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At the beginning of proofs, mathematicians will often write something like:

  • Let $x$ be an arbitrary integer

I understand that the point of this statement is to get the reader of the proof to think about "a new symbol $x$ which represents one of the integers, although the specific integer is unspecified". Because the specific integer is unspecified, nothing will be assumed about the object represented by $x$ throughout the proof, except that it is an integer. So the proof will apply for all integers.

The above is discussed here: Question about how to interpret arbitrary elements

My question is "Why is the term arbitrary used to invoke the above ideas?". This is my guess.

The term "arbitrary" generally means "done without any reasoning or logic". So when a mathematician says "Let $x$ be an arbitrary integer", they are saying "Let $x$ be an integer, where the integer is chosen without any reasoning or logic". Because the integer is chosen withou any reasoning, it is unknown / unspecified. So a reader will then think about "a new symbol $x$ which represents one of the integers, although the specific integer is unspecified".

Am I correct in understanding why the term "arbitrary" is used?

Thank you for your time and please feel free to elaborate.

Pranav Jain
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  • Yes, "arbitrary" here just means $x\in \mathbb Z$ without restrictions. – lulu Jan 05 '21 at 03:38
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    The word "arbitrary" is supposed to emphasize that $x$ could be any integer. But I almost never use the word arbitrary myself. I would just say "Let $x$ be an integer." – littleO Jan 05 '21 at 03:40

1 Answers1

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Yes, "arbitrary" is just a context clue to the reader. It reminds the reader that when we introduce $x$, we're not assuming any particular features about it beyond those explicitly stated (e.g. that $x$ is an integer). At the formal level this is totally unnecessary, but for readers new to proofs it can help the intuition.

(Incidentally, note that one should not use "random" in this context!)

Noah Schweber
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