NetNews Usenet Archive 1993 #3/NN_1993_3.iso/spool/sci/math/18641

Text File | 1993-01-22 | 1.6 KB | 51 lines

Newsgroups: sci.math
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From: [e-mail] (Gurel-gurevitch Ori)
Subject: Re: Integers
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Organization: School of Math & CS - Tel Aviv University , Tel Aviv , ISRAEL.
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Date: Fri, 22 Jan 1993 09:43:18 GMT
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 Magnus Olsson wrote:
 >>One way of defining integers is the following:
 >>
 >>a) There exists an integer called 1.
 >>b) Every integer n has a successor n'.
 >>c) Every integer n except 1 has a predecessor m, i.e. there exists
 an
 >>   integer m such that m' = n.
 >>
 >>This defines the non-negative integers.
 ...What I wrote down above is a set of axioms for the integers (known

 as the Peano axioms)

 This is not true.
 the correct Peano's Postulate are:
 1)there exist an integer called 0(sometimes 1).
 2)every integer n has a successor S(n).
 3)for every intgers n and m S(n+m)=S(n)+m
 4)if A is a setof integers 0 belongs to A and for every n which
 belongs
   to A S(n) belongs to A too then A is N(the set of all integers).

   the axioms Magnus gives define a structure of the integers which
   can be
   very different from what we know, for example:R-(-N) (the set of
   all
   the real numbers except the non-positive integer).
   the third axiom is neaded to make use of + and the fourth one is
   known
   as the principle of induction (which make induction possible),
                                   J.

--

Gurel-Gurevitch Ori            [e-mail]