Tuesday, February 26, 2013

Ordinal numbers.

If you are in Hong Kong and you need help for university mathematics courses, please visit  www.all-r-math.com.

Giuseppe Peano says 1 is the smallest natural number, while John von Neumann says that 0 is the smallest natural number?
Yes. In the past, or even nowadays, 0 as a number is not quite natural in many people's eyes. However, when mathematicians want to give a vigorous definition for ordinal numbers, they figure out that 0 should be the starting point.

What is the vigorous definition for ordinal numbers?
An ordinal number α is a set of ordinal numbers that are smaller then α.

So 3={0,1,2}, 5={0,1,2,3,4} are ordinal numbers?
True. All natural numbers are ordinal numbers. Indeed the set

is also an ordinal number. It is the smallest infinite ordinal number.

The second smallest infinite ordinal is {0,1,2,...,ω}?
Yes, it is ω+1. Note that it is different from 1+ω which equals to ω. The definition of addition is like this--
{0,1,2,...}+{0,1,2,...}={0,1,2,..., 0,1,2,...},
where the symbols with different colors are considered different, and that the elements are listed in the order.
By renaming the elements, we see that
The addition for finite ordinals is the usual addition of natural numbers.
However, we see that
ω+1={0,1,2,...}+{0}={0,1,2,...}+{ω}={0,1,2,...,ω} but
The two are different as ω+1 has a largest element but ω does not.

I doubt we still ignore 0 with we deal with ordinal numbers in daily language?
Well, we have invented a word "zeroth", denote that something should come earlier before the first object in a series. For example, the zeroth chapter or the zeroth law of thermodynamics.