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It's just a wording question:

How does one tell - by a simple adjective - that a collection is "of the size of a proper class"?

Their might be several sizes of proper classes, but on the other side, it's not a problem that there are several sizes of infinite/uncountable classes to call all of them "infinite/uncountable".

The context is: How do I have to proceed with "there are finitely many, infinitely many, countably many, uncountably many, weakly inaccessibly many, strongly inaccessibly many, ..." (see Inaccessible Cardinals)?

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  • $\begingroup$ @Harry: Thanks for renaming the tag. I'll keep that in mind. $\endgroup$
    – Hans-Peter Stricker
    Commented Mar 9, 2011 at 0:40
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    $\begingroup$ To be honest: I cannot see why this question should be closed. Is it trivial, "not a real question", subjective,... whatever? I, personally, see it as a problem that one cannot talk of collections being "of the size of a proper class" by a single, simple adjective - may it be composed like "weakly/strongly inaccessible" or not. $\endgroup$
    – Hans-Peter Stricker
    Commented Mar 9, 2011 at 1:31
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    $\begingroup$ Of the answers given so far, "proper-class-many" wins for being completely unambiguous, although it's not as simple as some of the others. But what I fail to see now is why any of this is an issue. Just say something at the outset of the paper or talk, as in my edited answer. $\endgroup$
    – Todd Trimble
    Commented Mar 9, 2011 at 7:03

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Proper-class-many.

“We show that if there exist proper class many Woodin cardinals, then the set of reals x for which there is exists an ordinal α with {a ∈ Pω1 (α) | x ∈ L[a]} stationary is countable.”    —Paul Larson, Reals constructible from many countable sets of ordinals.

It’s grammatically ugly, but mathematically transparent and unambiguous, and rolls off the tongue reasonably well. I’ve heard it used by and among set theorists, category theorists, and homotopy theorists, without confusion — I’m pretty sure it’s as widely understood as anything will be for this distinction.

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    $\begingroup$ In that quote, the author could just say "arbitrarily large Woodin cardinals". Every set of cardinals is bounded, so any unbounded class of cardinals is a proper class. $\endgroup$
    – Carl Mummert
    Commented Mar 9, 2011 at 4:28
  • $\begingroup$ +1, I was just typing that as an answer and was going to reference Larson as well. Also do a Google search for: "proper class many" cardinals. $\endgroup$
    – Jason
    Commented Mar 9, 2011 at 4:30
  • $\begingroup$ @Carl Mummert: that’s true — but similarly, Euclid could have said “there are arbitrarily large prime numbers”. In each case, there’s a slight difference in connotations; I can see why Larson might have considered both options and chosen what he did. [Oblig. note: IANAHOM; I do not know how Euclid actually worded that theorem.] $\endgroup$
    – Peter LeFanu Lumsdaine
    Commented Mar 9, 2011 at 13:37
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The word that immediately comes to mind is "large". "Large category", etc.

Edit: Carl Mummert suggested this one, which I should have remembered myself and which is definitely widely used: "unbounded". "Unboundedly many Woodin cardinals", "unbounded rank", etc.

It seems to me that the simplest solution would be to say at the outset something like, "for us, 'large' will mean 'proper-class-many'..." or something similar.

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    $\begingroup$ Guys: this terminology is quite standard. If someone emphasizes that a category is large, you can be quite sure that he means the class of morphisms is a proper class. This answers Hans's question. $\endgroup$
    – Todd Trimble
    Commented Mar 9, 2011 at 1:50
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    $\begingroup$ @Todd: I do agree that this terminology is standard among category theorists. But I am looking for terms that are understood among general mathematicians. $\endgroup$
    – Hans-Peter Stricker
    Commented Mar 9, 2011 at 2:04
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    $\begingroup$ Hans, when I said this is quite standard, I meant it. The usage is not restricted to category theorists: I am sure the usage is widely understood among general mathematicians who have reason to care about set-class distinctions (homotopy theorists, algebraic geometers, etc.). If you want to debate this, then show me evidence that the term is liable to be misunderstood by those who care about set-class distinctions. Why are you asking this question, anyway? $\endgroup$
    – Todd Trimble
    Commented Mar 9, 2011 at 2:25
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    $\begingroup$ Unfortunately, "large" also has another meaning, in the context "large cardinal." $\endgroup$
    – Andreas Blass
    Commented Mar 9, 2011 at 3:55
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    $\begingroup$ @Andreas: sure. The word "regular" also has more than one meaning. But generally people pick up which meaning is meant from context. @Peter: okay, that's true. But if the speaker hadn't mentioned such set-theoretic assumptions which would thus qualify the meaning of "large", the audience would no doubt interpret his "large collection" as referring to a proper class, and if that's not what the speaker meant, it would be his fault for not mentioning his assumptions. $\endgroup$
    – Todd Trimble
    Commented Mar 9, 2011 at 6:43
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I've heard the term "absolutely infinite" used to describe classes of size $\Omega$, the class of all ordinals.

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  • $\begingroup$ Sounds good: "absolutely infinite" - there seems nothing to be beyond. $\endgroup$
    – Hans-Peter Stricker
    Commented Mar 9, 2011 at 0:37
  • $\begingroup$ But the class of all ordinals might be strictly smaller than the class of all sets... $\endgroup$
    – François G. Dorais
    Commented Mar 9, 2011 at 1:01
  • $\begingroup$ @Francois: You name it: might be, but doesn't have to be (so far as I have understood). $\endgroup$
    – Hans-Peter Stricker
    Commented Mar 9, 2011 at 1:08
  • $\begingroup$ @François: I realize that things get kinda murky at this point, but at least up to "essential size", the skeleton of $\Omega$ has more objects than the skeleton of $Set$. $\endgroup$
    – Harry Gindi
    Commented Mar 9, 2011 at 1:09
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    $\begingroup$ I think "absolutely infinite" was already used by Cantor to indicate the size of inconsistent totalities (which correspond pretty well to the newer concept of proper classes). $\endgroup$
    – Andreas Blass
    Commented Mar 9, 2011 at 3:57
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I nominate "too big".

Gerhard "Ask Me About System Design" Paseman, 2011.03.08

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