Natural number

Mathematical statement

Under this entry’s convention, the natural-number set N contains 0 and the numbers 1, 2, 3, and so on obtained by finitely many successor steps from 0.

Representative formula (exact TeX)

n\in N

Assumptions

• The symbol N in this entry is explicitly defined to include 0.
• n denotes an arbitrary number whose membership in N is under consideration.

Symbol scope

• n — candidate natural number
• N — set of natural numbers
• ∈ — membership relation

Intuition, definition, and misconceptions

Natural numbers provide the basic language of counting and discrete order. The number 0 represents the quantity of no objects; taking the successor of any natural number produces the next natural number.

Mathematical traditions differ on whether 0 belongs to the natural numbers. For consistency with Lean Nat and later OCME objects, this entry fixes N to include 0. External sources should still be checked for their convention.

A natural number is distinct from a numeral used to write it. For example, decimal 2 and binary 10 can denote the same natural number.

A common mistake is to assume every text uses the same convention about 0, or to confuse a number with the notation used to write it.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

Verification boundary

This entry is a conceptual candidate. Its research sources are not OCME Evidence, and no computational check or Lean formal proof has yet been registered.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Build an explicit semantic bridge from OCME N, zero, and successor to Lean Nat, then generate content-addressed Evidence after validation.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.