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Although there is a standard meaning of ‘finite’ in constructive mathematics, it’s helpful to have a way to indicate that one really means this and is not just sloppily writing ‘finite’ in a situation where it is correct classically, without having to make a circumlocution like ‘finite (even in constructive mathematics)’. Based on Mike’s notation at finite set and drawing an analogy with ‘-finite’, I’ve invented the term ‘-finite’. (So now the circumlocution is simply ‘finite (-finite)’ or ‘finite (F-finite)’, assuming that one wishes to relegate constructivism to parenthetical remarks.)
I’ve added this to finite set, added redirects, and used the new abbreviated circumlocution at dual vector space.
The notation K-finite and -finite are not mine, they came from the Elephant.
Having a specific name for “really truly finite” is a good idea; thanks! Although “F-finite” isn’t obviously suggestive to me of its meaning. I don’t object to F-finite, but what about something like “cardinally finite” to indicate that it has a finite (natural number) cardinality?
-finite would be great, if it didn’t sound so much like “infinite”…
If I were the name dictator, I might use “nat-finite”, “subfinite” (as people do), “quofinite”, and “subquofinite”, all the better to indicate what the names mean even to those who have not heard them before. But, perhaps other people would find these names silly.
@Mike: I attribute the notation and from finite set#Finitist to you.
According to the history, you were the one who introduced them in revision 4.
Wow, I could have sworn that I wrote that passage but that you later edited it to introduce that notation. I guess that it’s all me, then! (The ‘F’s, that is.)
Unless I was circumventing the nLab’s fancy authentication system to impersonate you when I edited the page, I guess. (I wasn’t.)
Any thoughts about “cardinally finite”?
Any thoughts about “cardinally finite”?
To mean what? (All of these standards of ‘finite’ are isomorphism-invariant properties of sets and therefore also properties of cardinal numbers.)
That’s true, of course, but somehow I would still be very surprised to hear the cardinality of a K-finite set referred to as a “finite number”. I feel like there’s a difference in that F-finiteness can be defined in terms of cardinal numbers (assuming the axiom of infinity, there’s a canonical set of representatives for the F-finite cardinal numbers, and a set can be defined to be F-finite if it is bijective to one of those), which is not true (so far as I know) for any of the other kinds of finiteness. So saying that that sort of finiteness is “characterized by cardinality” seems a bit more valid.
Why not use a name like “nat-finite” (or “natural number-ly finite” or whatever) if the intent is to highlight the connection with natural numbers?
@ Mike: No, one might not refer to the set as a number, but if one speaks of the set’s cardinal number, then that number is a cardinal number just like any other cardinal number. But it’s not a natural number (aka an -finite cardinal number).
I sometimes think about truth values as (-finite, not -finite) numbers.
Another term that I might have seen (or might have made up myself, as Google suggests) is ‘Bishop-finite’ for -finite. (One might even use ‘’ instead of ‘’.)
I dislike ‘nat-finite’ for using an abbreviation, but ‘-finite’ might work.
I can’t help pronouncing “-finite” as “f-finite”, as if I had a stutter. And, for that matter, when people write “oo-category”, what I hear in my head is “ooh-category”: a category that makes you go “ooh”.
I added a section “properties and relationships” to finite set, along with an alternative formulation of Dedekind-finiteness (pointed out to me by Ross Tate) which is classically equivalent, but probably distinct constructively.
Since I introduced all of the ‘’ stuff, I feel empowered to get rid of it. The term ‘Bishop-finite’ may not exist in the literature, but it makes as much sense as ‘Kuratowski-finite’, and I would like to promote it. So I’ve changed all of the ‘’s to ‘’s (and put in some explicit references to Bishop’s name).
Is “as much sense as Kuratowski-finite” a lot of sense?
Well, “Kuratowski-finite” is unambiguous and derives from a definition written down by Kuratowski. “Bishop-finite” works the same way.
Was Bishop really a first or notable person to have written down that definition?
Certainly notable; his book was notable and had wide impact. His school was (is) influential and adopted his meaning of ‘finite’ (but not, thankfully, his meaning of ‘subfinite’, which was Kuratowski-finite), in contrast to topologically motivated constructivists, who used Kuratowski’s definition (since it essentially means compact).
I would be surprised if he was the first to write it down. Obviously not the first, but probably not even the first with the intention that it be interpreted constructively. I could ask on the constructive-news mailing list about the history.
Hmm, okay.
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