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From: Archimedes Plutonium on 9 Jan 2010 00:47
First I am going to quote Wikipedia on what the Peano Axioms are, so
as to give
a basis for the discussion. And the Peano Axioms are so riddled with
contradiction
and inconsistency that I need to replace them with a new system.

The major flaws of the Peano Axioms are the assumption of finite and
infinite
with never a well-defined finite, nor infinite. This causes the Peano
Axioms to
be inconsistent. There are minor flaws in the Peano Axioms such as
never a
"metric ruler" established for the creation of the Successor Function
as endless
adding of 1. So we need to create as given axiom, not just the
existence of
0 but the existence of a metric ruler of 0 and 1 so that we use the
metric distance
of "1 unit" for the Successor Function. Otherwise, the Peano Axioms
could just as well
be the set {0, 1/2, 1, 1.5, ...}. This is a minor error. Another error
is the need for inclusion
of a Mathematical Induction. Math Induction comes out of the Successor
Axiom and no
need to have a redundant axiom.

So in the next few days I shall offer the AP-Natural-Numbers-Axioms.


--- quoting Wikipedia on the Peano Axioms ---

The first four axioms describe the equality relation.
1. For every natural number x, x = x. That is, equality is
reflexive.
2. For all natural numbers x and y, if x = y, then y = x. That is,
equality is symmetric.
3. For all natural numbers x, y and z, if x = y and y = z, then x =
z. That is, equality is transitive.
4. For all a and b, if a is a natural number and a = b, then b is
also a natural number. That is, the natural numbers are closed under
equality.

The remaining axioms define the properties of the natural numbers. The
constant 0 is assumed to be a natural number, and the naturals are
assumed to be closed under a "successor" function S.
5. 0 is a natural number.
6. For every natural number n, S(n) is a natural number.

Peano's original formulation of the axioms used 1 instead of 0 as the
"first" natural number. This choice is arbitrary, as axiom 5 does not
endow the constant 0 with any additional properties. However, because
0 is the additive identity in arithmetic, most modern formulations of
the Peano axioms start from 0. Axioms 5 and 6 define a unary
representation of the natural numbers: the number 1 is S(0), 2 is S(S
(0)) (= S(1)), and, in general, any natural number n is Sn(0). The
next two axioms define the properties of this representation.
7. For every natural number n, S(n) ≠ 0. That is, there is no
natural number whose successor is 0.
8. For all natural numbers m and n, if S(m) = S(n), then m = n.
That is, S is an injection.

These two axioms together imply that the set of natural numbers is
infinite, because it contains at least the infinite subset { 0, S(0), S
(S(0)), … }, each element of which differs from the rest. The final
axiom, sometimes called the axiom of induction, is a method of
reasoning about all natural numbers.
9. If K is a set such that:
▪ 0 is in K, and
▪ for every natural number n, if n is in K, then S(n) is in K,
then K contains every natural number.

The induction axiom is sometimes stated in the following form:
If φ is a unary predicate such that:
▪ φ(0) is true, and
▪ for every natural number n, if φ(n) is true, then φ(S(n)) is true,
then φ(n) is true for every natural number n.

--- end quoting Wikipedia on what the Peano Axioms are ---

And funny how the concept of finite and infinite are never mentioned
in
any of the axioms of Peano.

Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
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