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The following is quoted from
Edmund Landau, Foundations of Analysis, Chelsea, 1951,
pp. 1-18.
We assume the following to be given:
A set (i.e. totality) of objects called natural numbers, possessing
the properties--called axioms--to be listed below.
Before formulating the axioms we make some remarks about the symbols
= and which will be used.
Unless otherwise specified, small italic letters will stand for
natural numbers...
If x is given and y is given, then
either x and y are the same number; this may be written x=y (=
to be read ``equals'');
or x and y are not the same number; this may be written
( to be read ``is not equal to'').
Accordingly, the following are true on purely logical grounds:
- x=x for every x.
- If x=y then y=x.
- If x=y, y=z then x=z.
Thus a statement such as a=b=c=d, which on the face of it means
merely that a=b, b=c, c=d, contains the additional information
that, say, a=c, a=d, b=d....
Now, we assume that the set of all natural numbers has the following
properties:
- Axiom 1:
- 1 is a natural number. That is, our set is not
empty; it contains an object called 1 (read ``one'').
- Axiom 2:
- For each x there exists exactly one natural
number, called the successor of x, which will be denoted by x'.
- Axiom 3:
- We always have . That is, there
exists no number whose successor is 1. That is, there exists no
number whose successor is 1.
- Axiom 4:
- If x'=y' then x=y. That is, for any given number
there exists either no number or exactly one number whose successor is
the given number.
- Axiom 5 (Axiom of Induction):
- Let there be given a set
M of natural numbers, with the following properties:
- I.
- 1 belongs to M.
- II.
- If x belongs to M then so does x'.
Then M contains all the natural numbers.
Notice that there is no mention of such things as addition or
multiplication. How are these to be defined?
Next: Some Theorems Derivable from
Up: The Natural Numbers
Previous: Some Historical Notes on
Carl Lee
Wed Sep 16 09:26:16 EDT 1998