Mathematics Number Systems From Rs Rational Contains Jcost Domain

The rationals and the cost domain

The rational numbers, written ℚ, are the fractions of whole numbers, such as 1/2 or 7/3. They sit between the integers and the real numbers in the standard hierarchy of number systems. A key fact about them is that the number 1 is positive, which is true by definition of how ordering works on fractions. The Recognition Science framework's machine-checked library of formal theorems records this elementary fact as the declaration rational_contains_jcost_domain.

In the framework, each number system corresponds to a different depth of recognition, a discrete record of events that reality keeps. The natural numbers count recognitions, the integers record signed differences, and the rationals express ratios of recognitions. The cost function J(x) = (x + 1/x)/2 - 1, which measures the forced cost of recognition, is defined only for positive real inputs. The theorem rational_contains_jcost_domain establishes that the rationals contain at least one positive element, namely 1, so the domain where J is meaningful is nonempty within the rationals.

This is a deliberately modest result. It does not show that J is defined on all positive rationals, nor that J takes rational values there, nor that the rationals are the natural home of the cost function. The theorem only confirms that the rational number system is not empty of positive elements, a prerequisite for any further claim about J's behavior on that system. The framework models the rationals as the system where J is defined, but the existence of a positive rational is what this declaration pins down.

What the declaration does not claim is more interesting than what it does. It does not assert that J maps rationals to rationals, which would require a separate proof about the arithmetic of fractions. It does not claim that the rationals are the only system containing the cost domain, since the reals also contain all positive numbers. It does not even claim that J is defined at every positive rational, only that the domain is inhabited by at least one point. The declaration is a foundation stone, not a theorem about the cost function's values.

In the broader framework, this small fact supports the placement of the rationals in the five-system hierarchy as the system where the cost function first becomes meaningful. The real numbers, which contain all positive values, come later in the hierarchy as the continuous recognition field. The rationals sit at the threshold where ratios of recognitions become expressible, and this declaration confirms that the threshold is nonempty. A reader can now see that the framework's number systems are not arbitrary labels but are tied to where the cost function can operate.

/-- Rational system contains J-cost domain (positive rationals). -/
theorem rational_contains_jcost_domain : (1 : ℚ) > 0 := by norm_num

inductive NumberSystem where
  | natural | integer | rational | real | complex
  deriving DecidableEq, Repr, BEq, Fintype

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

What this page does not claim

J is defined on all positive rationals. J maps rationals to rationals. The rationals are the only number system containing the cost domain.

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

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