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I separated Cauchy filter from Cauchy space.
Now with nonstandard analysis.
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…
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, $x \approx y$ iff $d(x,y)$ is infinitesimal. If $x$ is standard, then $x \approx y$ 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).
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.)
I did so pronounce it, but now that you mention ‘adequate’, maybe I won't. I've only seen it written.
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”.
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.
Truth is the adequation of mind and thing, is that what it says?
That’s right.
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