The set of natural numbers is infinite
This result can be understood by recalling the five axioms that characterize the natural numbers, together with the sixth axiom that introduces zero. By following these axioms step by step, we can see how the sequence of natural numbers is generated and why it never ends.
We begin by considering the set of natural numbers as empty.
$$ N = \{ \} $$
According to the sixth axiom defining the set of non-negative integers N0, zero is the additive identity.
This means that we must include the number zero in the set of non-negative integers N0.
$$ N_0 = \{ 0 \} $$
The fifth axiom of the natural numbers states that the multiplicative identity is the number 1.
We therefore introduce 1 into the sets N0 and N.
$$ N_0 = \{ 0 \ , \ 1 \ \} $$
$$ N = \{ \ 1 \ \} $$
Note. The set N0 represents the non-negative integers and includes zero, whereas the set N represents the positive natural numbers and does not include zero.
Successor of 1
The successor of 1 is obtained by adding 1:
$$ 1 + 1 = 2 $$
We now show, by elimination, that the symbol 2 denotes a new natural number.
- If 2 = 0, then 1+1=0. This contradicts the closure property of addition, since 1 belongs to N while 0 belongs to N0. In particular, zero 0∉N is not an element of N. By closure, the sum of two natural numbers such as 1+1 must itself be a natural number. Therefore, 2 cannot be equal to 0.
- If 2 = 1, then 1+1=1. This contradicts the axiom stating the existence of an additive identity in N0. If 1+1=1, then 1 would act as the additive identity. This is impossible, because zero is the additive identity and it is unique. Hence, 2 is different from 1.
Having established that 2 is different from both 0 and 1, and observing that no other elements are present in the sets N and N0, we conclude that 2 is a new number.
We therefore add the symbol 2 to the sets N and N0.
$$ N_0 = \{ 0 \ , \ 1 \ , \ 2 \} $$
$$ N = \{ 1 \ , \ 2 \} $$
Successor of 2
The next successor is obtained in the same way:
$$ 2 + 1 = 3 $$
Once again, we verify by elimination that the symbol 3 denotes a new natural number.
- For the same reasons discussed above, 3 is different from both 0 and 1.
- If 3 = 2, then 2+1=3=2. This would force 2 to behave as the additive identity. Such a situation is impossible, since zero is the additive identity and it is unique. Therefore, 3 is different from 2.
After showing that 3 is different from 0, 1, and 2, and noting that no other elements belong to the sets N and N0, we conclude that 3 is a new number.
We then add the symbol 3 to the sets N and N0.
$$ N_0 = \{ 0 \ , \ 1 \ , \ 2 \ , \ 3 \} $$
$$ N = \{ 1 \ , \ 2 \ , \ 3 \} $$
The same reasoning applies to every subsequent successor. Each time a new successor is formed, it cannot coincide with any of the numbers already introduced.
As a result, the sets N and N0 do not terminate and contain infinitely many elements.
And so on.
