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I want to ask my first question at MathOverflow. Since sometimes it can be hard to attract answers there (or so I’ve heard), I want to write it well. This is what I have so far. Advice on good phrasing is solicited. (I want to ask at M.O because I don’t expect an answer from this crowd, but you certainly may answer it if I’m wrong!)
Cardinal numbers vs collections of cardinal numbers
Is there any literature, especially when doing mathematics without the axiom of choice, that discusses using collections of cardinal numbers in place of individual cardinal numbers, when discussing cardinal numbers with certain properties?
That’s a little vague, and I will presently give a few motivational examples that should help to clarify what I’m thinking of. But first, a quick definition of the translation between cardinal numbers and collections of cardinal numbers assuming the axiom of choice: If K is a cardinal number, then {L | L < K} is a collection of cardinals which is small and downward-closed. Conversely, if C is a collection of cardinals which is small and downward-closed, then there exists a unique cardinal K (to wit, the smallest cardinal that does not belong to C) such that C = {L | L < K}. So the correspondence between these two perspectives is straightforward —if we assume choice.
Now I’ll consider some types of cardinal numbers. A weak limit cardinal is an infinite cardinal K such that L+ < K whenever L < K. A strong limit cardinal is an infinite cardinal K such that 2L < K whenever L < K. A regular cardinal is an infinite cardinal K such that Σi ∈ I Li < K whenever |I| < K and each Li < K. An inaccessible cardinal is an uncountable regular limit cardinal. A weakly compact cardinal is an inaccessible cardinal K such that the height of a tree is less than K whenever every level has width less than K and every branch has length less than K (the tree property). Etc.
Although these are all properties of an individual cardinal number K, they all refer to that cardinal only through the collection of smaller cardinal numbers. (The exceptions are the adjectives ‘infinite’ and ‘uncountable’, which are really only there to rule out trivial cases.)
If you don’t assume the axiom of choice, then it’s easy to still consider these conditions on a small, downward-closed collection of cardinals, but now this is more general than a collection of the form {L | L < K}. If you go beyond doubting the axiom of choice and drop the law of excluded middle (so doing constructive mathematics, in a moderate sense), you can get more flexibility by messing with the interpretation of ‘downward-closed’; for example, the collection {0,1,2,…} of all finite (in the strictest sense) cardinal numbers is closed under taking decidable sub-cardinals but not arbitrary sub-cardinals and so gives a ‘regular’ collection of cardinals which is different from {L | L < ℵ0}.
I’m interested in understanding what regular cardinals and inaccessible cardinals should be in constructive mathematics. (Bigger than that, I don’t really even understand them in classical mathematics.) It seems to me that they have to be collections of cardinals rather than individual cardinal numbers. But this perspective already makes sense in classical mathematics, especially without the axiom of choice. Has anybody studied this?
I think it is phrased well. My only issue is where you say
An inaccessible cardinal is an uncountable regular limit cardinal.
because you haven’t defined a limit cardinal for your purposes. But perhaps this is standard terminology that I do not know. It just seems strange after talking about weak and strong limit cardinals to just say ’limit cardinal’, to my untrained eye. Since Andrej Bauer is a regular MO user, you are in with a good chance to get an answer.
And everyone else, you might as well answer Toby there as well as here.
Well, I hadn’t submitted it yet, because I wasn’t sure whether it was a well written question.
But now I have, since you like it. Here it is.
By the way, inaccessible cardinals come in strong and weak forms, corresponding to the two kinds of limit cardinals, but I didn’t want to write too much. (I figured that anybody who might answer my question would know about this.)
If you go beyond doubting the axiom of choice
What does it mean “go beyond doubting” ? Not believing in, doubting forever, believing in/assuming ? Even the opposite options come to mind…
Oops, you’re right!
I actually had once, then changed it because some people have complained that they don’t know what it means. But here they are different!
There’s another mistake, which was caught on M.O. I’ve fixed both mistakes, both here and there.
Another error (about von Neumann ordinals), caught by Mike, fixed.
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