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Let R be the set of all sets that are not members of themselves. (This set is sometimes called "the Russell set".) If R is not a member of itself, then its definition entails that it is a member of itself; yet, if it is a member of itself, then it is not a member of itself, since it is the set of all sets that are not members of themselves. The ...
After all this, the version of the "set of all sets" paradox conceived by Bertrand Russell in 1903 led to a serious crisis in set theory. Russell recognized that the statement x = x is true for every set, and thus the set of all sets is defined by {x | x = x}. In 1906 he constructed several paradox sets, the most famous of which is the set of ...
Russell's paradox concerns the impossibility of a set of sets, whose members are all sets that do not contain themselves. If such a set could exist, it could neither contain itself (because its members all do not contain themselves) nor avoid containing itself (because if it did, it should be included as one of its members). [2]
A lexical set is a group of words that share a particular vowel or consonant sound. A phoneme is a basic unit of sound in a language that can distinguish one word from another. Most commonly, following the work of phonetician John C. Wells, a lexical set is a class of words in a language that share a certain vowel phoneme.
A phraseme, also called a set phrase, fixed expression, multiword expression (in computational linguistics), or idiom, [1] [2] [3] [citation needed] is a multi-word or multi-morphemic utterance whose components include at least one that is selectionally constrained [clarification needed] or restricted by linguistic convention such that it is not freely chosen. [4]
A set of sentences is called a (first-order) theory, which takes the sentences in the set as its axioms. A theory is satisfiable if it has a model M ⊨ T {\displaystyle {\mathcal {M}}\models T} , i.e. a structure (of the appropriate signature) which satisfies all the sentences in the set T {\displaystyle T} .
The bag-of-words model (BoW) is a model of text which uses an unordered collection (a "bag") of words. It is used in natural language processing and information retrieval (IR). It disregards word order (and thus most of syntax or grammar) but captures multiplicity .
Within set theory, many collections of sets turn out to be proper classes. Examples include the class of all sets (the universal class), the class of all ordinal numbers, and the class of all cardinal numbers. One way to prove that a class is proper is to place it in bijection with the class of all ordinal numbers