Often during informal discussion with colleagues, the following pattern emerges when we are stuck trying to prove a theorem about $x \in X$.


**A**: "*let's assume this hypothesis $H$ on $x$*"  

**B**: "*most elements of $X$ do not satisfy $H$*".  

**A**: (censored to protect the innocent).


Of course this makes sense when one restricts to a set $X$ equipped with a sigma algebra and measure, but what happens when you want to measure much larger things? Here are some example off-hand statements that sound reasonable:


1. Most topological spaces are not Hausdorff.
2. Most categories contain a non-identity morphism.



> 
> Is there any logical framework that makes this sort of informal statement precise? 
> 
> 
> 


I suppose that one of the most basic statements of this type would be "Most sets are infinite", so one sane approach would involve some correspondence of the form


$$\Omega: \text{[Classes or Large Categories]} \to \text{[Cardinals or Ordinals]}$$


that copies the basic properties of a measure on $\mathbb{R}$. There has been some prior discussion [here](https://mathoverflow.net/questions/17473/measure-theory-for-regular-cardinals) on extending the range of $\Omega$, but (as you can see from the informal statements above) my primary interest is in extending the domain beyond Sets. Sorry about the fuzzy question, but hopefully there are others who share my experience and want to know.



Measuring big stuff
Since any measure on a set can be transferred isomorphically to a
measure on any set of the same cardinality, and furthermore every
nonempty set has full measure with respect to *some* measure, your
initial remarks have a greater force only in a context where we
have an agreed-upon or natural measure.


From this perspective, I take the question to be: what kind of
natural measures do we have for measuring extremely large sets or
even proper classes?


For this, set theory has a lot to say. First of all, for any
cardinal $\kappa$ with uncountable cofinality, we have the measure
arising from the club filter, which gives measure one to any set
containing a closed unbounded subset of $\kappa$ and measure zero
to those omitting a club. This is a $\sigma$-additive (and indeed
$\kappa$-additive for regular $\kappa$ or more generally
$\text{cof}(\kappa)$-additive) two-valued measure. The [stationary
sets](https://mathoverflow.net/questions/37502/what-is-the-idea-behind-stationary-sets/37504#37504) are precisely the sets that are positive with respect to
this measure, and it is interesting to note that these are also precisely the sets that
could become measure one with respect to the club filter measure
in a forcing extension of the universe. The club and stationary
concepts are extremely robust concepts of largeness that are used
throughout set theory.


Meanwhile, second, much of large cardinal set theory is about the
possibility of various kinds of measures on extremely large sets.
For example, every measurable cardinal has what is called a
*normal* measure, a $\kappa$-additive two-valued measure on
$\kappa$ measuring every subset of $\kappa$, with the additional
property that every regressive function is constant on a measure
one set. Similar notions of normality and fineness of measures
arise with other large cardinals, such as for strongly compact
cardinals and supercompact cardinals. For example, $\kappa$ is
strongly compact if and only if every $\kappa$-additive two-valued
measure on a set can be extended to a $\kappa$-additive two-valued
measure measuring every subset of the set. We have hierarchies of
measures in the Mitchell order and the Rudin-Keisler orders.


As for proper classes, many of these same ideas still apply. For
example, the club filter measure concept still makes sense for
proper classes (although this becomes a second-order or
scheme-expressable notion), since we can say in first order that a
given proper class $C\subset\text{Ord}$ is closed and unbounded.
The hypothesis [Ord is Mahlo](https://mathoverflow.net/questions/37502/what-is-the-idea-behind-stationary-sets/37504#37504) is the hypothesis that the class of regular cardinals have outer measure
one with respect to that natural measure. One can similarly extend
the notion to concepts of ``Ord is hypermahlo'' and so on. All
these notions provide natural concepts of largeness for proper
classes.


