Concrete and abstract

Abstraction is so common in mathematics that we usually don’t bat an eye when jumping between different levels of abstraction. There are many cases in which such an abstraction makes concepts clearer, as it cuts away all unneccesary bits of information, and also many cases in which something more concrete makes things easier to work with, as we have more information about how our objects of study actually look like. I’ll give a few well-known examples of this phenomenon from mathematics, and argue that it occurs in several (perhaps subtle) places in set theory as well.

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                                                                    Art by Patrick D. Connors

Just to be clear on the terminology that I’ll use in this blog post, I’ll call the two modes of thinking the concrete method/viewpoint and the abstract method/viewpoint. This terminology comes from the theory of C*-algebras, in which the concrete C*-algebras are all algebras of operators on a Hilbert space, and the abstract C*-algebras are complex Banach algebras with an isometric involution that satisfies the C*-equality (for more information on this, see Ilijas Farah’s book). The point here is that the concrete ones are more hands-on, being explicit operators, and we can’t a priori say much about the elements of an abstract C*-algebra. They turn out to be exactly the same class of structures however, so it is really about a change in viewpoint.

The thesis that I’ll propose here is that it’s easier to build concrete objects and also easier to work internally with concrete objects, like working with operators rather than arbitrary elements of a C*-algebra, but it’s easier to work with abstract objects, like producing structure-preserving maps between abstract C*-algebras is easier as it’s more clear which conditions we need to impose on them to preserve the structure.

Another mathematical example comes from group theory, where the concrete groups would be subgroups of the permutations of some set X, and the abstract groups are the algebras satisfying the well-known group laws. Here Cayley’s Theorem informs us that the class of concrete groups and the class of abstract groups are just the same. If we restrict ourselves to finite groups then we could even take the concrete groups as groups of matrices (the so-called linear groups), still arriving at the same class as the class of all finite abstract groups.

In set theory I’d like to propose three instances of this concrete/abstract phenomenon: one in forcing, one in hierarchies and one in inner model theory.

Concrete forcing would be the approach using partially ordered sets, where abstract forcing uses complete boolean algebras; once again it’s well-known that the two approaches are equivalent (See e.g. Jech). I’ve previously written a post on some differences between these two approaches, where I also emphasise that forcing iterations are a lot easier to deal with in the abstract approach, but constructing forcings are a lot easier in the concrete setting — I don’t think I’ve ever seen a forcing which wasn’t built either as a partially ordered set or built from other forcings (like various (co)limits).

The second example is how we build and work with (set-theoretical) universes. I’d argue that the concrete hierarchy is the V_\alpha-hierarchy and the abstract hierarchy is the H_\kappa-hierarchy — I’ve also written a blog post on these hierarchies. Again, both hierarchies yield the same universe. A main difference between these two approaches is that the V_\alpha-hierarchy is built from below and the H_\kappa from above, meaning that we kind of have to build new universes using the V_\alpha‘s (assuming that these are the two hierarchies are the ones available; in more structured settings we also have the L_\alpha-hierarchy, the J_\alpha-hierarchy and their relativised versions). The V_\alpha approach is also great with it’s associated notion of rank, which allows us to move between ordinals and sets. When it comes to working with the hierarchies however, I find the H_\kappa approach a lot simpler. An example of this could be the difference between working with elements of V_{\kappa+1} and H_{\kappa^+}, where working with the former requires us to work with coding all the time, which is just not needed in the latter.

The last example of this phenomenon that I’d like to mention is in inner model theory, more precisely in terms of how we work with canonical iterable structures, known as mice. These structures are basically built as the constructible universe L, but where we also add on extenders coding elementary embeddings. The question then becomes where we add on these extenders: every extender has an index, also known as it’s length, which is the ordinal at which the extender appears in the model. There are two main indexing schemes: the Mitchell-Steel indexing and the Jensen indexing.

I’d say that the MS indexing qualifies as the concrete approach and Jensen-indexing as the abstract one. Firstly, when we construct new mice we usually find the appropriate extenders as derived from true extenders in V, and a Jensen-indexed extender in V would be a witness to the existence of a superstrong cardinal, but all our mice are (currently) built below a superstrong cardinal! MS indexing doesn’t imply that, which is why all mice (as far as I know) are built using MS indexing. If we then want to work with Jensen indexing, we take an MS indexed mouse and re-index it to Jensen indexing. It’s usually a lot more convenient to work with \lambda-indexed extenders however, one instance being that we get amenability for free, which in the MS indexed case we have to work with the so-called amenable coding of the extender. There are many (technical) points at which Jensen indexing just makes life a lot easier, including the proof of \Box_\kappa in certain mice.

The point I want to make here is not to set up a polarisation of concrete versus abstract, but rather the opposite: by identifying two different, but equivalent, approaches as the concrete and abstract approach, I’d argue that it becomes clearer to which contexts we may want to use one approach over the other, rather than seeing them as either/or, or even good/bad.