@Chris
You really bring out the best in me.

Every time I re-visit what I had previously studied I realise how flawed all the theorems and proofs are.
Here's the contradiction.
The set of natural numbers can be either defined as
N1 = {1,2,3...}
N2 = {0,1,2,3...}
depending on which definition of natural numbers you follow.
http://mathworld.wolfram.com/NaturalNumber.html
A set S is said to be countable if there exists a bijection from S to the set of natural numbers N
Let S = N2 and N = N1
Clearly, S here has a cardinality that is greater than N and there exists no bijection from S to N
Therefore the set of natural numbers N2 is uncountable if we assume that N1 is the set of natural numbers
But the set of natural numbers N2 is supposed to be countable and infinite.
It appears therefore that the method of enumeration and countability as suggested by Cantor is flawed because it is not even clear whether the set of natural numbers is N1 or N2
I further refer you to a statement by Poincare about the apparent folly of Cantor. In Poincares exact words "Later generations will regared Cantor's
Mengenlehre(Set Theory) as a disease from which one has recovered."
Quote: "A set that is countable has 'the same number' of elements as the integers, and a set that is uncountable has 'the same number' of elements as the real numbers.
"
No, a countable set S is a bijection from the set S to the set of natural numbers and has the cardinality of the set of natural numbers depending on which definition of natural numbers you follow.
An uncountable set is one that is not countable. i.e. has a cardinality greater than the set of natural numbers. That's all. The set of Real numbers just happens to be one example of an uncountable set.
Quote: "Quote: "However, the set of all numbers, positive and negative, is uncountably infinite."
No, they are countable like this; 1, -1, 2, -2, 3, -3, 4, -4, ...
"
No, the set of all numbers, positive and negative, is the set of Real numbers minus the singleton set containing a single element 0.
This set is uncountable and infinite.