# [FOM] Dedekind's theorem

Richard Heck rgheck at brown.edu
Mon Jun 20 10:17:31 EDT 2011

```On 06/17/2011 03:28 AM, meskew at math.uci.edu wrote:
> In "Was sind und was sollen die Zahlen?" Dedekind gives a noncircular
> proof of the statement that a set is finite if and only if it cannot be
> put in bijective correspondence with a proper subset.  By "circular" I
> mean in this context that you should not prove it by simply saying that a
> proper subset of a finite set will have a smaller cardinality; this
> theorem should be taken as the ground for the well-definedness of the
> finite cardinals.
>
> Regarding the "only if" direction, which establishes that finite ordinals
> are cardinals, was Dedekind the first to publish a proof of this?  Did
> Frege give a proof independently?  Galileo?  Leibniz?  Some medieval monk
> perhaps?  It would seem strange if this basic aspect of the concept of
> number was not reflected upon for so many centuries.
>
In section 83 of /Foundations of Arithmetic/, Frege mentions the need
for the following result: No finite number follows after itself in the
natural sequence of numbers. I think it is clear from the context of the
book that Frege would have been aware that this implies the only if
direction. Frege later gives what he calls "Endlos" but we call Aleph
zero as an example of a number that does follow itself in the natural
sequence.

I don't know whether Frege was first. I'd suggest looking at Bolzano,
and especially at Paolo Mancosu's recent paper:

@ARTICLE{Mancosu:Measure,
author = {Paolo Mancosu},
title = {Measuring the Size of Infinite Collections of Natural
Numbers: Was
{C}antor's Theory Inevitable?},
journal = {Review of Symbolic Logic},
year = {2009},
volume = {2},
pages = {612--46},
}

Regarding the if direction, which of course needs (countable) choice,
Frege seems to have spent some time exploring it. There is
circumstantial but significant evidence that he eventually stumbled upon
the problem as he tried to formalize Dedekind's argument. For
discussion, see the final section of my paper "The Finite and the
Infinite in Frege's /Grundgesetze der Arithmetik/", available here:
http://frege.brown.edu/heck/pdf/published/FiniteAndInfinite.pdf
in an old version.

Richard Heck

--
-----------------------
Richard G Heck Jr
Romeo Elton Professor of Natural Theology
Brown University

--
-----------------------
Richard G Heck Jr
Romeo Elton Professor of Natural Theology
Brown University

```