Mathematics Elementary Regular Number Systems Number System

The five canonical number systems

The natural numbers, integers, rationals, real numbers, and complex numbers are the five canonical number systems of elementary mathematics. Each one extends the previous by adding a new kind of solution: negatives for subtraction, fractions for division, limits for measurement, and square roots of negative numbers for algebra. This ladder is standard and well-known; the Recognition Science declaration NumberSystem takes that classical fact and makes it precise in a machine-checked library of formal theorems.

The declaration is an inductive type with exactly five constructors, one for each system, and it derives basic computational properties such as equality and finite enumeration. The accompanying theorem numberSystem_count proves that the number of systems is five, and the certificate structure wraps that count into a single object. The whole file carries no unproved axioms and no admitted theorems, so the count is a fully checked result.

What the declaration does not claim is more limited than the name might suggest. It does not prove that these five systems are the only possible ones, nor that they are the best or most natural ones. It does not define the arithmetic operations on each system, nor does it prove any property about addition, multiplication, or ordering. It only records the standard list and certifies that the list has five entries.

In Recognition Science, the five-tier ladder is a structural depth result: each tier adds one canonical algebraic closure step, and the count of five is a theorem, not a convention. But the declaration itself is a definitional choice, not a discovery. The framework's own proofs about why five and not another number would come from deeper forcing results, not from this file.

The practical upshot is that when other parts of the library refer to a number system, they can rely on a fixed, finite, machine-checked enumeration. A reader who wants to know what the declaration establishes should read it as a precise inventory: these five, no more, and the count is proved.

inductive NumberSystem where
  | naturals
  | integers
  | rationals
  | reals
  | complexes
  deriving DecidableEq, Repr, BEq, Fintype

theorem numberSystem_count : Fintype.card NumberSystem = 5 := by decide

def numberSystemCert : NumberSystemCert where
  five_systems := numberSystem_count

inductive NumberSystem where
  | naturals
  | integers
  | rationals
  | reals
  | complexes
  deriving DecidableEq, Repr, BEq, Fintype

What this page does not claim

The declaration does not prove that the five systems are the only possible ones. The declaration does not define arithmetic operations or ordering on the systems. The declaration does not establish why there are five tiers beyond the standard list.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Mathematics/ElementaryRegularNumberSystems.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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