Foundation Logic Real Transcendentals

The recovered real line

The real numbers are the continuum we use for measurement, from the length of a table to the value of a physical constant. They include the rational numbers, like 1/2, and the irrational ones, like the square root of 2, which cannot be written as a simple fraction. Transcendental numbers are a special class of irrationals: they are not the solution to any polynomial equation with integer coefficients. The most famous examples are π, the ratio of a circle's circumference to its diameter, and the base of natural logarithms, e, which governs growth and decay. These numbers are not just curiosities; they are the vocabulary of calculus, physics, and engineering.

The history of these numbers is long. The ancient Greeks knew of irrationals, but the existence of transcendentals was only proved in the 19th century. In 1844, Joseph Liouville constructed the first explicit examples. In 1873, Charles Hermite proved that e is transcendental, and in 1882, Ferdinand von Lindemann proved the same for π, settling the ancient question of whether one could square the circle with a compass and straightedge. The answer was no. These proofs showed that the real line is far richer than the algebraic numbers, and they set the stage for modern analysis.

In Recognition Science, the framework models reality as a discrete ledger of recognition events, and from that ledger it derives a real line, called LogicReal, which is equivalent to the standard real numbers. This means that the continuous real line, with all its transcendental functions, is not an assumption but a consequence of the framework's starting point. The module LogicRealTranscendentals establishes this by transporting the standard definitions through a proved equivalence. It defines the square root, exponential, logarithm, real power, π, sine, cosine, and hyperbolic functions on LogicReal, and it proves that these transported functions behave exactly like their classical counterparts.

The key fact is the equivalence itself: the recovered real line is equivalent to the standard real numbers. The module then defines each transcendental function by transport through this equivalence. For example, the exponential on LogicReal is defined as the function that maps a recovered real x to the recovered real corresponding to the standard exponential of the corresponding standard real. The module proves that this transported exponential is always positive, and that it is the inverse of the transported logarithm. It also proves the standard identity that the hyperbolic cosine is the average of the exponential and its reciprocal. These are not new results; they are the familiar facts of real analysis, restated in the framework's own terms.

What this establishes in plain language is that the framework does not need to invent a new kind of mathematics. It recovers the same real line that mathematicians have used for centuries, with the same π and the same e, and it proves that the recovery is exact. This is the foundation layer: later modules can reason over LogicReal and reduce analytic identities to the established real-analysis library, secure in the knowledge that the ground beneath them is the same ground classical mathematics stands on. The framework's claim is not that it has found a new number system, but that it has shown the old one is already implied by its own primitive structure.

@[simp] theorem toReal_expL (x : LogicReal) :
    toReal (expL x) = Real.exp (toReal x) :=
  toReal_fromReal _

@[simp] theorem toReal_sqrtL (x : LogicReal) :
    toReal (sqrtL x) = Real.sqrt (toReal x) :=
  toReal_fromReal _

/-- Transported positivity of exponential. -/
theorem expL_pos (x : LogicReal) : (0 : LogicReal) < expL x := by
  rw [lt_iff_toReal_lt, toReal_zero, toReal_expL]
  exact Real.exp_pos _

/-- Transported hyperbolic cosine definition. -/
theorem coshL_eq_exp (x : LogicReal) :
    coshL x = (expL x + expL (-x)) / fromReal 2 := by
  rw [eq_iff_toReal_eq, toReal_coshL, toReal_div, toReal_add, toReal_expL,
    toReal_expL, toReal_neg, toReal_fromReal]
  exact Real.cosh_eq _

What this page does not claim

This module does not prove that the real numbers exist; it assumes a real line is recovered and proves the transcendental functions transport correctly. This module does not derive the value of π or e; it transports the standard definitions from the classical real numbers. This module does not claim the framework's ledger is the only way to construct the real line.

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/Foundation/LogicRealTranscendentals.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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