Ad infinitum

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

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 1, 2, 3, ... continues ad infinitum."
• "The perimeter of a fractal may be iteratively drawn ad infinitum."

The 17th-century writer Jonathan Swift 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^[1]

The mathematician Augustus De Morgan included similar lines in his rhyme Siphonaptera.^[2]

Look up ad infinitum in Wiktionary, the free dictionary.

• Mathematical induction
• Recursion
• Self-reference
• "The Song That Never Ends"
• Turtles all the way down