Natural numbers: {1,2,3,4,...}
The set of
natural numbers, also known as the
counting numbers is equal to the set of all positive
integers.
Contrary to popular belief, zero is not a natural number. A good trick to remember this is to recall the phrase, It is not natural to start counting with zero.
The above inductive definition is not correct because step one states that zero is a natural number. Since zero is not positive (nor negative) is cannot be a natural number.
I am not familiar with Peano's postulate by I am guessing that this is an over-simplified version of it and I am also willing to bet that it goes something a little more like this (this is also over-simplified):
- 1 is a natural number because it is a positive (has value greater than zero) integer
- assume for some n=k, k is a natural number (where k>1)
- prove k+1 is a natural number
hence all integers greater than 1 are natural numbers.
After further research I have discovered that there are
indeed two definitions for natural numbers and it suggested
that when refering to number sets one should use the terms
integer, positive integer, and nonnegative integer
instead of natural number and counting number.