What is the Set of natural numbers and zero called?

Answer 1

There are two definitions of the natural numbers in use which differ slightly:

1) The natural numbers are the counting numbers, ie 1, 2, 3, ...

2) The natural numbers are the counting numbers and 0, ie 0, 1, 2, 3, ...

By your question I assume that the definition for are using is that ℕ = {1, 2, 3, ...}, and so are asking what the number of the set {0, 1, 2, 3, ...} is.

Either way, it doesn't matter as the number of elements in the set is infinite, denoted by Aleph-null: ℵ₀ which is the first level of infinity - the countable infinity.

It seems odd to say that infinity is countable, but what it means is that the elements of the infinite set can be put in a one-to-one correspondence with the counting numbers (ℤ⁺). One such set is the set of rational numbers (ℚ) - it is possible to create a method of listing ALL rational numbers, hence there can be a one-to-one relationship with the counting numbers.

One such method:

Take two number lines which are perpendicular - the x- and y-axes.

At every grid point off the axes that has x and y both integer, write the fraction y/x

As the y- and x-axes are infinite this will create every possible fraction (including equivalent fractions).

Now, start at the origin and move one step along the x-axis to (1, 0).

Now move up the (diagonal) line with slope -1 towards (0, 1).

From (0, 1) move up the y-axis to (0, 2) and move down the line with slope -1 to (2, 0)

Every time you reach an axis move one step away from the original and go along the line with slope -1 to reach the other axis.

When you pass through a point with integral x- and y-coordinates, there will be a fraction there, so note it down as the next fraction in the list.

This will collect every possible positive rational number

note that as ℤ⁺ ⊂ ℚ the whole numbers are included in every possible equivalent form: 1 = 1/1 = 2/2 = 3/3 = ...; 2 = 2/1 = 4/2 = 6/3 = ...; etc).

This will lead to the positive rational numbers being listed in the order: 1/1, 1/2, 2/1, 3/1, 2/2, 1/3, ...

The negative rational numbers can be included by inserting the negative version before or after the positive ones, eg: 1/1, -1/1, 1/2, -1/2, 2/1, -2/1, 3/1, -3/1, ... lists all positive and negative rational numbers.

zero can be included by noting the x-axis values with x > 0, ie 0/1, 0/2, 0/3, etc.

There are higher levels of infinity. The next one is Aleph-one (ℵ₁) which is the number of elements in the set of irrational numbers. A proof that the Irrational Numbers is not countable is fairly easy:

Assume we have listed ALL the irrational numbers and so put them is a one-to-one relationship with the counting numbers (ie there are ℵ₀ irrational numbers). - that is they are countable.

Add 1 (modulo 10) to the first digit of the first number;

Add 1 (modulo 10) to the second digit of the second number;

And so on down the list of ALL the irrational numbers - adding 1 (modulo 10) to the nth digit of the nth number.

The number created cannot be the first number as the first digits differ;

Nor can it be the second number as the second digits differ;

and so on: it cannot be the nth listed number as the nth digits differ.

Thus it cannot be a number in the list.

Thus the list cannot contain all irrational numbers.

This is a contradiction, so there cannot be a list of all the irrational numbers in a one-to-one relationship with the counting numbers; with such a list new numbers can be created which are not in the list.

Thus the list must have a higher number of elements than the counting numbers which has ℵ₀ elements.

This is higher infinity is called ℵ₁ such that ℵ₁ > ℵ₀.

If you are interested, for more information about levels of infinity, try a web search for Gregor Cantor.

Answer 2

The set of natural numbers. The set already has the zero in it.

Answer 3

According to Peano's axiomatic definition of our number system, the set of Natural numbers includes zero.

Answer 4

whole numbers