Encyclopedia/All topics/Foundation

Foundation

Articles 2,161–2,220 of 2,979. Alphabetical by title.

• Foundation Primitive Recognition Calculus Prcmodel Theory Non Forcing Real Has C

  Any first-order description of the real number line in a countable language also fits a countable structure that satisfies exactly the same sentences.

• Foundation Primitive Recognition Calculus Prcmodel Theory Non Forcing Real Not F

  A countable model of the real numbers agrees with them on every first-order sentence, yet has a different cardinality.

• Foundation Primitive Recognition Calculus Prcmonotone Dalembert

  A classical functional equation gets a new proof that uses order instead of continuity, shrinking the assumptions behind a core cost function.

• Foundation Primitive Recognition Calculus Prcmonotone Dalembert Composition Law

  A machine-checked proof shows that a simple monotonicity condition, not continuity, forces the recognition cost into a single family of curves.

• Foundation Primitive Recognition Calculus Prcmonotone Dalembert D Alembert Add O

  A single order property, monotonicity, replaces continuity in forcing the shape of a fundamental cost function.

• Foundation Primitive Recognition Calculus Prcmonotone Dalembert D Alembert Cosh

  A classic functional equation has a hidden order-only solution, and a machine-checked proof shows monotonicity alone can replace continuity.

• Foundation Primitive Recognition Calculus Prcmonotone Dalembert D Alembert Diff

  A single inequality, not calculus, decides which of two mirror-image curves a functional equation picks.

• Foundation Primitive Recognition Calculus Prcmonotone Dalembert D Alembert Ge On

  A monotone solution of a classical functional equation cannot dip below its starting value, and that simple fact replaces a whole analytic assumption.

• Foundation Primitive Recognition Calculus Prcmonotone Dalembert D Alembert S Add

  A single equation for the square-root part of a d'Alembert solution, proved with order alone and no reliance on continuity.

• Foundation Primitive Recognition Calculus Prcmonotone Dalembert Monotone Additiv

  A single, simple assumption about a function's shape, monotonicity, replaces the heavy analytic machinery of continuity in forcing a linear form.

• Foundation Primitive Recognition Calculus Prcnative Cost Minimality

  A machine-checked library proves that among all cost functions obeying its axioms, only one survives, and it is the same J(x) = (x + 1/x)/2 - 1.

• Foundation Primitive Recognition Calculus Prcnative Cost Minimality Certificate

  The framework's cost function is not just assumed: a machine-checked proof shows which axioms are essential, and which alternatives fail without them.

• Foundation Primitive Recognition Calculus Prcnative Cost Minimality Certificate

  A machine-checked proof shows that dropping one calibration condition lets a different cost function survive, so the uniqueness theorem needs every premise it uses.

• Foundation Primitive Recognition Calculus Prcnative Cost Minimality Character Pa

  A machine-checked theorem shows that fixing a cost function's value at the number two forces its values at every prime number, a step in a broader attempt to derive physics fr

• Foundation Primitive Recognition Calculus Prcnative Cost Minimality Constant Zer

  A machine-checked proof rules out the simplest possible cost function, the one that charges nothing, in a framework where recognition must carry a forced price.

• Foundation Primitive Recognition Calculus Prcnative Cost Minimality Native Cost

  A machine-checked ledger records which premises a cost-selection proof actually uses, and this declaration certifies that every entry carries the same minimal strength tag.

• Foundation Primitive Recognition Calculus Prcnative Cost Minimality Prcsigned St

  A proposed shortcut for deriving the fundamental cost function fails, and the machine-checked proof shows exactly why.

• Foundation Primitive Recognition Calculus Prcnative Cost Minimality Prczero Cali

  A machine-checked theorem pins down the only cost function that meets a strengthened set of recognition conditions, while a companion result shows why a simpler version fails.

• Foundation Primitive Recognition Calculus Prcnative Cost Selection

  A machine-checked proof that only one cost function survives five plain conditions, and the false candidates it rules out.

• Foundation Primitive Recognition Calculus Prcnative Cost Selection Canonical Sel

  A single theorem in a machine-checked library shows that the framework's chosen cost function is not an empty definition, and it pins down exactly where that proof's auth

• Foundation Primitive Recognition Calculus Prcnative Cost Selection Constant Zero

  A proposed rule that charges nothing for recognition fails the framework's own axioms, and the proof is a single line of arithmetic.

• Foundation Primitive Recognition Calculus Prcnative Cost Selection Native Cost S

  A machine-checked theorem certifies that every premise in a central cost-selection ledger carries the weakest possible evidential tag, a fact with sharp limits.

• Foundation Primitive Recognition Calculus Prcnative Cost Selection Native Deposi

  A machine-checked theorem ranks two kinds of evidence inside Recognition Science, and the ranking carries a precise limit.

• Foundation Primitive Recognition Calculus Prcnative Cost Selection Prcprime Sign

  A machine-checked proof shows that one proposed set of conditions for a cost function is too weak to single it out, and names the counterexample.

• Foundation Primitive Recognition Calculus Prcnative Cost Selection Zero Flat Nat

  A machine-checked proof shows a specific cost function satisfies the framework's strongest axioms, while a companion proof shows the uniqueness target itself fails.

• Foundation Primitive Recognition Calculus Prcnative Cost Structural Ledger

  A structural ledger is a bookkeeping rule that assigns a cost to every ratio, and Recognition Science's library proves that only one such rule can exist.

• Foundation Primitive Recognition Calculus Prcnative Cost Structural Ledger Canon

  A single rule about how cost changes when a ratio is flipped forces the entire cost function, and the proof is checked by machine.

• Foundation Primitive Recognition Calculus Prcnative Cost Structural Ledger Even

  A machine-checked proof shows that a cost function built from squaring ratios is the unique one satisfying a short list of structural conditions, and that dropping one condition ma

• Foundation Primitive Recognition Calculus Prcnative Cost Structural Ledger Prcsi

  A theorem in the Recognition Science library shows that any cost function obeying five natural conditions must be the same one, J(x) = (x + 1/x)/2 - 1.

• Foundation Primitive Recognition Calculus Prcnative Cost Structural Ledger Prcst

  A machine-checked proof shows that one proposed uniqueness claim for the recognition cost function is false, and exactly why it fails.

• Foundation Primitive Recognition Calculus Prcnative Cost Structural Ledger Struc

  A single cost function on ratios is forced by five plain conditions, and the proof shows why each condition is needed.

• Foundation Primitive Recognition Calculus Prcnative Cost Uniqueness

  A single cost function for recognition events is forced by five plain conditions, and the proof is checked by a machine.

• Foundation Primitive Recognition Calculus Prcnative Cost Uniqueness Prcprime Cal

  A machine-checked proof shows that if a recognition cost behaves correctly on two and three, it cannot secretly misbehave on composite numbers.

• Foundation Primitive Recognition Calculus Prcnative Cost Uniqueness Prcsigned St

  A formal target in the Recognition Science library that, if proved, would tie the unique cost function to a signed character factorization, but which the library currently refutes.

• Foundation Primitive Recognition Calculus Prcnative Cost Uniqueness Prcstrengthe

  A machine-checked library of formal theorems maps out exactly which extra conditions force a unique cost function in Recognition Science, and which combinations fail.

• Foundation Primitive Recognition Calculus Prcone Primitive

  A distinction between two things needs only one primitive act, not two: the act itself carries the comparison.

• Foundation Primitive Recognition Calculus Prcone Primitive Act Judgment

  A single primitive act can generate the ability to compare, without a second built-in rule for sameness or difference.

• Foundation Primitive Recognition Calculus Prcone Primitive Act Judgment Diff

  In Recognition Science, the act of comparing two things is not a separate primitive: it is a consequence of the act's own structure.

• Foundation Primitive Recognition Calculus Prcone Primitive Act Judgment Same

  In Recognition Science, the act of recognizing two things as the same or different is not a separate choice but a consequence of the act itself.

• Foundation Primitive Recognition Calculus Prcone Primitive Act Judgment Same Dec

  In Recognition Science, the act of distinguishing two things is not a separate primitive: it is a derived property of the act-generated structure itself.

• Foundation Primitive Recognition Calculus Prcone Primitive Comparison Is Derived

  The ability to tell two things apart is not a separate power in this framework; it is a consequence of the act that creates the things.

• Foundation Primitive Recognition Calculus Prcone Primitive Endpoint Eq Left Or R

  A distinction has exactly two sides, and the framework's library proves this is the only possibility for its primitive act of recognition.

• Foundation Primitive Recognition Calculus Prcone Primitive Genuine Judgment Same

  A theorem about the simplest possible act of comparison shows that saying 'same' and saying 'equal' are the same thing, with no second primitive needed.

• Foundation Primitive Recognition Calculus Prcone Primitive Structure

  A formal framework that starts with a single act of recognition and derives the ability to compare from it, rather than assuming comparison as a separate ingredient.

• Foundation Primitive Recognition Calculus Prcset Theory Parse

  A machine-checked library shows that the foundation's primitive recognition calculus can encode all of hereditarily finite set theory, using only the natural numbers.

• Foundation Primitive Recognition Calculus Prcset Theory Parse Distinguishes Iff

  In the framework's formal world, two sets are different exactly when they have different members, a fact that anchors all of set theory to a simple bit-level code.

• Foundation Primitive Recognition Calculus Prcset Theory Parse Hf Set Theory Real

  The hereditarily finite sets, built from nothing but the empty set, form a minimal universe that satisfies the basic axioms of set theory.

• Foundation Primitive Recognition Calculus Prcset Theory Parse Hf System Embeds D

  A machine-checked proof shows that the hereditarily finite sets, the universe built from the empty set by pairing, can be coded as ordinary numbers inside the Recognition Science f

• Foundation Primitive Recognition Calculus Prcset Theory Parse Hf System Expr Ref

  A single line of formal proof shows that the hereditarily finite sets can be ordered so that every set extends itself, a structural property with a precise scope.

• Foundation Primitive Recognition Calculus Prcset Theory Parse Hf System Expressi

  A machine-checked proof shows that the hereditarily finite sets, coded as natural numbers, form a system rich enough to support the framework's foundational claims.

• Foundation Primitive Recognition Calculus Prcset Theory Parse Hf System Not Dege

  A machine-checked proof shows that the hereditarily finite sets, the simplest universe of sets built from nothing, can serve as a non-degenerate foundation for recognition events.

• Foundation Primitive Recognition Calculus Prcset Theory Parse Mem One Iff

  In the framework's coding of set theory, the number 1 represents the set containing only the empty set, a fact with a precise proof.

• Foundation Primitive Recognition Calculus Prcset Theory Parse Not Mem Empty

  In the framework's coding of set theory, the empty set is the number zero, and the theorem not_mem_empty proves that nothing is a member of it.

• Foundation Primitive Recognition Calculus Prcshrunk Certificate

  A machine-checked certificate compresses the framework's seven load-bearing claims into one object, from a single primitive to a countable field for all constants.

• Foundation Primitive Recognition Calculus Prcshrunk Certificate Prc Shrunk Certi

  A machine-checked certificate bundles seven proved headlines about recognition, cost, and the countable field beneath them.

• Foundation Primitive Recognition Calculus Prctype Theory Parse

  A machine-checked library proves that a two-symbol alphabet, the simplest possible ledger, already contains the full expressive power of the recognition calculus.

• Foundation Primitive Recognition Calculus Prctype Theory Parse Canonicity

  A two-element type is the smallest possible discrete record: exactly two entries, and every entry is one of them.

• Foundation Primitive Recognition Calculus Prctype Theory Parse No Confusion

  The statement no_confusion pins down the most basic fact a two-valued logic needs: false and true are different.

• Foundation Primitive Recognition Calculus Prctype Theory Parse Tt System

  A two-element type with two distinct values is enough to encode the core of Martin-Löf type theory, the formal language underlying modern proof assistants.

• Foundation Primitive Recognition Calculus Prctype Theory Parse Tt System Embeds

  A single theorem in a machine-checked library shows that the simplest possible two-symbol system already contains the full expressive power of the framework's foundational cor