In

Stanford Encyclopedia of Philosophy entry on Infinitary Logic

Mathematical logic

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

and logic
Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and ar ...

, an operation is finitary if it has finite
Finite is the opposite of infinite
Infinite may refer to:
Mathematics
*Infinite set, a set that is not a finite set
*Infinity, an abstract concept describing something without any limit
Music
*Infinite (band), a South Korean boy band
*''Infin ...

arity
Arity () is the number of arguments
In logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logo ...

, i.e. if it has a finite number of input values. Similarly, an infinitary operation is one with an infinite number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

of input values.
In standard mathematics, an operation is finitary by definition. Therefore these terms are usually only used in the context of infinitary logicAn infinitary logic is a logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, though ...

.
Finitary argument

A finitary argument is one which can be translated into afinite set
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and 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
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical pr ...

s, e.g. the axiom schemes of propositional calculus
Propositional calculus is a branch of logic
Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fal ...

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

s. In other words, it is a proof
Proof may refer to:
* Proof (truth), argument or sufficient evidence for the truth of a proposition
* Alcohol proof, a measure of an alcoholic drink's strength
Formal sciences
* Formal proof, a construct in proof theory
* Mathematical proof, a co ...

(including all assumptions) that can be written on a large enough sheet of paper.
By contrast, infinitary logicAn infinitary logic is a logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, though ...

studies logics that allow infinitely long statements
Statement or statements may refer to: Common uses
*Statement (computer science) In computer programming, a statement is a Syntax (programming languages), syntactic unit of an Imperative programming, imperative programming language that expresses s ...

and proofs. In such a logic, one can regard the existential quantifier
In predicate logic
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal system
A formal system is an used for inferring theorems from axioms according to a s ...

, for instance, as derived from an infinitary disjunction
In logic
Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...

.
History

Logic
Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and ar ...

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
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician
This is a List of German mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, G ...

(referring to geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

), "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
In propositional logic
Propositional calculus is a branch of logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of things, applying logic
Logic (from ...

''. 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 di ...

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
Following the developments in formal logic with symbolic logic in the late nineteenth century and mathematical logic in the twentieth, topics traditionally treated by logic not being part of formal logic have tended to ...

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 reduction (philosophy), reducible ...

.
Notes

{{reflistExternal links

Stanford Encyclopedia of Philosophy entry on Infinitary Logic

Mathematical logic