Model theory

theorem that a wide class of logical systems cannot be both consistent and complete

collection of formal systems used in mathematics, philosophy, linguistics, and computer science

study of classes of mathematical structures from the perspective of mathematical logic

In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.

In logic, soundness can refer to either a property of arguments or a property of formal deductive systems.

theorem that, for any signature 𝜎, any infinite 𝜎-structure 𝑀 and any infinite cardinal 𝜅≥|𝜎|, there is a 𝜎‐structure 𝑁 of cardinality 𝜅 that is either an elementary substructure or an elementary extension of 𝑀

fundamental theorem in mathematical logic

theorem

assignment of meaning to the symbols of a formal language

study of the semantics, or interpretations, of formal and natural languages

fundamental concept in metalogic, and the term may be used without qualification with differing meanings depending on the context within mathematical logic

formalism of first-order logic

notion in mathematical logic

Concept in model theory

first-order theory of the natural numbers with addition

In mathematical logic, a formula is satisfiable if it is true under some assignment of values to its variables. For example, the formula x+3=y is satisfiable because it is true when x=3 and y=6, while the formula x+1=x is not satisfiable over the integers. The dual concept to satisfiability is validity; a formula is valid if every assignment of values to its variables makes the formula true. For example, x+3=3+x is valid over the integers, but x+3=y is not.

the paradox that there are countable models of theories that assert the existence of uncountable sets

semantics for modal logics

The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is the quotient set of the direct product of a family of structures. All factors need to have the same signature. The ultrapower is the special case of this construction in which all factors are equal.

set together with an interpretation of a given first-order language

technique to simplify formulas

consistent theory where every statement is provable or disprovable

area of research

term in model theory and related areas of mathematics

proof technique in model theory

model of (first-order) Peano arithmetic that contains non-standard numbers

in logic and model theory

n-ary relation on the domain of a structure whose elements satisfy some formula in the

In mathematical logic, indiscernibles are objects that cannot be distinguished by any property or relation defined by a formula. Usually only first-order formulas without equality are considered. ==Examples== If a, b, and c are distinct and {a, b, c} is a set of indiscernibles, then, for example, for each binary formula \beta , we must have

In mathematical logic (a subtopic within the field of formal logic), two formulae are equisatisfiable if the first formula is satisfiable whenever the second is and vice versa; in other words, either both formulae are satisfiable or neither is. The truth values of two equisatisfiable formulae may nevertheless disagree for a particular assignment of variables. As a result, equisatisfiability differs from logical equivalence, since two equivalent formulae always have the same models, whereas equisatisfiable ones need only share satisfiability status. More formally, the equisatisfiability meta fo

an injective polynomial function from an n-dim complex vector space to itself is bijective

given a model of a theory over a 1st-order language, the set of atomic sententences and negations thereof of that hold in the model, when the language is extended with constant symbols for each element in the domain in the model

subset of a structure that itself forms a structure

branch of logic