''Ad infinitum'' is a

Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Republic, it became ...

phrase
In syntax
In linguistics
Linguistics is the scientific study of language, meaning that it is a comprehensive, systematic, objective, and precise study of language. Linguistics encompasses the analysis of every aspect of language, as ...

meaning "to infinity
Infinity is that which is boundless, endless, or larger than any number
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything ...

" or "forevermore".
Description

In context, it usually means "continue forever, without limit" and this can be used to describe a non-terminating process, a non-terminating ''repeating'' process, or a set of instructions to be repeated "forever," among other uses. It may also be used in a manner similar to the Latin phrase ''et cetera
''Et Cetera'' (, ), abbreviated to ''etc.'', ''etc'', ''et cet.'', ''&c.'' or ''&c'' is a Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoke ...

'' to denote written words or a concept that continues for a lengthy period beyond what is shown. Examples include:
* "The sequence
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 ...

1, 2, 3, ... continues ''ad infinitum''."
* "The perimeter of a fractal
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 a ...

may be iteratively drawn ''ad infinitum''."
* The 17th-century writer Jonathan Swift
Jonathan Swift (30 November 1667 – 19 October 1745) was an Anglo-Irish
Anglo-Irish () is a term which was more commonly used in the 19th and early 20th centuries to identify an ethnic group
An ethnic group or ethnicity is a grouping of ...

incorporated the idea of self-similarity
__NOTOC__
has an infinitely repeating self-similarity when it is magnified.
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra ...

in the following lines from his satirical poem ''On Poetry: a Rhapsody'' (1733):
* The mathematicianThe vermin only teaze and pinch Their foes superior by an inch. So, naturalists observe, a flea Has smaller fleas that on him prey; And these have smaller still to bite 'em, And so proceed ''ad infinitum''. Thus every poet, in his kind, Is bit by him that comes behind

Augustus De Morgan
Augustus De Morgan (27 June 1806 – 18 March 1871) was a British mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), f ...

included similar lines in his rhyme ''Siphonaptera
Flea, the common name for the order Siphonaptera, includes 2,500 species of small flightless insect
Insects or Insecta (from Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-E ...

''.
See also

*Mathematical induction
Mathematical induction is a mathematical proof technique. It is essentially used to prove that a statement ''P''(''n'') holds for every natural number ''n'' = 0, 1, 2, 3, . . . ; that is, the overall statement is a ...

* Recursion
Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics
Linguistics is the scientific study of language, meaning tha ...

* Self-reference
Self-reference occurs in natural language, natural or formal languages when a Sentence (linguistics), sentence, idea or Well-formed formula, formula refers to itself. The reference may be expressed either directly—through some intermediate s ...

* " The Song That Never Ends"
* Turtles all the way down
"Turtles all the way down" is an expression of the problem of infinite regress
An infinite regress is an infinite series of entities governed by a recursive principle that determines how each entity in the series depends on or is produced by i ...

References

External links

* __notoc__ {{Infinity Infinity Latin logical phrases Latin words and phrases