Re: Thinking about MINUS

DBMS_Plumber wrote:
> Aloha Kakuikanu wrote:
> > DBMS_Plumber wrote:
> > > Marshall wrote:
> > > > Even if we consider a system consisting only of the natural numbers,
> > > > zero, successor and predecessor, we have to wrestle with the question
> > > > of what the predecessor of zero is.
> > >
> > > Zero is not an element of the domain of natural numbers.
> >
> > It certainly is.
>
> From: http://en.wikipedia.org/wiki/Natural_number
> "The natural numbers presumably had their origins in the words used to
> count things, beginning with the number one."
>

I have seen both definitions, and if you start from the beginning of the article:

In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i.e the positive integers) or an element of the set {0, 1, 2, 3, ...} (i.e. the non-negative integers). The former is generally used in number theory, while the latter is preferred in set theory and computer science. See below for a formal definition.

> From: http://mathworld.wolfram.com/NaturalNumber.html
> "A positive integer 1, 2, 3, ... (Sloane's A000027). The set of natural
> numbers is denoted N."
>
> >From Wordnet:
> "the number 1 and any other number obtained by adding 1 to it
> repeatedly"
>
> To be fair, what's confusing is that Peano axioms are often set down
> starting with '0 as a natural number', which (IIRC) simplifies some of
> the subsequent exposition. But this is a notational, not a logical,
> difference. Intuitively, the first 'symbol' (1, or 0, or |) is an
> anchor with special properties (see axiom # 3).
>

I believe there are some differences except notation. In one case we have an identity element for addition, in the other there is no identity.

/Lennart

> Also, you will note that Peano nowhere mentions the concept of a
> 'predecessor'. Induction points one way.
Received on Mon Jan 08 2007 - 20:42:49 CET