Ad infinitum

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{{Cite book |url=https://www.gutenberg.org/files/14353/14353-h/14353-h.htm#link2H_4_0112|title=On Poetry: a Rhapsody |last=Swift |first=Jonathan |year=1733 |access-date=8 February 2025}}

The mathematician Augustus De Morgan included similar lines in his rhyme Siphonaptera.{{Cite book |url=https://www.gutenberg.org/files/26408/26408-h/26408-h.htm |title=A Budget of Paradoxes |last=De Morgan |first=Augustus |editor-last=Smith |editor-first=David Eugene |edition=2nd |volume=II |publication-date=1915 |page=191}}

{{wiktionary|ad infinitum}}

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

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Category:Infinity

Category:Latin logical phrases

Category:Latin words and phrases