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Is there a notation for the smallest inaccessible cardinal? Is the concept even coherent?

I read on Wiki that under certain assumptions even Aleph0 is strongly inaccessible. I mean the kind of inaccesbility you get if you start with Aleph0 and cannot obtain the cardinal from repeated applying of power-set and union.

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There is no notation for the smallest inaccessible cardinal. In fact, there is no notation "agreed" for any large cardinal property. One may abbreviate things like the tree property, or so, but those are usually properties that can occur at small cardinals.

We would usually say something like,

Let $\kappa$ be the least inaccessible cardinal...

The concept, of course, is very coherent. Every non-empty class of ordinals has a least element. Therefore if the class of inaccessible cardinals is non-empty, it has a least element.

As for the issue of $\aleph_0$, large cardinals are often called "strong axioms of infinity". The reason is that you need the axiom of infinity in order to prove that $\aleph_0$ exists, and you need stronger axioms to prove the existence of inaccessible cardinals, weakly compact, measurable, Woodin, supercompact, huge, and so on.

However, when saying inaccessible cardinals we almost always include "uncountable" in the definition, as to avoid the $\aleph_0$ case.

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  • $\begingroup$ I read that when we cannot use sets because of size restrictions, we use classes. Does it make sense to talk about a cardinality of a class? $\endgroup$ – Adam Dec 5 '13 at 14:01
  • $\begingroup$ @Adam: A little bit, yes. We can extend the notion of cardinality to a class, saying that two classes $A$ and $B$ are of the same cardinality if there is a class bijection between them. We can then order them using injections or surjections. But then we have the following issue: the axiom of global choice is equivalent to saying that all classes have the same size, and without it class-cardinals may behave uncontrollably, like cardinals usually do when the axiom of choice fails. $\endgroup$ – Asaf Karagila Dec 5 '13 at 14:03
  • $\begingroup$ So if the class-cardinals either behave uncontrollably or are all of the same size, they are probably not very interesting study object, are they? $\endgroup$ – Adam Dec 5 '13 at 14:11
  • $\begingroup$ Adam, it can be interesting, however the main interest of set theory, as the name suggests, is sets and not necessarily classes (there are exceptions, of course). In any case, I am not aware of any deep research that has been done in that direction. $\endgroup$ – Asaf Karagila Dec 5 '13 at 14:20

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