''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 by the Latins (Italic tribe), Latins in Latium (now known as Lazio), the lower Tiber area aroun ...
phrase
In grammar, a phrasecalled expression in some contextsis a group of words or singular word acting as a grammatical unit. For instance, the English language, English expression "the very happy squirrel" is a noun phrase which contains the adject ...
meaning "to
infinity
Infinity is something which is boundless, endless, or larger than any natural number. It is denoted by \infty, called the infinity symbol.
From the time of the Ancient Greek mathematics, ancient Greeks, the Infinity (philosophy), philosophic ...
" 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'' to denote written words or a concept that continues for a lengthy period beyond what is shown. Examples include:
* "The
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
1, 2, 3, ... continues ''ad infinitum''."
* "The perimeter of a
fractal
In mathematics, a fractal is a Shape, geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scale ...
may be iteratively drawn ''ad infinitum''."
The 17th-century writer
Jonathan Swift
Jonathan Swift (30 November 1667 – 19 October 1745) was an Anglo-Irish writer, essayist, satirist, and Anglican cleric. In 1713, he became the Dean (Christianity), dean of St Patrick's Cathedral, Dublin, and was given the sobriquet "Dean Swi ...
incorporated the idea of
self-similarity in the following lines from his satirical poem ''On Poetry: a Rhapsody'' (1733):
The 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
The mathematician
Augustus De Morgan
Augustus De Morgan (27 June 1806 – 18 March 1871) was a British mathematician and logician. He is best known for De Morgan's laws, relating logical conjunction, disjunction, and negation, and for coining the term "mathematical induction", the ...
included similar lines in his rhyme ''
Siphonaptera''.
See also
*
Mathematical induction
Mathematical induction is a method for mathematical proof, proving that a statement P(n) is true for every natural number n, that is, that the infinitely many cases P(0), P(1), P(2), P(3), \dots all hold. This is done by first proving a ...
*
Recursion
Recursion occurs when the definition of a concept or process depends on a simpler or previous version of itself. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in m ...
*
Self-reference
Self-reference is a concept that involves referring to oneself or one's own attributes, characteristics, or actions. It can occur in language, logic, mathematics, philosophy, and other fields.
In natural or formal languages, self-reference ...
* "
The Song That Never Ends"
*
Turtles all the way down
References
{{Infinity
Infinity
Latin logical phrases
Latin words and phrases