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    • CommentRowNumber1.
    • CommentAuthorTobyBartels
    • CommentTimeOct 14th 2012

    I separated Cauchy filter from Cauchy space.

    • CommentRowNumber2.
    • CommentAuthorTobyBartels
    • CommentTimeOct 16th 2012

    Now with nonstandard analysis.

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeOct 16th 2012

    I’ve never encountered the word “adequality” before, and I thought I’d read a fair amount of NSA. I’m looking forward to seeing the page created…

    • CommentRowNumber4.
    • CommentAuthorTobyBartels
    • CommentTimeOct 16th 2012
    • (edited Oct 16th 2012)

    I could have sworn that we already had it, which is initially why I linked to it. Since the internal search no longer works, I can't easily check whether the string adequal appears anywhere in the Lab, but Google has neither word ‘adequal’ nor ‘adequality’ indexed, and I can't imagine how else that string would appear.

    There is a Wikipedia page, but that's a historical treatment focussing on Fermat (who invented the term), not really what I remember seeing before. Google is no help. But I know that I was reading about this (in the context of NSA) somewhere.

    Anyway, it simply means that two points are infinitely close together. For example, in a metric space, xyx \approx y iff d(x,y)d(x,y) is infinitesimal. If xx is standard, then xyx \approx y iff yy belongs to the monad of xx, which makes sense in any topological space (and in fact defines the topology); the relation between arbitrary hyperpoints makes sense in any uniform space (and in fact defines the uniformity). When both xx and yy are standard, then adequality reduces to the specialisation order (or its symmetrisation; I'm not sure how the term should be used in non-symmetric spaces).

    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 16th 2012

    Do you pronounce ’adequal’ as “ad+equal”, with approximately equal emphasis on the first and second syllables? (Up until now, it never occurred to me that there was an etymological connection between ’adequate’ and ’equate’, but a quick check in one of my dictionaries seems to indicate that.)

    • CommentRowNumber6.
    • CommentAuthorTobyBartels
    • CommentTimeOct 16th 2012

    I did so pronounce it, but now that you mention ‘adequate’, maybe I won't. I've only seen it written.

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeOct 17th 2012

    Got it, thanks! I’m familiar with the concept, but I don’t think I’ve heard a name for it before, aside from “infinite closeness”.

    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 19th 2012

    Re #5, the Latin ancestor of adequate appears in Aquinas’s

    Veritas est adaequatio intellectus et rei.

    There’s a lot packed into that ’adequation’ of mind and thing.

    • CommentRowNumber9.
    • CommentAuthorTobyBartels
    • CommentTimeOct 19th 2012

    Truth is the adequation of mind and thing, is that what it says?

    • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 21st 2012

    That’s right.