# Finitary

In

mathematics orlogic , a**finitary operation**is one, like those ofarithmetic , that takes a finite number of input values to produce an output. An operation such as taking anintegral of a function, incalculus , 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 forquantum mechanics , depending on the use of subspaces ofHilbert space asproposition s, operations such as taking the intersection of subspaces are used; this in general cannot be considered a finitary operation. What fails to be finitary can be called**"infinitary**".A

**finitary argument**is one which can be translated into afinite set of symbolic propositions starting from a finite [*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*] set ofaxiom scheme s, as for example the axiom schemes ofpropositional calculus .axiom s. In other words, it is a proof that can be written on a large enough sheet of paper (including all assumptions).The emphasis on finitary methods has historical roots.

studies logics that allow infinitely longInfinitary logic statement s andproofs . In such a logic, one can regard theexistential quantifier , for instance, as derived from an infinitarydisjunction .In the early 20th century,

logic ians aimed to solve the 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 ofDavid Hilbert (referring togeometry ), "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 thenumeral 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 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.The aim itself was proved impossible by

Kurt Gödel in 1931, with hisIncompleteness Theorem , but the general mathematical trend is to use a finitary approach, arguing that this avoids considering mathematical objects that cannot be fully defined.**Notes****External links*** [

*http://plato.stanford.edu/entries/logic-infinitary/ Stanford Encyclopedia of Philosophy entry on Infinitary Logic*]

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