I heard one mathematician who said “induction on 𝑛” and another who said “induction with respect to 𝑛”. Do these two expressions mean exactly the same thing mathematically?

  • If so, then are they both correct ways to say this and which is more common?

  • If not, then what shade of difference is there between the two phrases?

  • 9
    $\begingroup$ As far as I know, both are the same. Both sound correct to me. I'd say the shorter myself. $\endgroup$
    – Sue VanHattum
    Commented May 13, 2023 at 15:32
  • $\begingroup$ You can have induction on other, possibly finite sets, e.g. $X=\{0,~1\},$ with $S:X\to X,$ $S(0)=1,$ $S(1)=0,$ and $X=\{ 0, ~S(0)\}.$ $\endgroup$ Commented May 14, 2023 at 16:00

1 Answer 1


Yes, these expressions mean exactly the same thing. Both are correct; in my experience, ``induction on n'' is more common. Usually, the shorter the phrase, the more likely it is to be widely adopted.

  • 4
    $\begingroup$ For that matter, when the parameter to-be-inducted-upon is clear, one would simply say "by induction", without reference to the parameter, which is surely just a "dummy", in any case. :) $\endgroup$ Commented May 15, 2023 at 1:27

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.