dimpase / dedekind_2nd_def

This is a translation from the original German of an 1889 note from Dedekind's Nachlass, edited by Emmy Noether

Dedekind's "Zweite Definition des Endlichen und Unendlichen."

(tl;dr please help improve the translation if you can!)

In between the first (1887) and second (1893) editions of Was sind und was sollen die Zahlen (Wooster Woodruff Beman's 1901 English translation of the second edition, called by him "The nature and meaning of numbers"), Dedekind wrote a short unpublished note in March 1889 that ultimately was only hinted at in the foreword to the second definition (I give Beman's translation):

For the benefit of those who are interested in and understand the difficulties of such an investigation, I add the following remark. We can lay down an entirely different definition of the finite and infinite, which appears still simpler since the notion of similarity of transformation is not even assumed, viz.:

“A system S is said to be finite when it may be so transformed in itself (36) that no proper part (6) of S is transformed in itself; in the contrary case S is called an infinite system.”

Now let us attempt to erect our edifice upon this new foundation! We shall soon meet with serious difficulties, and I believe myself warranted in saying that the proof of the perfect agreement of this definition with the former can be obtained only (and then easily) when we are permitted to assume the series of natural numbers as already developed and to make use of the final considerations in (131); and yet nothing is said of all these things in either the one definition or the other! From this we can see how very great is the number of steps in thought needed for such a remodeling of a definition.

Even in the third edition Dedekind makes no edits to his monograph, and only adds another small preface explaining why. So this second definition of finite set lay fallow for a couple of decades, as far as the public knew. However Dedekind had in fact worked out, as far as his naive set theory framework allowed, the relationship between his second definition of fintie set, and the notion of finiteness that is "of cardinality a natural number".

The "entirely different definition of the finite and the infinite" is the subject of the 1889 note, which to my knowledge has only appeared at page 450 on volume 3 of Dedekind's collected works, (Gesammelte Mathematische Werke) edited by Robert Fricke, Emmy Noether and Øystein Ore, in the original German.

The note is followed by a short editorial commentary by Noether, and she cites the then-forthcoming note "Sur la deuxième définition des ensembles finis donnée par Dedekind" by Jean Cavaillès, which is published in Fundamenta Mathematicae vol 19 that (I believe) fills in a gap in the proof given by Dedekind (I plan to read this paper next; note that Noether and Cavaillès co-edited the Cantor–Dedekind correspondence). This paper is included in this repository under the publisher's CC-By license. Noether also gives a long-ish quotes from a letter from Dedekind to Weber on the matter.

I have used machine translation to give a first-pass translation of this text to English (which is, understandably, extremely rough and in places tortured), and made some small obvious edits (Satz > Theorem, and not "sentence", and so on). I hope to gradually improve this and also to do a careful proof-reading of the transcription of the mathematics, so all comments on various choices of wording are greatly appreciated.