Concept of Natural Numbers

∀n n ∈ ℕ ⇒ n + 1 ∈ ℕ

The universal quantification ∀n asserts we are referring to all of the Domain.

The double struck ℕ indicates that the rule applies to the domain of natural numbers only. ℕ defines what we mean by “every number”.

The set membership statement n ∈ ℕ, declares that whatever the variable n indexes belongs to the set of natural numbers.

The rule says that for any natural number, there is a natural number that is one unit larger.

The final set membership statement, n + 1 ∈ ℕ, declares that the successor number n + 1 is also a member of the set of natural numbers.

Not only does the rule define a valid transformation (we can add one to any number), it also generates all admissible numbers. Combined with other rules, it contributes to the definition of our concept of natural numbers.