In sequent calculus, the completeness of atomic initial sequents states that initial sequents A ⊢ A (where A is an arbitrary formula) can be derived from...

redundant: the completeness of atomic initial sequents states that the rule can be restricted to atomic formulas without any loss of provability. Observe...

maintain the property that arbitrary initial sequents can be derived from atomic initial sequents, and that whenever a sequent is provable it can be given a...

then Γ ⊢ A. Completeness of ⇒ wrt. ⊢ If Γ ⊢ A, then Γ ⇒ A. It is clear by these theorems that the sequent calculus does not change the notion of truth, because...

For first-order logic, the most important case, it follows from the completeness theorem that the two meanings coincide. In other logics, such as second-order...

|\ \alpha =_{c}U\},} where ON is the class of ordinals. This ordinal is also called the initial ordinal of the cardinal. That such an ordinal exists and...

sequent calculus was developed to study the properties of natural deduction systems. Instead of working with one formula at a time, it uses sequents,...

with semantic completeness, which means that the set of axioms proves all the semantic tautologies of the given language. In his completeness theorem (not...

in which the value of an atomic formula embodying the membership relation is not simply True or False. The Boolean-valued models of ZFC are a related subject...

symbolism of that language. Thus, not all sound deductive systems are complete in this special sense of completeness, in which the class of models (up...

cardinality of any infinite ordinal number is an aleph number. Every aleph is the cardinality of some ordinal. The least of these is its initial ordinal....

and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted by V, is the class of hereditary well-founded...

Thus, soundness and completeness together describe a system whose notions of validity and entailment line up perfectly. Systems of logic are theoretical...

free in f. η-conversion can be seen to be the same as the concept of local completeness in natural deduction, via the Curry–Howard isomorphism. For the...

Kaustuv; Marin, Sonia; Straßburger, Lutz (2016), Focused and Synthetic Nested Sequents, Lecture Notes in Computer Science, vol. 9634, Berlin, Heidelberg: Springer...

arithmetic. By the completeness theorem, this means that G is false in some model of Peano arithmetic. However, G is true in the standard model of arithmetic...

can prove the compactness theorem using Gödel's completeness theorem, which establishes that a set of sentences is satisfiable if and only if no contradiction...

S2CID 42822367. Carnielli, Walter A. (1991). "On sequents and tableaux for many-valued logics" (PDF). The Journal of Non-Classical Logics. 8 (1): 59–76. Archived...

subsets of X , Y 1 {\displaystyle X,Y_{1}} and Y 2 {\displaystyle Y_{2}} are subsets of Y {\displaystyle Y} : for two regions where the initial function...

example, the set of the real numbers is uncountable. A set is finite if it can be enumerated by means of a proper initial segment {1, ..., n} of the natural...

0 or 1) holds for all values of n. The proof consists of two steps: The base case (or initial case): prove that the statement holds for 0, or 1. The...

topological space is the unique initial object in the category of topological spaces with continuous maps. In fact, it is a strict initial object: only the empty...

called atoms, and a term whose type is an atom is known as an atomic term. Common atomic terms included in type theories are natural numbers, often notated...

(and evidently also of constants) of higher orders and types is to make it possible to assert more complicated truth-functions of atomic propositions" (Gödel...

statement by arriving at a contradiction, even when the initial assumption is not the negation of the statement to be proved. In this general sense, proof...

especially in proof theory, a rule that concerns the manipulation of the components of sequents or deductions without reference to their internal logical structure...

Gödel in the proofs of his completeness theorem and incompleteness theorems. Gödel's proofs show that the set of logical consequences of an effective first-order...

machines, use random-access memory. Turing completeness is the ability for a computational model or a system of instructions to simulate a Turing machine...

unsatisfiability problem of first-order logic, providing a more practical method than one following from Gödel's completeness theorem. The resolution rule...

is sequent calculus, which has two sorts, propositions as in ordinary propositional calculus, and pairs of lists of propositions called sequents, such...