Ad infinitum  

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Ad infinitum is a Latin phrase meaning "to infinity."

In context, it usually means "continue forever, without limit" and thus 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 2, 4, 6, 8, 10 ... continues ad infinitum."
• "The perimeter of a fractal may be iteratively drawn ad infinitum."
• The 17th century writer Jonathan Swift mocked the idea of self-similarity in natural philosophy with the following lines in his poem 'On Poetry: A Rhapsody':

"So nat'ralists observe, a flea

Hath smaller fleas that on him prey,

And these have smaller fleas that bite 'em,

And so proceed ad infinitum."

• The Victorian era mathematician Augustus De Morgan expanded on this with a similar verse, A Budget of Paradoxes

"Great fleas have little fleas upon their backs to bite 'em,

And little fleas have lesser fleas, and so ad infinitum.

And the great fleas themselves, in turn, have greater fleas to go on,

While these again have greater still, and greater still, and so on."

See also

• The song that never ends
• Turtles all the way down
• Recursion
• Self-reference
• Induction