Measuring big stuff

Question

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 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.

Answer 1

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 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 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.

Answer 2

One can construct a framework to analyze these questions, but not without giving up something.

Almost all finite algebras (of finite similarity type, on a labeled set, and other technicalities which apply) generate finitely based varieties in universal algebra. This is justified by looking at multiplication tables on an underlying set of n elements, choosing a subset of them that satisfy a certain property, computing a ratio R(n), and then showing that this ratio goes to 1 as n grows. Note that it is no longer looking at "all" finite algebras, but at representatives of such, and the ratio R(n) may have a limit which may bear no relation to an analogous quantity defined for countably infinite algebras: R() might not be continuously extended to all cardinals or even enough ordinals.

One can try a similar framework for arbitrary cardinalities, using the appropriate notions of measure as suggested in Joel Hamkins's post. Let me suggest an alternative idea which may make sense. Consider a set sized collection of structures, as well as the associated theory describing a class containing that set. Ideally the collection can be embedded in a large model of the same theory, and one can now ask if there are models that exist in which the set is embedded and the ratio R of some subset is as large or as small as possible, relative to a measure on the large model and using certain minimality considerations to avoid trivial and uninteresting consequences.

If you are still interested in looking at "all" structures, you will still need a container to hold them. I do not know about functors from the "power category" of a category to the reals, but if I were to attempt measuring inside a category, that is where I would start.

Gerhard "Ask Me About System Design" Paseman, 2012.11.22