Encyclopedia/All topics/Foundation

Foundation

Articles 2,881–2,940 of 2,979. Alphabetical by title.

• Foundation Universal Forcing Self Reference Meta Cost Eq Zero Iff

  A theorem in a machine-checked library shows that a framework for deriving mathematics can compare its own building blocks, but it stops short of Gödel-style self-proof.

• Foundation Universal Forcing Self Reference Meta Cost Self

  A small formal lemma says that comparing a logical structure with itself costs zero, a step in showing the framework's own method fits its own shape.

• Foundation Universal Forcing Self Reference Meta Cost Symm

  A small theorem about comparing logical structures shows that the act of comparison itself obeys a basic law of thought.

• Foundation Universal Forcing Self Reference Meta Cost Total

  A small formal theorem says that comparing two versions of the universe's arithmetic always yields a definite answer, but it does not claim the framework can prove itself.

• Foundation Universal Forcing Self Reference Meta Forced Arithmetic Invariance Se

  A theorem that compares systems of arithmetic turns out to obey the same structural law it describes, a self-reference the framework treats as closure, not paradox.

• Foundation Universal Forcing Self Reference Meta Meta Theorem

  A theorem that checks its own shape: the framework's central result, applied to itself, comes out unchanged.

• Foundation Universal Forcing Self Reference Meta Realization Cert Inhabited

  The framework's central theorem about logic itself fits the framework's own shape, a structural self-reference proved without claiming Gödel-style self-proof.

• Foundation Universal Forcing Strict Canonical Iso

  When two systems each obey the same minimal laws of logic, their number systems must be the same, and there is only one way to match them up.

• Foundation Universal Forcing Strict Canonical Iso Strict Peano Equiv Unique

  When two systems each generate their own arithmetic from pure law, there is exactly one way to translate between them.

• Foundation Universal Forcing Strict Canonical Iso Strict Universal Forcing Iso C

  A machine-checked certificate that any two strict realizations of the framework's logic have one and only one structure-preserving bridge between their derived arithmetics.

• Foundation Universal Forcing Strict Canonical Iso Strict Universal Forcing Peano

  Two different sets of primitive laws still force the same arithmetic structure, and the bridge between them is unique.

• Foundation Universal Forcing Strict Categorical

  A machine-checked bridge shows that the framework's discrete ledger can be realized as the natural numbers, the same counting structure behind arithmetic.

• Foundation Universal Forcing Strict Categorical Logic Nat Cost

  A cost function that charges 0 for equality and 1 for difference is the simplest possible ledger of recognition events.

• Foundation Universal Forcing Strict Categorical Logic Nat Cost Symm

  A machine-checked theorem shows that a simple two-valued cost function treats both sides of a comparison identically.

• Foundation Universal Forcing Strict Categorical Mathlib

  A machine-checked bridge shows that the framework's own counting numbers behave exactly like the familiar natural numbers, with the same universal recursion property.

• Foundation Universal Forcing Strict Categorical Mathlib Categorical Mathlib Cert

  A machine-checked certificate shows that the framework's counting numbers behave exactly like the ordinary natural numbers, no more and no less.

• Foundation Universal Forcing Strict Categorical Mathlib Nno Universal Existence

  A machine-checked proof shows the framework's counting numbers behave exactly like the natural numbers: any step-by-step process has one and only one way to run along them.

• Foundation Universal Forcing Strict Categorical Mathlib Nno Universal Uniqueness

  The declaration proves that the framework's counting numbers are the only way to count: any two counting processes that start the same and step the same must be the same proce

• Foundation Universal Forcing Strict Categorical Mathlib Recursor Succ

  A formal theorem about counting numbers shows that one step of a recursive process is exactly what the next number means, no more and no less.

• Foundation Universal Forcing Strict Categorical Mathlib Recursor Zero

  A single equation about counting from zero, and why it matters for building mathematics on a ledger of events.

• Foundation Universal Forcing Strict Categorical Strict Categorical Arith Equiv L

  A machine-checked bridge identifies the natural numbers built inside categorical logic with the framework's own counting structure.

• Foundation Universal Forcing Strict Categorical Strict Categorical Realization

  A machine-checked bridge shows that the framework's arithmetic can be built on the simplest possible number system: natural numbers with equality and a one-step cost.

• Foundation Universal Forcing Strict Discrete Boolean

  A two-valued logic with a simple cost rule turns out to generate the same natural-number arithmetic as a continuous system of ratios.

• Foundation Universal Forcing Strict Discrete Boolean Strict Boolean Arith Equiv

  A machine-checked library shows that two different starting points, Boolean logic and positive ratios, force the same counting structure.

• Foundation Universal Forcing Strict Discrete Boolean Strict Boolean Realization

  A tiny two-value logic system, with just true and false, already forces the same arithmetic as the positive ratios, a result about the foundations of counting.

• Foundation Universal Forcing Strict Discrete Boolean Strict Positive Ratio Arith

  Within the framework's machine-checked library, a comparison of positive ratios and a two-valued Boolean logic turn out to force the very same arithmetic.

• Foundation Universal Forcing Strict Discrete Boolean Xor Bool

  A two-valued logic gate turns out to define a complete arithmetic, and the same arithmetic that positive ratios force.

• Foundation Universal Forcing Strict Invariance

  A machine-checked proof that any universe with a strict discrete ledger must derive the same arithmetic, no matter how it starts.

• Foundation Universal Forcing Strict Invariance Strict Arith Universal Initial

  A machine-checked proof shows that any strict logical system, however it is built, must generate the same natural numbers.

• Foundation Universal Forcing Strict Invariance Strict Peano Surface

  Every strict realization of the framework's logic yields the same arithmetic structure, canonically, no matter which realization you start from.

• Foundation Universal Forcing Strict Invariance Strict Universal Forcing

  No matter which strict version of the framework's logic you start from, the numbers it produces are the same numbers, in exactly one canonical way.

• Foundation Universal Forcing Strict Mathlib Nno

  A machine-checked bridge shows that the framework's counting numbers satisfy the same universal property that defines the natural numbers in category theory.

• Foundation Universal Forcing Strict Mathlib Nno Logic Nat Has Type Nno Universal

  A natural number system is the one where every counting process, however strange, is forced to exist and to be unique.

• Foundation Universal Forcing Strict Mathlib Nno Logic Nat Nno Uniqueness

  A machine-checked theorem pins down the natural numbers as the unique structure that supports recursive definition, and says nothing about what those numbers are.

• Foundation Universal Forcing Strict Mathlib Nno Mathlib Nnocert

  A compact formal certificate says the framework's counting numbers behave exactly like the natural numbers, no more and no less.

• Foundation Universal Forcing Strict Mathlib Nno Mathlib Nnocert Holds

  A formal certificate proves that the framework's counting numbers behave exactly like the natural numbers of standard mathematics.

• Foundation Universal Forcing Strict Modular

  A finite clock face can carry the same forced arithmetic as the infinite number line, if the cost of telling two positions apart is simply 0 or 1.

• Foundation Universal Forcing Strict Modular Strict Modular Realization

  In modular arithmetic, a strict recognition cost that charges 0 for equality and 1 for any difference still forces the same free arithmetic structure as the integer case.

• Foundation Universal Forcing Strict Modular Zmod Cost

  A simple rule for counting differences on a clock face, and the precise limits of what that rule proves.

• Foundation Universal Forcing Strict Music

  A musical scale as a discrete record of events, where the only cost is whether two notes are the same.

• Foundation Universal Forcing Strict Music Music Arith Equiv Logic Nat

  A machine-checked library of formal theorems shows that a musical scale built from octaves can carry the same arithmetic as the natural numbers.

• Foundation Universal Forcing Strict Music Music Is Positive Ratio Subrealization

  A formal proof that musical intervals, taken as positive frequency ratios, form a valid model of arithmetic logic.

• Foundation Universal Forcing Strict Music Octave

  In the Recognition Science framework, the octave is not a musical accident but a primitive unit of a discrete recognition ledger, proven to cost nothing to recognize as identical.

• Foundation Universal Forcing Strict Music Perfect Fifth

  The perfect fifth is the musical interval between two notes whose frequencies stand in a 3 to 2 ratio, a definition that predates any framework.

• Foundation Universal Forcing Strict Music Perfect Fourth

  In music, the perfect fourth is the interval between two notes whose frequencies stand in a 4:3 ratio, a definition that needs no theory of recognition.

• Foundation Universal Forcing Strict Music Ratio Cost

  In the Recognition Science framework, ratioCost is the simplest possible way to compare two musical frequency ratios: 0 if they are the same, 1 if they differ.

• Foundation Universal Forcing Strict Music Ratio Cost Symm

  A musical interval costs the same whether you ascend or descend; this theorem records that symmetry as a formal rule.

• Foundation Universal Forcing Strict Music Strict Music Realization

  A machine-checked construction shows how musical intervals, built from octave stacking, form a complete arithmetic system.

• Foundation Universal Forcing Strict Ordered

  A minimal model of recognition cost on the integers shows how the framework's core axioms can be satisfied by a simple equality test.

• Foundation Universal Forcing Strict Ordered Int Cost

  A simple rule that charges 0 for equality and 1 for any difference turns the integers into a recognition ledger with a proved symmetry.

• Foundation Universal Forcing Strict Ordered Strict Ordered Arith Equiv Logic Nat

  A machine-checked proof shows that a ledger whose only rule is 'same entry costs nothing, different entries cost one' still contains the natural numbers.

• Foundation Universal Forcing Strict Ordered Strict Ordered Realization

  A minimal formal model shows how a strict ordering and a unit step can realize the framework's logic, and where that model stops.

• Foundation Universal Forcing Strict Positive Ratio

  A strict, continuous model of comparison built directly from the laws of logic, and the arithmetic it forces is exactly the natural numbers.

• Foundation Universal Forcing Strict Positive Ratio Positive Ratio Arith Equiv Lo

  A formal bridge shows that the arithmetic forced by one recognition structure is exactly the same as the arithmetic forced by another, and nothing more.

• Foundation Universal Forcing Strict Positive Ratio Positive Ratio Strict Equiv E

  Two different routes to the same forced arithmetic produce the same natural numbers, a machine-checked bridge inside the Recognition Science framework.

• Foundation Universal Forcing Strict Positive Ratio Strict Positive Ratio Realiza

  A machine-checked library shows that the strict positive-ratio model of comparison yields the same arithmetic as the natural numbers.

• Foundation Universal Forcing Strict Realization

  Universal forcing says any system that obeys a few laws of logic must contain the natural numbers; strict realization proves it without letting the system secretly supply them.

• Foundation Universal Forcing Strict Realization Arith

  A machine-checked proof shows that any system obeying the basic laws of comparison and composition must contain the natural numbers, with no extra structure supplied by hand.

• Foundation Universal Forcing Strict Realization Arith Equiv Logic Nat

  A machine-checked theorem shows that any system satisfying a minimal set of logical laws must contain a structure indistinguishable from the natural numbers.

• Foundation Universal Forcing Strict Realization Free Orbit

  In the Recognition Science framework, the free orbit is the counting numbers, and it is the only orbit a strict realization is allowed to have.