html %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%