Mathematics Elementary Regular Number Systems Number System Count

The five tiers

The familiar tower of number systems, the naturals, integers, rationals, reals, and complexes, is a standard piece of mathematics. Each tier adds one canonical algebraic closure step: the integers close subtraction, the rationals close division, the reals close limits, and the complexes close polynomial roots. This ladder is a standard structural fact, taught in every undergraduate course.

The Recognition Science framework's machine-checked library of formal theorems records this fact as a formal theorem named numberSystem_count. It states that the cardinality of the set of these five systems is exactly 5. The proof is a direct computation, and the declaration carries no axioms and no unfinished proof obligations. It is a small theorem, but it anchors a structural claim: the framework's mathematics treats these five systems as a complete, discrete family, not an open-ended list.

In Recognition Science, this count is a definitional choice about how the framework organizes its number systems. The framework models the five tiers as a closed set, with each tier a distinct member of an inductive type. The theorem numberSystem_count proves the count of that type is 5. It does not derive the existence of these systems from first principles, and it does not claim that no other number system could be defined. It records a structural fact about the framework's chosen vocabulary.

The practical upshot is that when the framework's mathematics speaks of the elementary regular number systems, it means exactly these five, in this order, with no gaps and no extras. This gives a fixed reference point for later constructions that build on the naturals, integers, rationals, reals, or complexes. The count is a small but load-bearing part of the framework's mathematical infrastructure.

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

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

What this page does not claim

The theorem does not derive the existence or necessity of these five systems from deeper principles. The theorem does not claim that no other number system could be defined or considered. The theorem does not assign any algebraic or analytic properties to the individual systems.

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.

Derived articles

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