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Axiom Of Heredity
In mathematics and logic, Ackermann set theory (AST) is an axiomatic set theory proposed by Wilhelm Ackermann in 1956. The language AST is formulated in first-order logic. The formal language, language L_ of AST contains one binary relation \in denoting set membership and one constant (mathematics), constant V denoting the Von Neumann universe, class of all sets (Ackermann used a predicate M instead). The axioms The axioms of AST are the following: # axiom of extensionality, extensionality # axiom of heredity, heredity: (x \in y \lor x \subseteq y) \land y \in V \to x \in V # axiom of comprehension, comprehension on V: for any formula \phi where x is not free variable, free, \exists x \forall y (y \in x \leftrightarrow y \in V \land \phi) # Ackermann's schema: for any formula \phi with free variables a_1, \ldots, a_n, x and no occurrences of V, a_1, \ldots, a_n \in V \land \forall x (\phi \to x \in V) \to \exists y V \forall x (x \in y \leftrightarrow \phi) An alternative axio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
