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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?

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    $\begingroup$ As far as I know, both are the same. Both sound correct to me. I'd say the shorter myself. $\endgroup$
    – Sue VanHattum ♦
    May 13 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 $X=\{ 0, ~S(0)\}.$ $\endgroup$
    – Dan Christensen
    May 14 at 16:00

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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.

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    $\begingroup$ 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. :) $\endgroup$
    – paul garrett
    May 15 at 1:27

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