What's so well about the principle of well-ordering? Why is this pattern of order named like this? I am unable to trace a connection between well-ordering and a set that has a least element so I presume that there must be some mathematical becakground I'm unaware of. I've searched some wikipedia links but they describe only what the principle is, not the reason why it's named like this.
2026-04-07 17:15:49.1775582149
On
On
What's so well in having a least element in a set?
198 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
3
There are 3 best solutions below
2
On
It's just a name, I wouldn't spend too much time pondering it.
It's not totally arbitrary though - for example, choosing the minimal element of a subset of some set $X$ is well-defined exactly if $X$ is well-ordered. Similarly, an inductively defined function is well-defined only if the domain is well-ordered.
7
On
It's a good order to do induction over, because we always know what's next. If you are given a bounded subset, you know what the "next step".
Therefore you can perform induction over a linear order when it is well-ordered. That's good, that's very well.
(Another analogy is to a line to the bathroom. If the line is well-ordered everybody knows their place, and whoever wants to get in line knows exactly where they need to stand, and no one can cut in line. If the line is ordered like the rational numbers then everything is dense and people can cut in line wherever they feel like it, but no one is moving along anyway! That's not very good.)
Other than that, don't put too much emphasis on the actual name. Focus on the definition itself, it's much better.
Remember that Cantor was led to introduce the basic concepts of set theory (especially ordinal an cardinal numbers) by the need for transfinite inductive definitions in treating some aspects of trigonometric series. So it's not surprising that he should call a set well-ordered (wohlgeordnet) when it serves his purpose of supporting transfinite inductions.