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In mathematics and logic, an operation is finitary if it has finite arity, 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 of symbolic propositions starting from a finite^[1] 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, 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.

History

Logicians 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), "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 ^{[citation needed]} 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

^ The 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.

External links

Stanford Encyclopedia of Philosophy entry on Infinitary Logic

[1] The 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.

[1]

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+{{Short description|Qualifies an operation with a finite number of arguments}}
-In [[mathematics]] or [[logic]], a '''finitary operation''' is one, like those of [[arithmetic]], that takes a number of input values to produce an output. An operation such as taking an [[integral]] of a [[function (mathematics)|function]], in [[calculus]], is defined in such a way as to depend on all the values of the function (infinitely many of them, in general), and is so not ''[[prima facie]]'' finitary. In the logic proposed for [[quantum mechanics]], depending on the use of subspaces of [[Hilbert space]] as [[proposition]]s, operations such as taking the [[intersection (set theory)|intersection]] of subspaces are used; this in general cannot be considered a finitary operation. What fails to be finitary can be called '''''infinitary'''''.
+{{refimprove|date=April 2012}}
+In [[mathematics]] and [[logic]], an [[Operation (mathematics)|operation]] is '''finitary''' if it has [[Finite cardinality|finite]] [[arity]], i.e. if it has a finite number of input values.  Similarly, an '''infinitary'''<!--boldface per WP:R#PLA--> operation is one with an [[Infinite set|infinite number]] of input values.
-A '''finitary argument''' is one which can be translated into a [[finite set|finite]] set of symbolic propositions starting from a finite set of [[axiom]]s. In other words, it is a [[Mathematical proof|proof]] that can be written on a large enough sheet of paper (including all assumptions).
+In standard mathematics, an operation is finitary by definition. Therefore, these terms are usually only used in the context of [[infinitary logic]].
-The emphasis on finitary methods has historical roots. '''[[Infinitary logic]]''' studies logics that allow infinitely long [[statement]]s and [[proofs]]. In such a logic, one can regard the [[existential quantifier]], for instance, as derived from an infinitary [[disjunction]].
+== Finitary argument ==
-In the early [[20th century]], [[logic]]ians aimed to solve the [[foundations of mathematics | problem of foundations]]; that is, answer the question: "What is the true base of mathematics?" The program was to be able to rewrite all mathematics starting using an entirely syntactical language ''without semantics''. In the words of [[David Hilbert]] (referring to [[geometry]]), "it does not matter if we call the things ''chairs'', ''tables'' and ''cans of beer'' or ''points'', ''lines'' and ''planes''."
+A '''finitary argument''' is one which can be translated into a [[finite set]] of symbolic propositions starting from a finite<ref>The 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]]s, e.g. the axiom schemes of [[propositional calculus]].</ref> set of [[axiom]]s. In other words, it is a [[Mathematical proof|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 [[statement (logic)|statements]] and [[Mathematical proof|proofs]]. In such a logic, one can regard the [[existential quantifier]], for instance, as derived from an infinitary [[disjunction]].
-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 [[numeral]]s 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 [[rule of inference|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.
+== History ==
-The aim itself was proved impossible by [[Kurt Gödel]] in [[1931]], with his [[Incompleteness Theorem]], but the general mathematical trend is to use a finitary approach, arguing that this avoids considering mathematical objects that cannot be fully defined.
+[[Logic]]ians in the early 20th century aimed to solve the [[foundations of mathematics|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]]), "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 {{proveit|date=April 2013}} and all the reasonings follow essentially one rule: the ''[[modus ponens]]''. The project was to fix a finite number of symbols (essentially the [[Numerical digit|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 [[rule of inference|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]].
-==External links==
+== Notes ==
-*[http://plato.stanford.edu/entries/logic-infinitary/ Stanford Encyclopedia of Philosophy entry on Infinitary Logic]
+{{reflist}}
+== External links ==
+{{wiktionary|finitary}}
+* [http://plato.stanford.edu/entries/logic-infinitary/ Stanford Encyclopedia of Philosophy entry on Infinitary Logic]
 [[Category:Mathematical logic]]