In mathematics and

logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premise ...

, an operation is finitary if it has

finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marke ...

arity, i.e. if it has a finite number of input values. Similarly, an infinitary operation is one with an

infinite number In mathematics, transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite. These include the transfinite cardinals, which are cardinal numbers used to qua ...

of input values. In standard mathematics, an operation is finitary by definition. Therefore these terms are usually only used in the context of

infinitary logic An infinitary logic is a logic that allows infinitely long statements and/or infinitely long proofs. Some infinitary logics may have different properties from those of standard first-order logic. In particular, infinitary logics may fail to be co ...

Finitary argument

A finitary argument is one which can be translated into a

finite set In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, :\ is a finite set with five elements. T ...

of symbolic propositions starting from a finiteThe number of axioms ''referenced'' in the argument will necessarily be finite since the proof is finite, but the number of axioms from which these are ''chosen'' is infinite when the system has

axiom scheme In mathematical logic, an axiom schema (plural: axiom schemata or axiom schemas) generalizes the notion of axiom. Formal definition An axiom schema is a formula in the metalanguage of an axiomatic system, in which one or more schematic variables ap ...

s, e.g. the axiom schemes of

propositional calculus Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations ...

. set of axioms. In other words, it is a proof (including all assumptions) that can be written on a large enough sheet of paper. By contrast,

studies logics that allow infinitely long

statements Statement or statements may refer to: Common uses *Statement (computer science), the smallest standalone element of an imperative programming language *Statement (logic), declarative sentence that is either true or false *Statement, a declarative ...

and proofs. In such a logic, one can regard the

existential quantifier In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by the logical operator symbol ∃, which, w ...

, for instance, as derived from an infinitary disjunction.

History

Logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premise ...

ians in the early 20th century aimed to solve the problem of foundations, such as, "What is the true base of mathematics?" The program was to be able to rewrite all mathematics using an entirely syntactical language ''without semantics''. In the words of David Hilbert (referring to

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

), "it does not matter if we call the things ''chairs'', ''tables'' and ''beer mugs'' or ''points'', ''lines'' and ''planes''." The stress on finiteness came from the idea that human ''mathematical'' thought is based on a finite number of principles and all the reasonings follow essentially one rule: the '' modus ponens''. The project was to fix a finite number of symbols (essentially the

numerals A numeral is a figure, symbol, or group of figures or symbols denoting a number. It may refer to: * Numeral system used in mathematics * Numeral (linguistics), a part of speech denoting numbers (e.g. ''one'' and ''first'' in English) * Numerical d ...

1, 2, 3, ... the letters of alphabet and some special symbols like "+", "⇒", "(", ")", etc.), give a finite number of propositions expressed in those symbols, which were to be taken as "foundations" (the axioms), and some

rules of inference In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions). For example, the rule of ...

which would model the way humans make conclusions. From these, ''regardless of the semantic interpretation of the symbols'' the remaining theorems should follow ''formally'' using only the stated rules (which make mathematics look like a ''game with symbols'' more than a ''science'') without the need to rely on ingenuity. The hope was to prove that from these axioms and rules ''all'' the theorems of mathematics could be deduced. That aim is known as

logicism In the philosophy of mathematics, logicism is a programme comprising one or more of the theses that — for some coherent meaning of 'logic' — mathematics is an extension of logic, some or all of mathematics is reducible to logic, or some or all ...

Notes

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External links

Stanford Encyclopedia of Philosophy entry on Infinitary Logic
Mathematical logic