implies
P1661
predicate
Indicates that the truth of one statement guarantees or leads logically to the truth of another statement.
Aliases (16)
- hasImplication ×8
- implication ×8
- impliesIdentity ×4
- inferredFrom ×4
- implicates ×2
- implied ×2
- impliesProperty ×2
- induces ×2
- bowlImplication ×1
- epistemicImplication ×1
- hasImpliedConcept ×1
- ifAheadHumanityImplies ×1
- ifBehindHumanityImplies ×1
- impliesExistenceOf ×1
- imply ×1
- mathematicalConsequence ×1
Sample triples (showing first 500 of 506)
| Subject | Object |
|---|---|
| "Haile Selassie I is divine" | Haile Selassie I fulfills messianic prophecies → |
| "Haile Selassie I is divine" | Haile Selassie I is an incarnation of God → |
| 2008 Oregon–Oregon State football game | Oregon State Rose Bowl berth at stake ("bowlImplication") → |
| A Chicken in Every Pot and a Car in Every Garage | affordability of consumer goods → |
| A Chicken in Every Pot and a Car in Every Garage | rising standard of living → |
| A Chicken in Every Pot and a Car in Every Garage | widespread economic well-being → |
| A Dynamical Theory of the Electromagnetic Field | finite speed of propagation of electromagnetic interactions ("implied") → |
| A Dynamical Theory of the Electromagnetic Field | speed of light determined by electromagnetic constants ("implied") → |
| Abel–Ruffini theorem | there is no radical expression for the roots of a generic polynomial of degree five or higher → |
| AdS/CFT correspondence | correspondence between bulk fields and boundary operators → |
| AdS/CFT correspondence | equivalence of partition functions between bulk and boundary theories → |
| AdS/CFT correspondence | matching of symmetries between AdS isometries and CFT conformal group → |
| Advocate | legal and supportive imagery → |
| Al-Quddus | freedom from all defect → |
| Al-Quddus | freedom from all imperfection → |
| Almighty | God’s control over all events → |
| Almighty | no limitation on God’s power → |
| Am Segula | a special mission for the Jewish people → |
| Am Segula | a unique responsibility toward divine commandments → |
| Ampère–Maxwell law | existence of electromagnetic waves in vacuum → |
| Anthropocene | humans as major geological force ("hasImplication") → |
| Anthropocene | long-term persistence of human impacts in geological record ("hasImplication") → |
| Anthropocene | need for global environmental governance ("hasImplication") → |
| Apostolic Succession | continuity of church governance → |
| Apostolic Succession | continuity of sacramental ministry → |
| Apostolic Succession | continuity of teaching from the apostles → |
| Atiyah–Singer index theorem | integrality of certain characteristic numbers → |
| Banach space | metric via the norm ("induces") → |
| Bekenstein bound | entropy is bounded by area and energy scales → |
| Bekenstein bound | limit on information storage density → |
| Bekenstein bound | maximum information content for given energy and size → |
| Bekenstein–Hawking entropy | information content of a black hole scales with area not volume → |
| Berserker hypothesis | humanity may be at risk from future berserker probes → |
| Berserker hypothesis | intelligent life is frequently exterminated before becoming spacefaring → |
| Bianchi identities | conservation laws in general relativity → |
| Bianchi identities | ∇_μ G^{μν} = 0 → |
| Bianchi identities | ∇_μ T^{μν} = 0 under Einstein field equations → |
| Big Bang | cosmic time has a beginning → |
| Big Bang | space itself is expanding → |
| Big Bang | universe has finite age → |
| Big Bang cosmology | the universe has been expanding over time → |
| Boltzmann distribution | higher energy states are less populated than lower energy states → |
| Boltzmann equation | Boltzmann H-theorem → |
| Born rule | interference patterns in double-slit experiments → |
| Born rule | probabilities depend on relative phases of amplitudes → |
| Born rule | total probability equals one → |
| Brouwer fixed-point theorem | Borsuk–Ulam theorem in certain formulations → |
| Burali-Forti paradox | hierarchical cumulative universe of sets ("implication") → |
| Burali-Forti paradox | no universal set of all sets of standard set theory ("implication") → |
| CLT | distribution of standardized sums tends to normal distribution → |
| CLT | sample mean is approximately normally distributed for large samples → |
| Candidate Recommendation | no major feature additions are expected → |
| Cantor’s paradox | collection of all sets must be a proper class in axiomatic set theory → |
| Cantor’s paradox | universe of sets is not itself a set → |
| Cantor’s theorem | the existence of infinitely many distinct infinite cardinalities → |
| Cantor’s theorem | the real numbers are uncountable → |
| Cantor’s theorem | the set of all subsets of the natural numbers is uncountable → |
| Cantor’s theorem | there is no largest cardinal number → |
| Cantor’s theorem | there is no largest infinity → |
| Charles Lyell | 19th-century Scottish geologist description ("inferredFrom") → |
| Chocolate City | strong African American cultural institutions in Washington, D.C. → |
| Chocolate City | strong African American political leadership in Washington, D.C. → |
| Church–Turing thesis | any algorithmic computation can be simulated by a Turing machine → |
| Church–Turing thesis | no stronger notion of algorithmic computability than Turing computability exists → |
| Coase theorem | government intervention is not necessary for efficiency under zero transaction costs → |
| Coase theorem | initial allocation of rights affects wealth distribution → |
| Coase theorem | initial allocation of rights does not affect efficiency under ideal conditions → |
| Coase theorem | legal rule affects distribution but not efficiency under zero transaction costs → |
| Coase theorem | parties can internalize externalities through bargaining → |
| Commerce Clause | dormant Commerce Clause ("hasImpliedConcept") → |
| Courant–Friedrichs–Lewy condition | information must not travel more than one spatial cell per time step → |
| Crown-in-Parliament | courts cannot question validity of properly enacted Acts of Parliament → |
| Deus sive Natura | knowledge of Nature is knowledge of God ("epistemicImplication") → |
| Dirac equation | spin–orbit coupling → |
| Drake Shake | more comfortable crossing of Drake Passage → |
| Dyson shell | extremely advanced technology → |
| Dyson shell | large-scale material resources → |
| Dyson sphere concept | civilization with energy use comparable to total output of its star ("implication") → |
| Einstein coefficients | existence of stimulated emission ("imply") → |
| Einstein synchronization convention | relativity of simultaneity between inertial frames → |
| Einstein synchronization convention | standard Einstein time coordinate in each inertial frame → |
| Entscheidungsproblem | limits of mechanical reasoning → |
| Entscheidungsproblem | no algorithm decides validity for all first-order formulas → |
| Euler product formula for the Riemann zeta function | non-vanishing of the zeta function on the half-plane Re(s) > 1 → |
| Euler product formula for the Riemann zeta function | zeta function has no zeros for Re(s) > 1 → |
| Euler’s formula for complex exponentials | Im(e^{i\theta}) = \sin\theta → |
| Euler’s formula for complex exponentials | Re(e^{i\theta}) = \cos\theta → |
| Euler’s formula for complex exponentials | e^{i(\theta + 2\pi)} = e^{i\theta} → |
| Euler’s formula for complex exponentials | periodicity of complex exponential on the imaginary axis → |
| Euler’s polyhedron formula | V + F = E + 2 → |
| European Union customs union | common commercial policy towards non-EU countries → |
| European Union customs union | harmonised customs legislation among member states → |
| FLRW cosmological models | Hubble–Lemaître law on large scales → |
| Faddeev’s axioms | Shannon entropy is the unique solution up to scale → |
| Faraday's law of induction | changing magnetic fields create electric fields → |
| Father, Son, and Spirit are not distinct eternal persons | Father, Son, and Spirit are modes of one divine person → |
| Father, Son, and Spirit are not distinct eternal persons | temporal manifestations rather than eternal distinctions in God → |
| Fermi paradox | advanced civilizations may deliberately avoid contact ("implication") → |
| Fermi paradox | at least one common assumption about life and intelligence may be wrong ("implication") → |
| Fermi paradox | intelligent life may be rare ("implication") → |
| Fermi paradox | our search methods may be inadequate ("implication") → |
| Fermi paradox | technological civilizations may be short-lived ("implication") → |
| Fermi–Dirac statistics | maximum of one fermion per single-particle quantum state → |
| Filioque clause in the Nicene Creed | the concept of the monarchy of the Father ("implicates") → |
| Filioque clause in the Nicene Creed | the relationship between economic and immanent Trinity ("implicates") → |
| Formula of Humanity | duty not to exploit others → |
| Formula of Humanity | duty of beneficence → |
| Formula of Humanity | duty of self-respect → |
| Formula of Humanity | duty to respect others → |
| Franck–Condon principle | most probable transitions are vertical on a potential energy diagram → |
| Franck–Condon principle | nuclei are effectively fixed during an electronic transition → |
| Galilean relativity | form of Newton’s laws is invariant in all inertial frames → |
| Galilean relativity | no preferred inertial frame for mechanics → |
| Galilean relativity | relative velocities add linearly → |
| Galilean relativity | simultaneity is absolute → |
| Galilean relativity | spatial distances are frame-independent up to rigid translations and rotations → |
| Galilean relativity | time intervals are the same in all inertial frames → |
| Game 6 | at least five games have already been played in the series → |
| Game 6 | series score is either 3–2 or 2–3 before the game → |
| Ganges of the South | Godavari River is as sacred as the Ganges River → |
| Gauss's law | electric field lines originate and terminate on charges → |
| Gauss's law for magnetism | magnetic field lines form closed loops → |
| Gauss's law for magnetism | magnetic monopoles do not exist in classical electromagnetism → |
| Gaussian integral | ∫_{0}^{∞} e^{−x^2} dx = √π / 2 → |
| Gaussian law of error | errors are symmetrically distributed around zero → |
| Gaussian law of error | small errors are more probable than large errors → |
| Gaussian law of error | sum of many small independent errors is approximately normal → |
| Gauss–Markov theorem | ordinary least squares is efficient within the class of linear unbiased estimators → |
| Gauss’s lemma (number theory) | Legendre symbol takes values ±1 for a coprime to p → |
| Gell-Mann–Low theorem | Dyson series expansion for the S-matrix → |
| Gell-Mann–Nishijima formula | quantization of electric charge in hadrons → |
| Gibbons–Hawking temperature | de Sitter horizon has entropy → |
| Girsanov theorem | drift terms can be removed or introduced by changing measure → |
| Girsanov theorem | local martingales under one measure may become martingales under another → |
| God is in the details | carelessness in minor points can undermine a project → |
| God is in the details | small details can determine overall success or failure → |
| Golus | lack of full national sovereignty → |
| Golus | living under foreign rule → |
| Golus | separation from the Land of Israel → |
| Grand Cru (Burgundy classification) | limited vineyard area → |
| Grand Cru (Burgundy classification) | stricter production standards than lower classifications → |
| Great Filter | Human civilization is unusually lucky or early ("ifBehindHumanityImplies") → |
| Great Filter | Humanity faces high existential risk before becoming interstellar ("ifAheadHumanityImplies") → |
| Great Filter | Most civilizations never reach long-lived interstellar expansion → |
| Great Filter | The observable universe may contain very few advanced spacefaring civilizations → |
| Great Silence | civilizations may avoid detectable communication → |
| Great Silence | civilizations may be short-lived → |
| Great Silence | either intelligent life is rare or hard to detect → |
| Great Silence | our assumptions about life and intelligence may be wrong → |
| Grothendieck universe | if x ∈ U then the transitive closure of x is contained in U → |
| Gödel's incompleteness theorems | a sufficiently strong consistent theory cannot prove its own consistency → |
| Gödel's incompleteness theorems | existence of true but unprovable statements → |
| Gödel's incompleteness theorems | limitations of Hilbert's program → |
| Gödel's incompleteness theorems | no complete and consistent extension of Peano arithmetic is recursively axiomatizable → |
| H-theorem | monotonic decrease of H-function over time → |
| H-theorem | monotonic increase of entropy-like quantity → |
| Hawking radiation | black holes have entropy → |
| Hawking radiation | black holes have temperature → |
| Her Imperial Highness | precedence over non-imperial titles → |
| Hilbert basis theorem | ascending chain condition on ideals holds in polynomial rings over Noetherian rings → |
| Hilbert basis theorem | coordinate rings of affine varieties over a field are Noetherian → |
| Hilbert basis theorem | every ideal in R[x_1,\dots,x_n] is finitely generated when R is Noetherian → |
| Hilbert basis theorem | every ideal in k[x_1,\dots,x_n] is finitely generated for any field k → |
| Hilbert basis theorem | k[x_1,\dots,x_n] is Noetherian for any field k → |
| Hilbert basis theorem | polynomial rings over Noetherian rings are Noetherian → |
| Hilbert space | norm via the inner product ("induces") → |
| Hilbert’s Nullstellensatz | correspondence between maximal ideals and points of affine space → |
| Hilbert’s Nullstellensatz | every proper ideal in k[x1,…,xn] has a common zero in some field extension of k → |
| Hilbert’s irreducibility theorem | existence of infinitely many specializations preserving Galois group in many cases → |
| Hilbert’s irreducibility theorem | existence of infinitely many specializations preserving irreducibility → |
| Hilbert’s syzygy theorem | global dimension of a polynomial ring in n variables over a field is n → |
| Hilbert’s syzygy theorem | polynomial rings over a field are regular rings → |
| Hilbert’s syzygy theorem | polynomial rings over a field have finite global dimension → |
| Huygens–Fresnel principle | light behaves as a wave → |
| Hölder inequality | Cauchy–Schwarz inequality → |
| Hölder inequality | Minkowski inequality → |
| In Service to the Church in the World | integration of faith and public life → |
| In Service to the Church in the World | mission beyond the local congregation → |
| In Service to the Church in the World | public witness of the church → |
| Incarnation of Christ | God’s solidarity with humanity → |
| Incarnation of Christ | possibility of deification or theosis in some traditions → |
| Infante of Portugal | membership in the Portuguese royal family → |
| Inner Light | every person has access to divine truth → |
| Inner Light | spiritual equality of all persons → |
| Israel–Carter–Robinson uniqueness theorems | no-hair property of black holes → |
| Israel–Carter–Robinson uniqueness theorems | uniqueness of the Kerr solution for rotating uncharged black holes → |
| Israel–Carter–Robinson uniqueness theorems | uniqueness of the Kerr–Newman family for rotating charged black holes → |
| Jagaban | informal leadership within Nigerian political networks → |
| Jagaban | kingmaker role in Nigerian politics → |
| Janet–Cartan theorem | local realization of analytic Riemannian metrics as induced metrics from Euclidean space → |
| Jensen's inequality | E[|X|^p] ≥ |E[X]|^p for p ≥ 1 → |
| Jensen's inequality | log E[X] ≥ E[log X] for positive X and concave log → |
| John Whitehurst | 18th-century English scientific history ("inferredFrom") → |
| Junior Lords of the Treasury | role as government whips in the British House of Commons ("inferredFrom") → |
| Kakutani fixed-point theorem | fixed point of a correspondence ("impliesExistenceOf") → |
| Karush–Kuhn–Tucker conditions | zero product between each inequality constraint and its multiplier at optimum → |
| Kennedy–Thorndike experiment | time dilation must accompany length contraction → |
| Kepler’s second law | conservation of angular momentum for orbital motion → |
| Khinchin–Kahane type inequalities | stability of random series under change of p-norm → |
| King of Kings | no rival to God's rule → |
| King of Kings | transcendent rule → |
| King of Kings | ultimate supremacy → |
| King of the Universe | cosmic legitimacy → |
| King of the Universe | divinely sanctioned rule → |
| Kingship of Jesus | allegiance of the Church to Christ alone → |
| Kingship of Jesus | obedience of believers to Christ → |
| Kingship of Jesus | universal sovereignty of Christ → |
| Kirchhoff's law of thermal radiation | good absorbers are good emitters → |
| Kirchhoff's law of thermal radiation | poor absorbers are poor emitters → |
| Kähler manifold | underlying manifold is Riemannian → |
| Kähler manifold | underlying manifold is complex → |
| Kähler manifold | underlying manifold is symplectic → |
| LSZ reduction formula | equivalence between field-theoretic description and particle scattering description → |
| Laffer curve | there is a revenue-maximizing tax rate between 0% and 100% → |
| Landauer's principle | no computation can be completely dissipationless if it includes erasure → |
| Landauer's principle | reversible computation can in principle avoid Landauer heat for logical operations → |
| Lax equivalence theorem | convergence of a finite difference scheme under stability and consistency → |
| Lemaître–Hubble law | more distant galaxies recede faster → |
| Light of the World | exposure of sin and error → |
| Light of the World | guidance toward salvation → |
| Light of the World | revelation of God’s will → |
| London equations | exponential decay of magnetic field inside a superconductor → |
| Lonely Assassins | stealthy killers → |
| Lonely Assassins | victims often die alone → |
| Longuet-Higgins theorem in molecular symmetry | degenerate electronic states are associated with non-totally symmetric irreducible representations → |
| Longuet-Higgins theorem in molecular symmetry | degenerate electronic states in nonlinear molecules are unstable with respect to symmetry-lowering distortions → |
| Lord | authority of Christ over believers → |
| Lord | kingship of Christ → |
| Lord | ownership of Christ over the Church → |
| Lord of the worlds | Allah’s sovereignty over angels → |
| Lord of the worlds | Allah’s sovereignty over humans → |
| Lord of the worlds | Allah’s sovereignty over jinn → |
| Lord of the worlds | Allah’s sovereignty over the entire universe → |
| Lord of the worlds | Allah’s sovereignty over the seen world → |
| Lord of the worlds | Allah’s sovereignty over the unseen world → |
| Lorentz contraction | moving rods appear shorter along motion direction → |
| Lorentz transformation | length contraction → |
| Lorentz transformation | relativity of simultaneity → |
| Lorentz transformation | time dilation → |
| Lorentz transformation | velocity addition formula of special relativity → |
| Lucas critique | historical correlations may not be valid under new policy regimes → |
| Lucas critique | macroeconomic models should be derived from optimizing behavior → |
| Lucas critique | policy-invariant parameters must be structural and microfounded → |
| Martingale representation theorem | any L2-martingale is an Itô integral with respect to Brownian motion → |
| Martingale representation theorem | uniqueness of integrand up to indistinguishability → |
| Mascot Hall of Fame inductee | broad public recognition → |
| Mascot Hall of Fame inductee | historical significance in mascot field → |
| Mascot Hall of Fame inductee | sustained performance excellence → |
| Maxwell's equations | existence of electromagnetic waves → |
| Maxwell's equations | finite speed of propagation of electromagnetic interactions → |
| Meissner effect | magnetic susceptibility of −1 in SI units for ideal bulk superconductor → |
| Meissner effect | superconductors are not just perfect conductors → |
| Merkle–Damgård construction | hash function is collision-resistant if compression function is collision-resistant → |
| Michael I Cerularius | Byzantine historical sources ("inferredFrom") → |
| Migdal approximation | electron-phonon coupling can be treated perturbatively → |
| Migdal approximation | vertex corrections are of higher order in small mass ratio → |
| Minkowski inequality | L^p norm satisfies triangle inequality → |
| Minkowski space-time | length contraction → |
| Minkowski space-time | relativity of simultaneity → |
| Minkowski space-time | time dilation → |
| Most High | God’s superiority over all earthly rulers → |
| Most High | God’s superiority over all spiritual beings → |
| Most High | no higher authority than God → |
| Munich Betrayal | sacrifice of Czechoslovakia to Nazi Germany → |
| Nash embedding theorem | every abstract Riemannian manifold can be realized as a submanifold of Euclidean space → |
| Nature’s Air-Conditioned City | natural cooling without mechanical air conditioning → |
| Newton's first law of motion | acceleration is zero when net force is zero → |
| Newton's first law of motion | velocity remains constant when net force is zero → |
| Newton's second law of motion | acceleration is inversely proportional to mass → |
| Newton's second law of motion | acceleration is proportional to net force → |
| Newton's second law of motion | zero net force gives zero acceleration → |
| Newtonian absolute time | no relativity of simultaneity → |
| Newtonian absolute time | simultaneity is absolute → |
| Newtonian absolute time | time intervals are invariant between observers → |
| Noether normalization lemma | Krull dimension of a finitely generated k-algebra equals the transcendence degree of its field of fractions over k → |
| Noether normalization lemma | any affine k-algebra is a finite module over a polynomial subring → |
| Noether normalization lemma | any affine variety admits a finite surjective morphism to an affine space of the same dimension → |
| Noether normalization lemma | existence of a finite injective k-algebra homomorphism from a polynomial ring to a finitely generated k-algebra → |
| Noether's isomorphism theorems | correspondence between ideals containing the kernel and ideals of the image → |
| Noether's isomorphism theorems | correspondence between subgroups containing the kernel and subgroups of the image → |
| Noether's theorem | conservation of angular momentum from rotational symmetry → |
| Noether's theorem | conservation of electric charge from global gauge symmetry → |
| Noether's theorem | conservation of energy from time-translation symmetry → |
| Noether's theorem | conservation of linear momentum from spatial-translation symmetry → |
| Noether's theorem | existence of a conserved current for each continuous symmetry → |
| Noether's theorem | existence of a conserved quantity for each one-parameter Lie group of symmetries → |
| Noetherian induction | no infinite strictly descending sequence exists → |
| Noetherian module | every nonempty set of submodules has a maximal element → |
| Noetherian module | every submodule is finitely generated → |
| Noetherian module | finitely generated over a Noetherian ring is Noetherian → |
| Noetherian space | every closed subset is quasi-compact → |
| Noetherian space | every open subset is quasi-compact → |
| Noetherian space | every subset is a finite union of locally closed subsets → |
| Oppenheimer–Volkoff limit | collapse to a black hole above the limit → |
| Oppenheimer–Volkoff limit | no stable neutron star configuration above the limit → |
| Original sin | inability of humans to save themselves → |
| Original sin | inclination of humans toward sin → |
| Original sin | necessity of salvation through Christ → |
| Original sin | need for divine grace → |
| Original sin | spiritual death → |
| Original sin | universal sinfulness of human beings → |
| Osterwalder–Schrader axioms | Wightman axioms for the reconstructed Minkowski theory → |
| Osterwalder–Schrader axioms | existence of a Hilbert space of states → |
| Osterwalder–Schrader axioms | existence of a self-adjoint Hamiltonian → |
| Osterwalder–Schrader axioms | unitary representation of the Poincaré group after continuation → |
| PCP theorem | many NP-optimization problems are hard to approximate within certain constant factors → |
| PCP theorem | strong inapproximability bounds for Max-3SAT → |
| PCP theorem | strong inapproximability bounds for Max-Clique → |
| PCP theorem | strong inapproximability bounds for Set Cover → |
| PCP theorem | strong inapproximability bounds for Vertex Cover → |
| PCP theorem | strong inapproximability bounds for various constraint satisfaction problems → |
| PCP theorem | there exist fixed constants for which approximating some NP-hard problems within those factors is NP-hard → |
| Pascal's identity | each entry of Pascal's triangle equals the sum of the two entries above it ("impliesProperty") → |
| Pascal's identity | symmetry of Pascal's triangle along its vertical axis (together with boundary conditions) ("impliesProperty") → |
| Pauli exclusion principle | antisymmetric total wavefunction for identical fermions → |
| Pearl of the Aegean | high aesthetic value of İzmir → |
| Pearl of the Aegean | prominent status among Aegean cities → |
| Phillips curve framework | no long-run trade-off between inflation and unemployment under adaptive or rational expectations → |
| Planck length | possible discreteness of spacetime at very small scales → |
| Planck radiation law | energy quantization of electromagnetic modes → |
| Polar Bear Garden | cold and harsh environment → |
| Polar Bear Garden | sparsely populated area → |
| Prince Regent | delegated sovereignty → |
| Probabilistic Encryption | ciphertexts are not uniquely determined by plaintext → |
| Q source hypothesis | Luke did not use Matthew directly → |
| Q source hypothesis | Matthew and Luke were composed independently of each other → |
| Q source hypothesis | Matthew did not use Luke directly → |
| Rayleigh–Jeans law at low frequencies | energy density of radiation grows without bound with frequency in classical theory → |
| Ricardian equivalence | consumption depends on the present value of government spending not on tax timing → |
| Ricardian equivalence | fiscal deficits do not stimulate aggregate demand if conditions hold → |
| Ricardian equivalence | government debt is not net wealth for the private sector under its assumptions → |
| Ricardian equivalence | temporary tax cuts financed by debt are saved rather than consumed → |
| Ricardian rent | marginal land earns zero rent in competitive equilibrium → |
| Ricardian rent | rent does not enter into price determination under Ricardo's assumptions → |
| Riemann hypothesis | best possible error term in the prime number theorem up to constants → |
| Riemann hypothesis | bounds on the Chebyshev functions → |
| Riemann hypothesis | results on gaps between primes → |
| Riemann hypothesis | results on the Mertens function → |
| Riemann hypothesis | results on the Möbius function → |
| Riemann hypothesis | strong results on the distribution of prime numbers → |
| Riemann integral | function is Riemann integrable on the interval → |
| Riemann mapping theorem | All simply connected proper domains in the complex plane are conformally equivalent → |
| Riemann mapping theorem | The unit disk is a universal model for simply connected proper planar domains → |
| Riemann–Lebesgue lemma | Fourier coefficients of an L¹ function form a null sequence. → |
| Riemann–Lebesgue lemma | Fourier transform of an L¹ function is a continuous function that tends to 0 at infinity. → |
| Riemann–Roch theorem | Riemann–Hurwitz formula in certain cases → |
| Russia Without Putin | possibility of alternative leadership in Russia → |
| Russia Without Putin | post-Putin political transition in Russia → |
| Salam–Weinberg model | massive W and Z bosons → |
| Salam–Weinberg model | massless photon → |
| Saros cycle | similar eclipse duration → |
| Saros cycle | similar eclipse magnitude → |
| Saros cycle | similar eclipse path on Earth → |
| Savage Mountain | that K2 is more dangerous than Everest → |
| Schwarzschild radius | inside radius light cannot escape to infinity → |
| Seal of the Prophets | authority of Muhammad’s message for Muslims ("hasImplication") → |
| Seal of the Prophets | closure of prophetic office ("hasImplication") → |
| Seward’s Icebox | Alaska was a frozen wasteland → |
| Seward’s Icebox | Alaska was economically worthless → |
| Shannon–Khinchin axioms | adding an event with probability zero does not change entropy → |
| Shannon–Khinchin axioms | entropy is Schur-concave → |
| Shannon–Khinchin axioms | entropy is continuous in its arguments → |
| Shannon–Khinchin axioms | entropy is maximal for the uniform distribution → |
| Shannon–Khinchin axioms | entropy is nonnegative → |
| Shannon–Khinchin axioms | entropy is symmetric in its arguments → |
| Shannon–Khinchin axioms | entropy is uniquely determined up to a positive multiplicative constant → |
| Shannon–Khinchin axioms | entropy of a compound experiment is sum of entropies plus conditional entropy → |
| Snell’s law of refraction | light bends away from normal when entering lower index medium → |
| Snell’s law of refraction | light bends toward normal when entering higher index medium → |
| Snell’s law of refraction | ratio of sines of angles equals ratio of refractive indices → |
| Social Security wage base limit | earnings above the limit do not increase Social Security benefits ("hasImplication") → |
| Social Security wage base limit | high earners stop paying Social Security tax after reaching the limit ("hasImplication") → |
| Tarski's undefinability theorem | hierarchy between object language and metalanguage → |
| Tarski's undefinability theorem | nonexistence of an internal truth predicate for arithmetic → |
| Tarski's undefinability theorem | semantic notions like truth may require a stronger metalanguage → |
| The City | central importance within the region → |
| The City | primary urban core of the metropolitan area → |
| The House That Nat Built | Capitol Records Building was financed significantly by Nat King Cole’s record sales → |
| The Royal Road | state control of route → |
| The Royal Road | strategic importance → |
| The Southern Part of Heaven | idealized vision of Chapel Hill → |
| The Southern Part of Heaven | pleasant quality of life in Chapel Hill → |
| Theorema Egregium | Gaussian curvature can be computed from the first fundamental form alone → |
| Theorema Egregium | local isometry preserves Gaussian curvature → |
| Third World | economic underdevelopment → |
| Third World | marginalization in global power structures → |
| Third World | political dependency → |
| Thomas rules for multistationarity and oscillations | absence of negative circuits precludes sustained oscillations → |
| Thomas rules for multistationarity and oscillations | absence of positive circuits precludes multistationarity → |
| Transubstantiation | Christ is wholly present under each part of the species → |
| Transubstantiation | Christ is wholly present under each species → |
| Transubstantiation | change is not empirically detectable → |
| Tritheism | plurality of gods within Christian framework → |
| Tritheism | three separate divine essences → |
| Tucker’s lemma | discrete Borsuk–Ulam type results → |
| Ulam stability | continuous dependence of exact solutions on perturbations → |
| Uncle Joe | folksy familiarity → |
| Uncle Joe | paternal image → |
| Way, the Truth, and the Life | no one comes to the Father except through Jesus → |
| Whereof one cannot speak, thereof one must be silent | metaphysical, ethical, and mystical matters cannot be put into meaningful propositions → |
| Whereof one cannot speak, thereof one must be silent | philosophy should clarify language rather than state metaphysical theses → |
| Whereof one cannot speak, thereof one must be silent | some things lie beyond the limits of meaningful language → |
| Whitney approximation theorem | C^∞(M,N) is dense in C^0(M,N) for smooth manifolds M and N under suitable topologies → |
| Whitney approximation theorem | smooth maps are dense in the space of continuous maps between smooth manifolds → |
| Whitney embedding theorem | every smooth manifold is diffeomorphic to a submanifold of some R^N → |
| Whitney embedding theorem | every smooth n-dimensional manifold can be realized as a submanifold of some Euclidean space → |
| Wick’s theorem | higher n-point Green’s functions of free fields factorize into products of two-point functions → |
| Wick’s theorem | time-ordered vacuum expectation values can be expressed in terms of two-point functions → |
| Wien displacement law | cooler bodies emit peak radiation at longer wavelengths → |
| Wien displacement law | hotter bodies emit peak radiation at shorter wavelengths → |
| Wigner surmise | short-range correlations between eigenvalues → |
| Wigner’s theorem on symmetry transformations | symmetry transformations act by unitary or antiunitary operators on Hilbert space → |
| Wigner’s theorem on symmetry transformations | symmetry transformations are isometries of projective Hilbert space → |
| Winterpeg | Winnipeg experiences heavy snowfall → |
| Winterpeg | Winnipeg experiences low winter temperatures → |
| Winterpeg | Winnipeg has harsh winters → |
| Yukawa potential | finite range for forces mediated by massive particles → |
| Zermelo–Fraenkel set theory | existence of integers → |
| Zermelo–Fraenkel set theory | existence of many transfinite cardinals → |
| Zermelo–Fraenkel set theory | existence of natural numbers → |
| Zermelo–Fraenkel set theory | existence of rational numbers → |
| Zermelo–Fraenkel set theory | existence of real numbers → |
| act utilitarianism | each person’s happiness counts equally → |
| all-or-none principle in nerve excitation | information about stimulus intensity is encoded in firing frequency rather than spike size → |
| all-or-none principle in nerve excitation | stronger suprathreshold stimuli do not increase action potential amplitude → |
| axiom of choice | Banach–Tarski paradox → |
| axiom of choice | Hausdorff maximal principle → |
| axiom of choice | every field has an algebraic closure → |
| axiom of choice | every infinite set has a countable subset → |
| axiom of choice | every product of nonempty sets is nonempty → |
| axiom of choice | every set can be well-ordered → |
| axiom of choice | every vector space has a Hamel basis → |
| axiom of choice | existence of non-measurable sets of real numbers → |
| axiom of choice | well-ordering of every set → |
| axiom schema of separation | existence of subsets of any given set satisfying a given property → |
| binomial theorem | entries of Pascal's triangle are binomial coefficients → |
| binomial theorem | Σ_{k=0}^n (-1)^k C(n,k) = 0 for n > 0 ("impliesIdentity") → |
| binomial theorem | Σ_{k=0}^n C(n,k) = 2^n ("impliesIdentity") → |
| binomial theorem | Σ_{k=0}^n k C(n,k) = n 2^{n-1} ("impliesIdentity") → |
| binomial theorem | Σ_{k=0}^n k^2 C(n,k) = n(n+1)2^{n-2} ("impliesIdentity") → |
| black hole no-hair theorem | black holes have no additional independent multipole moments beyond those determined by mass, charge, and angular momentum → |
| black hole no-hair theorem | details of the matter that formed a black hole are not observable from outside except through mass, charge, and angular momentum → |
| black hole no-hair theorem | information about infalling matter is not encoded in additional classical observables outside the horizon ("hasImplication") → |
| central limit theorem | approximate normality of sample mean for large samples → |
| efficient market hypothesis | active portfolio management cannot systematically beat passive strategies after costs → |
| efficient market hypothesis | fundamental analysis cannot consistently outperform the market on a risk‑adjusted basis → |
| efficient market hypothesis | no free lunch in financial markets → |
| efficient market hypothesis | prices follow a martingale or random walk process under certain conditions → |
| efficient market hypothesis | technical analysis cannot consistently outperform the market → |
| equipartition theorem | Dulong–Petit law for molar heat capacity of many solids at high temperature → |
| equipartition theorem | average translational kinetic energy per particle is (3/2)kT in three dimensions → |
| equipartition theorem | each quadratic term in the Hamiltonian contributes (1/2)kT to the mean energy → |
| equipartition theorem | heat capacity of a monatomic ideal gas is (3/2)Nk → |
| equipartition theorem | internal energy of a monatomic ideal gas is (3/2)NkT → |
| equivalence principle | all test bodies fall with same acceleration in a given gravitational field → |
| equivalence principle | equivalence between uniform acceleration and uniform gravitational field locally → |
| equivalence principle | motion of free-falling test particles follows geodesics of spacetime ("mathematicalConsequence") → |
| equivalence principle | universality of free fall → |
| eternal inflation | different bubble universes can have different low-energy constants → |
| eternal inflation | different bubble universes can have different physical properties → |
| eternal inflation | different bubble universes can have different vacuum states → |
| eternal inflation | existence of a multiverse → |
| eternal inflation | formation of multiple bubble universes → |
| eternal inflation | inflation ends locally in bubble regions → |
| eternal inflation | space never stops inflating globally → |
| eternal inflation | universe is much larger than observable universe → |
| expected utility theory | choices can be represented as maximization of expected utility → |
| falsificationism | scientific knowledge is provisional → |
| falsificationism | scientific theories are conjectural → |
| fluctuation–dissipation theorem | Onsager reciprocal relations under certain conditions → |
| fluctuation–dissipation theorem | response functions are determined by equilibrium correlation functions → |
| generalized binomial theorem | analyticity of (1+z)^α on unit disk minus branch cut → |
| generalized binomial theorem | radius of convergence 1 for binomial series in z → |
| hypothesis of the primeval atom | space and time have a beginning → |
| hypothesis of the primeval atom | universe has a finite age → |
| identity of indiscernibles | if two entities share all properties, they are identical → |
| implicit function theorem | inverse function theorem in special cases → |
| inverse function theorem | continuity of the inverse map → |
| inverse function theorem | differentiability of the inverse map → |
| inverse function theorem | inverse map has derivative equal to the matrix inverse of the original derivative → |
| inverse function theorem | local bijectivity near the point → |
| inverse function theorem | openness of the map near the point → |
| kingdom of ends | dignity of rational nature → |
| kingdom of ends | each person is both legislator and subject of moral law → |
| kingdom of ends | moral obligations arise from rational will → |
| law of universal gravitation | gravitational force decreases with distance → |
| law of universal gravitation | gravitational interaction is universal → |
| local existence and uniqueness theorem | solutions depend continuously on initial data (locally) → |
| multinomial theorem | number of distinct terms in expansion equals C(n+m-1, m-1) → |
| permanent income hypothesis | marginal propensity to consume out of permanent income is high → |
| permanent income hypothesis | marginal propensity to consume out of transitory income is low → |
| permanent income hypothesis | predictable income changes should not systematically change consumption → |
| relativity of simultaneity | moving clocks that are synchronized in one frame are not synchronized in another → |
| relativity of simultaneity | no preferred inertial frame for defining simultaneity → |
| relativity of simultaneity | simultaneity is frame-dependent → |
| relativity of simultaneity | simultaneity is not absolute → |
| special relativity | Lorentz invariance → |
| special relativity | length contraction → |
| special relativity | mass–energy equivalence → |
| special relativity | relativity of simultaneity → |