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nLab > Latest Changes: Real number object

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  1.  
    • CommentRowNumber1.
    • CommentAuthorSridharRamesh
    • CommentTimeSep 1st 2012
    Am I correct in supposing that the first definition of Dedekind cuts at real numbers object is missing an openness condition (as given in the later, power object-using definition on the same page)?
    • CommentRowNumber2.
    • CommentAuthorTobyBartels
    • CommentTimeSep 1st 2012
    • (edited Sep 1st 2012)

    Quite right. I’ve fixed this and removed your query box. Presumably I would have discovered this if I’d ever finished writing out more proofs on that page!

    Actually, two other clauses (the roundedness ones) can be removed, but it would take a little rewriting of the Properties to do so.

    • CommentRowNumber3.
    • CommentAuthorSridharRamesh
    • CommentTimeSep 1st 2012
    • (edited Sep 1st 2012)
    Thanks for the confirmation/taking care of the update. But why did you rename "downwards/upwards closedness (of L/U)" into "downwards/upwards roundedness (of L/U)"? The former seems a more obvious name; I've only ever heard "roundedness" used to describe the (just-added) openness condition.
    • CommentRowNumber4.
    • CommentAuthorTobyBartels
    • CommentTimeSep 1st 2012

    H’m, because I had used ‘rounded’ at Dedekind cut for that condition, and I didn’t like the contrast between ‘closed’ (order-theoretic) and ‘open’ (topological). When I try to search Google to see how the word is used, the relevant hits are all by (or closely associated with) Paul Taylor, who uses it for the combination of the two conditions. So perhaps it is a bad word to use.

    • CommentRowNumber5.
    • CommentAuthorSridharRamesh
    • CommentTimeSep 1st 2012
    Yes, the two senses of "closed" are unfortunate... Ah, well, it's just terminology. I don't really have any strong opinions about it.
    • CommentRowNumber6.
    • CommentAuthorTobyBartels
    • CommentTimeSep 2nd 2012
    • (edited Sep 2nd 2012)

    I tried to use Google to find any hits for “rounded” and “Dedekind cut” together that were relevant and not affiliated with Taylor or us, but I had to give up. Of course, there were plenty of hits that talked about rounding real numbers, and while this is relevant to Dedekind cuts, I can’t make it correspond directly to either or both of these properties.

    But I did find one interesting hit. See pages 249&250 of this Google book and particularly Exercise III-4.18 on page 250. This matches Taylor’s usage.

1 to 6 of 6

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