69 ideas
10170 | While true-in-a-model seems relative, true-in-all-models seems not to be [Reck/Price] |
17749 | Post proved the consistency of propositional logic in 1921 [Walicki] |
17765 | Propositional language can only relate statements as the same or as different [Walicki] |
17764 | Boolean connectives are interpreted as functions on the set {1,0} [Walicki] |
17752 | The empty set is useful for defining sets by properties, when the members are not yet known [Walicki] |
17753 | The empty set avoids having to take special precautions in case members vanish [Walicki] |
10166 | ZFC set theory has only 'pure' sets, without 'urelements' [Reck/Price] |
17759 | Ordinals play the central role in set theory, providing the model of well-ordering [Walicki] |
17741 | To determine the patterns in logic, one must identify its 'building blocks' [Walicki] |
10175 | Three types of variable in second-order logic, for objects, functions, and predicates/sets [Reck/Price] |
17747 | A 'model' of a theory specifies interpreting a language in a domain to make all theorems true [Walicki] |
17748 | The L-S Theorem says no theory (even of reals) says more than a natural number theory [Walicki] |
17761 | A compact axiomatisation makes it possible to understand a field as a whole [Walicki] |
17763 | Axiomatic systems are purely syntactic, and do not presuppose any interpretation [Walicki] |
17758 | Ordinals are transitive sets of transitive sets; or transitive sets totally ordered by inclusion [Walicki] |
17755 | Ordinals are the empty set, union with the singleton, and any arbitrary union of ordinals [Walicki] |
17756 | The union of finite ordinals is the first 'limit ordinal'; 2ω is the second... [Walicki] |
17760 | Two infinite ordinals can represent a single infinite cardinal [Walicki] |
17757 | Members of ordinals are ordinals, and also subsets of ordinals [Walicki] |
10165 | 'Analysis' is the theory of the real numbers [Reck/Price] |
17762 | In non-Euclidean geometry, all Euclidean theorems are valid that avoid the fifth postulate [Walicki] |
10174 | Mereological arithmetic needs infinite objects, and function definitions [Reck/Price] |
10164 | Peano Arithmetic can have three second-order axioms, plus '1' and 'successor' [Reck/Price] |
17754 | Inductive proof depends on the choice of the ordering [Walicki] |
10172 | Set-theory gives a unified and an explicit basis for mathematics [Reck/Price] |
10167 | Structuralism emerged from abstract algebra, axioms, and set theory and its structures [Reck/Price] |
10169 | Relativist Structuralism just stipulates one successful model as its arithmetic [Reck/Price] |
10179 | There are 'particular' structures, and 'universal' structures (what the former have in common) [Reck/Price] |
10181 | Pattern Structuralism studies what isomorphic arithmetic models have in common [Reck/Price] |
10182 | There are Formalist, Relativist, Universalist and Pattern structuralism [Reck/Price] |
10168 | Formalist Structuralism says the ontology is vacuous, or formal, or inference relations [Reck/Price] |
10178 | Maybe we should talk of an infinity of 'possible' objects, to avoid arithmetic being vacuous [Reck/Price] |
10176 | Universalist Structuralism is based on generalised if-then claims, not one particular model [Reck/Price] |
10177 | Universalist Structuralism eliminates the base element, as a variable, which is then quantified out [Reck/Price] |
10171 | The existence of an infinite set is assumed by Relativist Structuralism [Reck/Price] |
15682 | Even fairly simple animals make judgements based on categories [Gelman] |
15691 | Children accept real stable categories, with nonobvious potential that gives causal explanations [Gelman] |
10173 | A nominalist might avoid abstract objects by just appealing to mereological sums [Reck/Price] |
15700 | In India, upper-castes essentialize caste more than lower-castes do [Gelman] |
15685 | Essentialism is either natural to us, or an accident of our culture, or a necessary result of language [Gelman] |
15684 | Children's concepts include nonobvious features, like internal parts, functions and causes [Gelman] |
15681 | Essentialism: real or representational? sortal, causal or ideal? real particulars, or placeholders? [Gelman] |
15678 | Essentialism says categories have a true hidden nature which gives an object its identity [Gelman] |
15683 | Sortals are needed for determining essence - the thing must be categorised first [Gelman] |
15697 | Kind (unlike individual) essentialism assumes preexisting natural categories [Gelman] |
15687 | Kinship is essence that comes in degrees, and age groups are essences that change over time [Gelman] |
15679 | Essentialism comes from the cognitive need to categorise [Gelman] |
15698 | We found no evidence that mothers teach essentialism to their children [Gelman] |
15709 | Essentialism is useful for predictions, but it is not the actual structure of reality [Gelman] |
15696 | Peope favor historical paths over outward properties when determining what something is [Gelman] |
17742 | Scotus based modality on semantic consistency, instead of on what the future could allow [Walicki] |
15707 | There is intentional, mechanical, teleological, essentialist, vitalist and deontological understanding [Gelman] |
15703 | Memories often conform to a theory, rather than being neutral [Gelman] |
15708 | Inductive success is rewarded with more induction [Gelman] |
15694 | Children overestimate the power of a single example [Gelman] |
15695 | Children make errors in induction by focusing too much on categories [Gelman] |
15692 | People tend to be satisfied with shallow explanations [Gelman] |
15680 | Folk essentialism rests on belief in natural kinds, in hidden properties, and on words indicating structures [Gelman] |
15686 | Labels may indicate categories which embody an essence [Gelman] |
15690 | Causal properties are seen as more central to category concepts [Gelman] |
15688 | Categories are characterized by distance from a prototype [Gelman] |
15689 | Theory-based concepts use rich models to show which similarities really matter [Gelman] |
15699 | Prelinguistic infants acquire and use many categories [Gelman] |
15693 | One sample of gold is enough, but one tree doesn't give the height of trees [Gelman] |
15701 | Nouns seem to invoke stable kinds more than predicates do [Gelman] |
15705 | Essentialism encourages us to think about the world scientifically [Gelman] |
15702 | Essentialism doesn't mean we know the essences [Gelman] |
15704 | Essentialism starts from richly structured categories, leading to a search for underlying properties [Gelman] |
15706 | A major objection to real essences is the essentialising of social categories like race, caste and occupation [Gelman] |