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Topic: Fermath's sake
Replies: 19   Last Post: Nov 12, 1997 11:27 PM

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Calvin Jongsma

Posts: 51
Registered: 12/3/04
Natural Numbers (Was: Fermath's sake)
Posted: Nov 11, 1997 6:39 PM
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First a comment on subject headings. Later on when I look through my
files (or the archives, I'm never going to know that "Fermath's sake"
contains a discussion of what counts as a natural number. Hence my
change of the topic heading. If the old topic is listed behind the
first such change, threads should still be visible, shouldn't they?

Now for the topic: what counts as a natural number, taking a more
historical perspective.

The Greeks start with 2, as Sam Kutler observed.

Dedekind starts with 1, as does Peano. Since they were responsible
for formalizing the natural numbers, we might follow their lead and
do likewise.

John Conway notes that it is more "natural" to start with 0, and that
he has done so since the 60s. Phil Parker says the same, noting its
connection with SMSG textbooks (also the 60s). This probably
coincides with New Math trends generally, where set theory was taken
as the foundation for arithmetic.

My conjecture, therefore, is that it was the set theoretic
cardinality approach to natural numbers (Cantor? Zermelo? von
Neumann?) that switched (many of us) to taking the natural numbers as
starting with 0: it's the cardinality of the empty set.

Footnote: Dedekind takes a set theoretic approach in 1889, but seems
not to consider the empty set. When did this first enter set theory?
Is it already in Cantor?

On 11 Nov 97 John Conway wrote:

> On Tue, 11 Nov 1997, Michael Button wrote:

> > >Usual notation is :
> > >
> > >Natural numbers = {0,1,2,3,......}
> > >
> > >positive integers = {1,2,3,......}

> >
> > This does not agree with what I learned as an undergraduate or as a
> > graduate, so I looked it up in some of my old texts, and the definitions I
> > learned are
> >
> > Natural numbers = {1,2,3,...}
> >
> > Whole numbers = {0,1,2,3,...}
> >
> > Thus I would have said (and this is what I teach) that the set of natural
> > numbers is the same as the set of positive integers. Is this non-standard?
> > Does anyone know the history behind the naming of these sets?
> >
> > Michael Button
> >

> The older nomenclature was that "natural number" meant
> "positive integer". This was unfortunate, because the set
> of non-negative integers is really much more `natural' in
> the sense that it has simpler properties. So starting in
> about the 1960s lots of people (including me) started to
> use "natural number" in the inclusive sense. After all, for
> the positive integers we have the much better term "positive integer".
> John Conway

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