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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, xy iff d(x,y) is infinitesimal. If x is standard, then xy iff y belongs to the monad of x, 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 x and y 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.