Talk:Cantor's Theorem

Currently this theorem is phrased negatively, asserting the non-existence of something, and is proved by contradiction. Is there anything to be said for phrasing it differently: for every function from a set into its power set, there is some member of the power set that is not in the image? Then one can make the proof constructive and not by contradiction. Michael Hardy (talk) 00:17, 27 July 2015 (UTC)


 * There might be a point to that. However, one important factor to take into consideration is that the current form is the most commonly used one. Therefore, rewriting it seems suboptimal.


 * Possibly one could simply introduce some cosmetic changes to the existing proofs to make them show that every function $S \to P(S)$ is not a surjection, which I believe would be an entirely constructive way of proving the theorem in its current formulation. What do you think? &mdash; Lord_Farin (talk) 15:51, 27 July 2015 (UTC)

How did you conclude that the current form is the most commonly used one? Michael Hardy (talk) 16:03, 27 July 2015 (UTC)


 * I don't recall another form ever being used. Simple as that. But that was not the point. &mdash; Lord_Farin (talk) 16:12, 27 July 2015 (UTC)


 * Actually, reading more closely, I think Proof 2 already carries the gist of this idea. &mdash; Lord_Farin (talk) 16:14, 27 July 2015 (UTC)