mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

and

logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...

, an operation is finitary if it has finite

arity In logic, mathematics, and computer science, arity () is the number of arguments or operands taken by a function, operation or relation. In mathematics, arity may also be called rank, but this word can have many other meanings. In logic and ...

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

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. Th ...

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 schemes, e.g. the axiom schemes of

propositional calculus The propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. Sometimes, it is called ''first-order'' propositional logic to contra ...

. set of

axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...

s. In other words, it is a proof (including all assumptions) that can be written on a large enough sheet of paper. By contrast, infinitary logic studies logics that allow infinitely long statements and proofs. In such a logic, one can regard the existential quantifier, for instance, as derived from an infinitary

disjunction In logic, disjunction (also known as logical disjunction, logical or, logical addition, or inclusive disjunction) is a logical connective typically notated as \lor and read aloud as "or". For instance, the English language sentence "it is ...

History

Logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...

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 a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

), "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 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 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.

Notes

External links

{{wiktionary, finitary
Stanford Encyclopedia of Philosophy entry on Infinitary Logic
Mathematical logic