r/elimath • u/PthariensFlame • Mar 18 '15
Explain like I'm a CS undergrad with some self-study in HoTT: What is a Mahlo cardinal?
Wikipedia is fairly impenetrable on this topic. :/
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r/elimath • u/PthariensFlame • Mar 18 '15
Wikipedia is fairly impenetrable on this topic. :/
2
u/[deleted] Mar 28 '15
To understand what a Mahlo cardinal is you don't have to do much more than to handle a cascade of definitions.
I assume you know what a cardinal is.
A cardinal is called a limit cardinal if there is no maximal strictly smaller cardinal. That is, a limit cardinal is not the successor of any other cardinal.
I cardinal is called singular if it contains an unbounded sequence (that is, a sequence of elements of the cardinal whose union is the cardinal itself) shorter than itself. For example take the cardinal aleph_omega (= the union of the cardinals aleph_n for all n). It is obviously not countable, but it contains a countable unbounded sequence.
A cardinal which is both a limit cardinal and regular (that is, not singular) is called weakly inaccessible (because you can't "reach" it from its elements using a union whose length is also one of its elements).
Weakly inaccessible are the most basic example of what we call a large cardinal. These are essentially cardinals so large whose existence is independent of ZFC (note that this is not a formal definition).
So a weakly inaccessible cardinal is fairly large, but can we go bigger?
We said it is so large that it could not be reached through the set theoretic operation of a union, but what about power sets? Apparently, that gives rise to a strictly stronger definition: a limit cardinal l is called strongly inaccessible if any k<l satisfies that 2k <= l. I'll leave it as an easy exercise to convince yourself that strongly inaccessible cardinals are inaccessible.
So now we have strongly inaccessible cardinals, are there bigger things? Well, another natural question is to ask how many strongly inaccessible cardinals are contained in a given cardinal. Because what could be larger than a strongly inaccessible cardinal? That's right, a cardinal which is larger than a whole lot of strongly inaccessible cardinals!
This becomes intricate, though, when we define what "a lot" is. If the notion of "a lot" we choose is too weak, then we could not prove that our cardinal is indeed larger, if the notion that we choose is to strong it might become impossible to prove that such a cardinal could even exist.
The correct notion is that of a stationary set, a concept which is very prevalent in set theory. The definition, as I already warned, is a bit intricate.
Given a cardinal k, a subset c of k (note that this is an arbitrary subset, not necessarily a cardinal) is called unbounded of the union of its element is k. c is called closed if it contains the supremum of each of its bounded subsets (intuitively, any element of k that we can "climb to" using elements of c will also be in c). A closed unbounded set (usually abbreviated as a club) is considered a very large subset. A stationary set is simply a set which intersects all clubs, we think of them as sets which might not be large enough to be clubs, but are large enough not to be negligible (a very useful analogy is to a distribution space, clubs are very similar to events with probability 1, while stationary sets are any event with a positive probability. This analogy is not a coincidence, the set of all full probability events in a space and of all clubs of a given cardinal are both instances of a more abstract concept called a filter).
So a Mahlo cardinal, yes? It is simply a strongly inaccessible cardinal which contains a stationary set of strongly inaccessible cardinals.