I was trying to formulate intuitive descriptions of some large cardinals.  

Roughly something equivalent to "A manifold is an object which looks like patches of $R^n$ glued together". Not perfectly rigorous, but hopefully conveys the basic picture.  

Here are three descriptions I have in mind:  

1) An inaccessible cardinal is a set so large that it can't be reached from smaller infinite sets using unions and power set operations.  

2) A measurable cardinal is a set so large that it can't be reached from smaller infinite sets using any set theoretic formula. (I am going for V $\neq$ L)  

3) A Reinhardt cardinal is a set so large that all possible properties of the entire set theoretic universe are also true of this set. (If I am correct, this would capture the intuition of why Reinhardt cardinals don't exist. They are simply too ambitious).  

I am pretty confident that description 1 is correct in essentials, but are 2) and 3) hopelessly off ? If so, how would one appropriately modify the descriptions ? Or are object like measurable cardinals just too abstract to be expressed in anything but technical definitions ?  

Apologies in advance if this question is too elementary for MathOverflow.



Intuitive descriptions of some large cardinals
(1) seems okay, but I'm afraid that most large cardinal properties beyond inaccessibility probably aren't going to admit such simple formulations.


I'm not sure I understand the precise meaning of (2), but in any case it doesn't seem right. For one thing, the existence of $0^\sharp$ is weaker than the existence of a measurable cardinal and still implies $V \ne L$. Also if $0^\sharp$ exists then there are many cardinals, namely indiscernibles for $L$, that seem to satisfy (2) but are not measurable.


I don't think (3) is right either. To me it sounds like it describes a cardinal $\kappa$ such that $V\_\kappa$ is an elementary substructure of $V$. The existence of such a cardinal is much weaker than the existence of a Reinhardt cardinal, and in fact is equiconsistent with $\mathsf{ZFC}$ by a compactness argument.


