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Wikipedia states that the least-upper-bound property “ is a fundamental property of the real numbers and certain other ordered sets. A set $X$ has the least-upper-bound property if and only if every non-empty subset of $X$ has a supremum in $X$.”

This seems to me to be a bad slip, because from that it follows that $\mathbb R$ does not have the least-upper-bound property. (Later in the article Wikipedia gives what I would consider to be the correct formulation of the property.)

So, am I missing something, or is this a glaring error in Wikipedia?

JMP
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    Should be "if and only if every non-empty subset of $X$ which has an upper bound has a supremum in $X$". –  Nov 10 '15 at 23:27
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    It should indeed say "every non-empty bounded subset of $X$ has a supremum in $X$." – Noah Schweber Nov 10 '15 at 23:27
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    And I have no idea why someone downvoted this intelligent question. – Brian M. Scott Nov 10 '15 at 23:28
  • Further down the correct statement appears. "The least-upper-bound property states that any non-empty set of real numbers that has an upper bound must have a least upper bound in real numbers." – quid Nov 10 '15 at 23:30
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    For anyone who goes to look, I've now fixed the Wikipedia article! – not all wrong Nov 10 '15 at 23:30
  • @Brian: I haven't voted, but I am guessing that this is some sort of a "gag reflex", whenever someone claims a "reasonably trusted source" to be wrong. Mathematics is about humility, and the assumption should always be "What have I misunderstood in ..." rather than "Could it be that ... is wrong?" – Asaf Karagila Nov 10 '15 at 23:35
  • @BrianM.Scott now, I did not downvote and it might not be that bad a question, however, on that very Wikipedia page latter the correct definition appears. It is thus rather clear that there is a misstatement there. It is not quite clear what the actual point of the question is. It is also already completely obsolete now after a couple minutes. WIkipedia has a talk page for such things. – quid Nov 10 '15 at 23:37
  • @Asaf: But in this case the response was am I missing something?, which is perfectly fine. You may well be right, but that would do nothing to improve my low opinion of the downvote. – Brian M. Scott Nov 10 '15 at 23:38
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    @Brian: We're not in disagreement here. I'm just saying why I would have downvoted something whose title is "Is X wrong about Y?". – Asaf Karagila Nov 10 '15 at 23:39
  • @quid: No. It is only clear that at least one of the two statements is wrong. And someone can reasonably expect to get a quicker and more authoritative response here than at the Wikipedia talk page. And no, the question is not really obsolete: its reference to the Wikipedia page is obsolete, but the underlying mathematical question remains perfectly reasonable. – Brian M. Scott Nov 10 '15 at 23:41
  • @BrianM.Scott first of all, I said it was clear "there is a misstatement there" (and not, for example, that "the statement is wrong", and purposefully so) which is the case when at least one of the two statements is wrong. Thus, your comment ought to start with "yes." The question as written is obsolete. Indeed, a better question would have been what you propose, what the definition is mentioning WIkipedia as context and motivation. – quid Nov 10 '15 at 23:46
  • @quid: As usual, you have a problem with pragmatics. The mere fact that a reader of the article could deduce that there was a misstatement somewhere – as indeed the OP obviously did – does nothing to support your contention that it is not clear what the actual point of the question is. Of course, it would be difficult to support that contention, since the point of the question is entirely clear. I’m done here. – Brian M. Scott Nov 10 '15 at 23:53
  • @BrianM.Scott so what is the point? I can imagine numerous different motivations for making that post. – quid Nov 10 '15 at 23:59
  • @AsafKaragila I have mixed feelings if Wikipedia is "reasonably trusted". It used to have the opposite reputation, even if this has improved a lot. The incorrect definition of Bernstein set survived from 2009 to 2015 when I spotted it: I revised it then someone (wikipedia enthusiastic, but not a mathematician) didn't like the form of my first revision and found it more appropriate to revert to the incorrect one, so I revised it again the proper way, .. I was not aware of "the correct procedure" or the discussion page – Mirko Nov 11 '15 at 00:13
  • Possible duplicate of Least-upper-bound property Rudin book – quid Nov 11 '15 at 02:03
  • Given that the reference to Wikipedia is obsolete and thus moot, the question is now a duplicate of a question asking to clarify the definition of the notion. By the way, for me that question was the second hit on the obvious Google search. – quid Nov 11 '15 at 02:06
  • Here is the link to the older version of the Wikipedia entry that prompted this question: https://en.wikipedia.org/w/index.php?title=Least-upper-bound_property&oldid=689237257 (the link provided in the question is to the current, corrected version, of the definition of the least upper bound property). – Mirko Nov 11 '15 at 09:39

1 Answers1

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(As stated by Noah above), another necessary condition is that every such subset $X$ be bounded from above. That is, every non-empty bounded subset of $X$ (be it closed, open, or neither) has a supremum in $X$.

Alekos Robotis
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  • Thank you everybody for the attention to this question. I WANT Wikipedia to be a “reasonably trusted source”, and I’m dismayed at how often they seem to shoot themselves in the foot. –  Nov 11 '15 at 03:50
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    @EsperantoSpeaker1 maybe this is a language issue, but I find your tone excessive. There was a minor error, basically a typo, textbooks and research articles are full of such things. Besides, it seems all but one version were correct. With some effort on your side, you could just have corrected it there. – quid Nov 11 '15 at 10:56
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    @quid: So, engagement with Wikipedia is compact? That is, if you detect a bug you are required to fix it - the middle ground of merely reporting it is not an option - or even of getting a second opinion before doing so? Given Asaf Karagila who says the OP should take the “What have I misunderstood in...” stance, and you who say the OP should just charge on ahead, I would say that the solution lies on the imaginary axis. –  Nov 11 '15 at 22:02
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    @EsperantoSpeaker1 no, you can report a bug, but you should report it where it occurs if possible. There is infrastucture at Wikipedia for just that. Also for asking for clarification. It is ultimately quite harmful if the infrastructure that a project provides for addressing its issues is not used but quasi-random other channels are used. – quid Nov 11 '15 at 22:08
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    @EsperantoSpeaker1 further on the perceived discrpancy between what Asaf and I say. Ultimately we both are not overly thrilled about your approach and tone related to the matter: "glaring error," "dismayed" etc. All that because somebody dropped a word when moving around some text a couple days ago. – quid Nov 11 '15 at 22:11
  • @quid: ““The difference between the right word and almost the right word is the difference between lightning and a lightning bug.” -- Mark Twain –  Nov 12 '15 at 00:55
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    @EsperantoSpeaker1 I take it you mean to chose your words more carefully next time. That's great. Anyway, I think we discussed this enough. Your question was not that bad, either. In some sense I made too much fuss about it. – quid Nov 12 '15 at 01:02