Integer

Mathematical statement

The integer set Z contains 0, every positive natural number, and its negative. The notation n∈Z states that n is an integer.

Representative formula (exact TeX)

n\in Z

Assumptions

• This entry follows the OCME convention that natural numbers include 0.

Symbol scope

• n — candidate integer
• Z — set of integers
• ∈ — membership relation

Intuition, definition, and misconceptions

Natural numbers support counting. Integers add a negative direction, allowing temperature differences, elevation differences, debt, and displacement to share one number system.

A minus sign is not a judgment about the quality of a number; it is a structural notation relative to zero and additive opposites.

This entry establishes membership and notation boundaries. Universal properties of addition, multiplication, order, and divisibility require separate statements and evidence.

Common mistakes include treating a minus sign as subtraction itself, classifying every decimal-looking value as an integer, or inferring all integer algebra laws from a few examples.

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

• Natural number — Positive integers and zero enter through natural-number language; negative integers then add the opposite direction.

Verification boundary

Finite classification and arithmetic examples cannot replace a universal proof of the integer construction and its algebraic properties.

not_formalized

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

Next obligation

Bind the construction of Lean Int, the natural-number embedding, negative values, and zero precisely, then separate later order and ring properties.

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.