infinitary logic造句

• Infinitary logic generalizes first-order logic to allow formulas of infinite length.
• Many different versions of infinitary logic were proposed in the late 20th century.
• In other infinitary logics, a subformula may be in the scope of infinitely many quantifiers.
• A pupil of Solomon Feferman at Stanford University, Barwise started his research in infinitary logic.
• See Infinitary logic and Beth number.
• By contrast, "'infinitary logic "'studies logics that allow infinitely long proofs.
• Concepts such as infinite proof trees or infinite derivation trees have also been studied, e . g . infinitary logic.
• A theory T in infinitary logic L _ { \ alpha, \ beta } is a set of sentences in the logic.
• Stronger classical logics such as second-order logic or infinitary logic are also studied, along with nonclassical logics such as intuitionistic logic.
• Finally, some questions arising from model theory ( such as compactness for infinitary logics ) have been shown to be equivalent to large cardinal axioms.
• As a consequence many theories, including Peano arithmetic, which cannot be properly axiomatised in finitary logic, can be in a suitable infinitary logic.
• I have encountered Infinitary logic, Fuzzy logic, and Set theory, but I get confused, which among these should be the focus of my readings?
• He axiomatised these fields and, using Shelah on categoricity in infinitary logics, proved that this theory of " pseudo-exponentiation " has a unique model in each uncountable cardinal.
• These include infinitary logics, which allow for formulas to provide an infinite amount of information, and higher-order logics, which include a portion of set theory directly in their semantics.
• Zermelo's further work on the foundations of set theory after Skolem's paper led to his discovery of the cumulative hierarchy and formalization of infinitary logic ( van Dalen and Ebbinghaus, 2000, note 11 ).
• A proof in infinitary logic from a theory T is a sequence of statements of length \ gamma which obeys the following conditions : Each statement is either a logical axiom, an element of T, or is deduced from previous statements using a rule of inference.
• A logic L _ { \ alpha, \ beta } is complete if for every sentence S valid in every model there exists a proof of S . It is strongly complete if for any theory T for every sentence S valid in T there is a proof of S from T . An infinitary logic can be complete without being strongly complete.
• Other examples of his results in pure model theory include : generalizing the Keisler Shelah omitting types theorem for \ mathit { L ( Q ) } to successors of singular cardinals; with Shelah, introducing the notion of unsuper-stability for infinitary logics, and proving a nonstructure theorem, which is used to resolve a problem of Fuchs and Salce in the theory of modules; with Hart, proving a structure theorem for \ mathit { L } _ { \ omega _ 1, \ omega }, which resolves Morley's conjecture for excellent classes; and the notion of relative saturation and its connection to Shelah's conjecture for \ mathit { L } _ { \ omega _ 1, \ omega }.