106 ideas
| 9724 | Until the 1960s the only semantics was truth-tables [Enderton] |
| 9703 | 'dom R' indicates the 'domain' of objects having a relation [Enderton] |
| 9705 | 'fld R' indicates the 'field' of all objects in the relation [Enderton] |
| 9704 | 'ran R' indicates the 'range' of objects being related to [Enderton] |
| 9710 | We write F:A→B to indicate that A maps into B (the output of F on A is in B) [Enderton] |
| 9707 | 'F(x)' is the unique value which F assumes for a value of x [Enderton] |
| 13201 | ∈ says the whole set is in the other; ⊆ says the members of the subset are in the other [Enderton] |
| 9712 | A relation is 'symmetric' on a set if every ordered pair has the relation in both directions [Enderton] |
| 9713 | A relation is 'transitive' if it can be carried over from two ordered pairs to a third [Enderton] |
| 13204 | The 'ordered pair' <x,y> is defined to be {{x}, {x,y}} [Enderton] |
| 13206 | A 'linear or total ordering' must be transitive and satisfy trichotomy [Enderton] |
| 9699 | The 'powerset' of a set is all the subsets of a given set [Enderton] |
| 9700 | Two sets are 'disjoint' iff their intersection is empty [Enderton] |
| 9702 | A 'domain' of a relation is the set of members of ordered pairs in the relation [Enderton] |
| 9701 | A 'relation' is a set of ordered pairs [Enderton] |
| 9706 | A 'function' is a relation in which each object is related to just one other object [Enderton] |
| 9708 | A function 'maps A into B' if the relating things are set A, and the things related to are all in B [Enderton] |
| 9709 | A function 'maps A onto B' if the relating things are set A, and the things related to are set B [Enderton] |
| 9711 | A relation is 'reflexive' on a set if every member bears the relation to itself [Enderton] |
| 9714 | A relation satisfies 'trichotomy' if all pairs are either relations, or contain identical objects [Enderton] |
| 9717 | A set is 'dominated' by another if a one-to-one function maps the first set into a subset of the second [Enderton] |
| 13200 | Note that {Φ} =/= Φ, because Φ ∈ {Φ} but Φ ∉ Φ [Enderton] |
| 13199 | The empty set may look pointless, but many sets can be constructed from it [Enderton] |
| 13203 | The singleton is defined using the pairing axiom (as {x,x}) [Enderton] |
| 9715 | An 'equivalence relation' is a reflexive, symmetric and transitive binary relation [Enderton] |
| 9716 | We 'partition' a set into distinct subsets, according to each relation on its objects [Enderton] |
| 13202 | Fraenkel added Replacement, to give a theory of ordinal numbers [Enderton] |
| 13205 | We can only define functions if Choice tells us which items are involved [Enderton] |
| 9722 | Inference not from content, but from the fact that it was said, is 'conversational implicature' [Enderton] |
| 9718 | Validity is either semantic (what preserves truth), or proof-theoretic (following procedures) [Enderton] |
| 9721 | A logical truth or tautology is a logical consequence of the empty set [Enderton] |
| 9994 | A truth assignment to the components of a wff 'satisfy' it if the wff is then True [Enderton] |
| 9719 | A proof theory is 'sound' if its valid inferences entail semantic validity [Enderton] |
| 9720 | A proof theory is 'complete' if semantically valid inferences entail proof-theoretic validity [Enderton] |
| 9995 | Proof in finite subsets is sufficient for proof in an infinite set [Enderton] |
| 9996 | Expressions are 'decidable' if inclusion in them (or not) can be proved [Enderton] |
| 9997 | For a reasonable language, the set of valid wff's can always be enumerated [Enderton] |
| 19482 | Current physics says matter and antimatter should have reduced to light at the big bang [New Sci.] |
| 19483 | CP violation shows a decay imbalance in matter and antimatter, leading to matter's dominance [New Sci.] |
| 9723 | Sentences with 'if' are only conditionals if they can read as A-implies-B [Enderton] |
| 19737 | A system can infer the structure of the world by making predictions about it [New Sci.] |
| 14367 | An explanation is a causal graph [Woodward,J, by Strevens] |
| 19736 | Neural networks can extract the car-ness of a car, or the chair-ness of a chair [New Sci.] |
| 16419 | No one has yet devised a rationality test [New Sci.] |
| 16417 | About a third of variation in human intelligence is environmental [New Sci.] |
| 16418 | People can be highly intelligent, yet very stupid [New Sci.] |
| 19484 | Psychologists measure personality along five dimensions [New Sci.] |
| 21167 | Gravity is unusual, in that it always attracts and never repels [New Sci.] |
| 19950 | Entropy is the only time-asymmetric law, so time may be linked to entropy [New Sci.] |
| 21176 | In the Big Bang general relativity fails, because gravity is too powerful [New Sci.] |
| 21147 | Quantum electrodynamics incorporates special relativity and quantum mechanics [New Sci.] |
| 21155 | Photons have zero rest mass, so virtual photons have infinite range [New Sci.] |
| 19478 | Light moves at a constant space-time speed, but its direction is in neither space nor time [New Sci.] |
| 21161 | In the standard model all the fundamental force fields merge at extremely high energies [New Sci.] |
| 21146 | Electrons move fast, so are subject to special relativity [New Sci.] |
| 19474 | Quantum states are measured by external time, of unknown origin [New Sci.] |
| 19473 | The Schrödinger equation describes the evolution of an object's wave function in Hilbert space [New Sci.] |
| 21148 | The strong force is repulsive at short distances, strong at medium, and fades at long [New Sci.] |
| 21151 | Gluons, the particles carrying the strong force, interact because of their colour charge [New Sci.] |
| 21152 | The strong force binds quarks tight, and the nucleus more weakly [New Sci.] |
| 21150 | Three different colours of quark (as in the proton) can cancel out to give no colour [New Sci.] |
| 21143 | Quarks in threes can build hadrons with spin ½ or with spin 3/2 [New Sci.] |
| 21142 | Classifying hadrons revealed two symmetry patterns, produced by three basic elements [New Sci.] |
| 21153 | The weak force explains beta decay, and the change of type by quarks and leptons [New Sci.] |
| 21154 | Three particles enable the weak force: W+ and W- are charged, and Z° is not [New Sci.] |
| 21156 | The weak force particles are heavy, so the force has a short range [New Sci.] |
| 21164 | Why do the charges of the very different proton and electron perfectly match up? [New Sci.] |
| 21170 | The Standard Model cannot explain dark energy, survival of matter, gravity, or force strength [New Sci.] |
| 21145 | The four fundamental forces (gravity, electromagnetism, weak and strong) are the effects of particles [New Sci.] |
| 21157 | Particles are spread out, with wave-like properties, and higher energy shortens the wavelength [New Sci.] |
| 21140 | Spin is a built-in ration of angular momentum [New Sci.] |
| 21149 | Quarks have red, green or blue colour charge (akin to electric charge) [New Sci.] |
| 21158 | Fermions, with spin ½, are antisocial, and cannot share quantum states [New Sci.] |
| 21165 | Spin is akin to rotation, and is easily measured in a magnetic field [New Sci.] |
| 21163 | The mass of protons and neutrinos is mostly binding energy, not the quarks [New Sci.] |
| 21168 | Gravitional mass turns out to be the same as inertial mass [New Sci.] |
| 21138 | Neutrons are slightly heavier than protons, and decay into them by emitting an electron [New Sci.] |
| 21144 | Top, bottom, charm and strange quarks quickly decay into up and down [New Sci.] |
| 21141 | Neutrinos were proposed as the missing energy in neutron beta decay [New Sci.] |
| 21169 | Only neutrinos spin anticlockwise [New Sci.] |
| 21166 | Standard antineutrinos have opposite spin and opposite lepton number [New Sci.] |
| 21171 | The symmetry of unified electromagnetic and weak forces was broken by the Higgs field [New Sci.] |
| 19954 | It is impossible for find a model of actuality among the innumerable models in string theory [New Sci.] |
| 21178 | String theory is now part of 11-dimensional M-Theory, involving p-branes [New Sci.] |
| 19953 | In string theory space-time has a grainy indivisible substructure [New Sci.] |
| 19476 | String theory needs at least 10 space-time dimensions [New Sci.] |
| 21179 | Supersymmetric string theory can be expressed using loop quantum gravity [New Sci.] |
| 21175 | String theory might be tested by colliding strings to make bigger 'stringballs' [New Sci.] |
| 21177 | String theory offers a quantum theory of gravity, by describing the graviton [New Sci.] |
| 21162 | Only supersymmetry offers to incorporate gravity into the scheme [New Sci.] |
| 21159 | Supersymmetry has extra heavy bosons and heavy fermions [New Sci.] |
| 21173 | Supersymmetry says particles and superpartners were unities, but then split [New Sci.] |
| 21172 | The evidence for supersymmetry keeps failing to appear [New Sci.] |
| 19947 | Hilbert Space is an abstraction representing all possible states of a quantum system [New Sci.] |
| 21160 | The Higgs field means even low energy space is not empty [New Sci.] |
| 19948 | Einstein's merging of time with space has left us confused about the nature of time [New Sci.] |
| 19475 | Relativity makes time and space jointly basic; quantum theory splits them, and prioritises time [New Sci.] |
| 19955 | Space-time may be a geometrical manifestation of quantum entanglement [New Sci.] |
| 19949 | Quantum theory relies on a clock outside the system - but where is it located? [New Sci.] |
| 19951 | Entropy is puzzling, so we may need to build new laws which include time directionality [New Sci.] |
| 19477 | General relativity predicts black holes, as former massive stars, and as galaxy centres [New Sci.] |
| 19952 | Black holes have entropy, but general relativity says they are unstructured, and lack entropy [New Sci.] |
| 16420 | 84.5 percent of the universe is made of dark matter [New Sci.] |
| 21174 | Dark matter must have mass, to produce gravity, and no electric charge, to not reflect light [New Sci.] |
| 17604 | We are halfway to synthesising any molecule we want [New Sci.] |
| 17603 | Chemistry just needs the periodic table, and protons, electrons and neutrinos [New Sci.] |