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Transitive Collapse
In mathematical logic, the Mostowski collapse lemma, also known as the Shepherdson–Mostowski collapse, is a theorem of set theory introduced by and . Statement Suppose that ''R'' is a binary relation on a class ''X'' such that *''R'' is set-like: ''R''−1 'x''= is a set for every ''x'', *''R'' is well-founded: every nonempty subset ''S'' of ''X'' contains an ''R''-minimal element (i.e. an element ''x'' ∈ ''S'' such that ''R''−1 'x''∩ ''S'' is empty), *''R'' is extensional: ''R''−1 'x''≠ ''R''−1 'y''for every distinct elements ''x'' and ''y'' of ''X'' The Mostowski collapse lemma states that for every such ''R'' there exists a unique transitive class (possibly proper) whose structure under the membership relation is isomorphic to (''X'', ''R''), and the isomorphism is unique. The isomorphism maps each element ''x'' of ''X'' to the set of images of elements ''y'' of ''X'' such that ''y R x'' (Jech 2003:69). Generalizations Every well-founded set-like relation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and analysis. In the early 20th century it was shaped by David Hilbert's program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in proving consistency. Work in set theory s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |