Why do mathematicians seek to go beyond the Gödel Universe of set theory?

Question

Gödel's constructible universe seems to have some attractive properties. Sets are constructed in a very regular, easy-to-understand way, and one has a definite answer on certain major set theoretic questions, such as the generalized continuum hypothesis. Alternative models posit the existence of objects like large cardinals, which (to my humble intelligence) seem esoteric and far removed from reality.

I am curious about what motivates mathematicians to study these alternative models. Are there "practical" reasons to explore models besides the constructible universe? Does the knowledge thus acquired lead to insights outside of mathematical logic and model theory? Or is it more of a "pure" inquiry, pursued for its own sake or for aesthetic reasons?

Answer 1

I think a realist philosophy accounts for much of the interest. (Of course, $L$ itself has been scrutinized thoroughly over the years.) Many (most?) set theorists take a so-called Platonist attitude: they believe that the universe of sets "really exists" in some sense. Quoting from Drake's Set Theory: an Introduction to Large Cardinals:

I have written this book from an uncompromisingly realist or platonist position; that is, I have taken the viewpoint that in some sense sets do exist, as objects to be studied, and that set theory is just as much about fixed objects as is number theory. ... It seems very difficult to me to give any reason for the study of large cardinals without taking a viewpoint of this sort.

You can find similar sentiments in Gödel's essay "What is Cantor's Continuum Problem?" and in the conclusion to Cohen's book Set Theory and the Continuum Hypothesis.

If you adhere to this philosophy, it seems very odd to suppose that all sets are constructible. Gödel's definition involves (speaking casually) describing sets using first-order logic, and repeating this process transfinitely. Why should we believe that all sets can be described this way, even all subsets of the integers? For many set theorists, intuition supports the opposite conclusion. Gödel supposedly believed that $c=\aleph_2$ (and of course $V=L$ implies $c=\aleph_1$). Cohen is on record about his belief:

A point of view which the author feels may eventually come to be accepted is that CH is obviously [his emphasis] false. The main reason one accepts the Axiom of Infinity is probably that we feel it absurd to think that the process of adding only one set at a time can exhaust the entire universe. Similarly with the higher axioms of infinity. Now $\aleph_1$ is the set of countable ordinals and this is merely a special and the simplest way of generating a higher cardinal. The set $C$ is, in contrast, generated by a totally new and more powerful principle, namely the Power Set Axiom. It is unreasonable to expect that any description of a larger cardinal which attempts to build up that cardinal from ideas deriving from the Replacement Axiom can ever reach $C$. Thus $C$ is greater than $\aleph_n$, $\aleph_\omega$, $\aleph_\alpha$ where $\alpha=\aleph_\omega$, etc. This point of view regards $C$ as an incredibly rich set given to us by one bold new axiom, which can never be approached by any piecemeal process of construction. Perhaps later generations will see the problem more clearly and express themselves more eloquently.

Here's quote from the Gödel essay, motivating large cardinal axioms:

...the axioms of set theory by no means form a system closed in itself, but, quite on the contrary, the very concept of set on which they are based suggests their extension by new axioms which assert the existence of still further iterations of the operation "set of". These axioms can be formulated also as propositions asserting the existence of very great cardinal numbers...

For a more recent discussion, see "Does V Equal L?" by Penelope Maddy, Journal of Symbolic Logic 58(1) (March 1993) pp.15-41.

Besides Platonism, the desire to explore logical structure motivates a lot of work in this field. For example, which consequences of the Axiom of Choice are actually equivalent to it? People explore this sort of thing, even without doubting the truth (whatever that means) of the axiom of choice.

Answer 2

Set theory is one of the standard ways to do higher order logic.

Sometimes, one wants to be able to use higher order logic to reason about universes of sets; e.g. category theory needs convenient ways to reason about "large" categories.

The simplest way is to do this is to use a model of set theory that contains a universe $U$, so that you get everything you want: not only does your model have the universe $U$ of interest, but your model also contains all of the sets you need for studying $U$ via higher order logic. (e.g. the power set $\mathcal{P}(U)$ corresponds to the second-order type of all subclasses of the unvierse $U$)