As a concrete concept, the notion of numeral is clear. The attempt to

formalize the

concept usually proceeds as follows:

(1) zero is a number

(2) the successor of a number is a number

(3) zero is not the successor of any number

(4) different numbers have different successors

(5) something is a number only if it is so by virtue of (1) and (2)

We shall refer to this as the usual dfinition. Sometimes (3) and (4)

are not stated explicitly, but it is the extremal clause (5) that is

unclear. What is the meaning of "by

virtue of"? It is obviously circular to define a number as something

constructible by

applying (1) and (2) any number of times. We cannot characterize

numbers from below,

so we attempt to characterize them from above.

[Edward Nelson: "Elements", p. 7, 10, Version 26. 9. 2011]

http://www.math.princeton.edu/~nelson/books/elem.pdf

The Peano Axioms are believed to define the natural numbers. They

don’t. If you take them literally, you will find that every sequence

starting with 1 (respectively 0) and without repeating terms will

comply with the axioms. That does not only hold for 1, 1/2, 1/4, …

(where the successor always is found in a very natural way, namely by

halving the cake) or for 1, 1/2^2, 1/3^2, … where we even can compute

the sum pi^2/6 of all natural numbers. Also all rational numbers (in

Cantor’s sequence) or all algebraic numbers (in Dedekind’s sequence)

comply with the Peano Axioms. Has anybody felt problems by that

imprecision?

In my books and in my math-lessons I use the axioms (although I

never met a student who really needed them to understand |N)

(1) 1 is in M

(2) If n is in M, then n+1 is in M

(3) Every set M that complys with (1) and (2), contains |N

(of course |N must also comply with (1) and (2))

Some mathematicians oppose that +1 is undefined as long as the real

numbers have not been introduced. But is +1 “defined” by the usual

axioms that fix commutativity and associativity of addition? No in the

least!

+1 must be known and in fact is known by every human. Without

knowing it, you cannot “define” the natural sequence. And knowing it,

you need no axioms at all, in particular you need not the notion of a

successor.

[Posted by: WM on October 2, 2011]

http://golem.ph.utexas.edu/category...ml#c039531
Gruß, WM

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