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[HM] Natural numbers
Posted:
Dec 13, 2001 6:19 AM
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Calvin Jongsma wrote:
>1. Dedekind's classic 1887 work on natural numbers explicitly excludes
>classes with no objects; he has to beg the reader's indulgence to accept
>classes with single elements as being genuine classes. So there is no
>idea of starting with 0 here; 1 is the base element of the natural number
>system.
>2. Peano's classic 1889 work independently also takes 1 as the least
>natural number. This seems, then, to be the standard approach at this
>time period.
Peano's work was not independent of Dedekind's. It was quite consciously and explicitly an attempt to formalize it.
William Tait wrote
>Very likely Russell was influenced by Frege in this regard: Unlike
>many authors writing about the concept of (natural) number in the
>late 19th century (e.g. Kronecker, Helmholz [as I recall] and
>Dedekind), who thought of the numbers in terms of counting, Frege
>conceived of them as cardinals, the basic concept being `the number
>of x such that F(x), where F is a concept. Since this would include
>empty concepts (and it would have been unnatural not to), 0 had to be
>included.
>
>But maybe there is a question of how much immediate influence this,
>and the work of Russell, had on the general mathematical practice.
IMHO not much -- directly. But what has had a lasting effect on maths practice was Zermelo's contemporaneous axiomatization of set theory which (with some additions due mainly to Skolem and Fraenkel) is now used by most mathematicians (knowingly or otherwise) as the ambient theory within which one works.
In ZF, whether you regard the natural numbers as finite cardinals or as finite ordinals, the natural starting point is 0, for reasons analogous tho that which Bill Tait has pointed out wrt Frege & Russell.
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