In brief though prescient remarks made in 1946 to the audience of the Princeton Bicentennial Conference on Problems in Mathematics, the logician Kurt Gödel suggested a new concept of definability. He proposed to extend the ordinary language of mathematics and set theory to a higher realm of expressivity that includes the ordinals themselves as logical constants—namely, a set is ordinal definable when we can define it from ordinal parameters. Gödel described his vision for the role that this concept might play in the foundations of mathematics and set theory, predicting that it could be used in independence results, for example, as an alternative to the constructible universe for showing the relative consistency of the axiom of choice. This vision has come to fruition, developed further by Myhill and Scott (1967) and more so in subsequent developments, which reveal Gödel’s foresight. The notion of ordinal definability is by now firmly established as a key set-theoretic tool, often used just as Gödel predicted.
Nevertheless, from the inception of the concept there was a certain troubling philosophical puzzle at its core, a philosophical problem lying in wait to upset those plans. Left unresolved, this issue could ultimately have formed a dangerous pitfall, preventing the success of the method. The issue to which I refer is:
Why is the notion of ordinal definability itself definable?
The question might have seemed especially worrisome and indeed a positive explanation unlikely in light of the fact that we already know from Tarski that the underlying notion of definability itself is not definable. A central lesson of many of the major developments of twentieth-century metamathematics—the Löwenheim-Skolem theorem, the related Skolem paradox, the compactness theorem, the existence of nonstandard models—is that first-order logic is weaker than one may initially expect, and we are often generally unable to express natural higher-order metamathematical notions within a given first-order object theory such as set theory.
So how is it that we entitled to express and refer to ordinal definability in set theory but not definability? Indeed we are able to refer to it—the notion of ordinal definability is definable—but how?
This is the first in a series of essays on the topic of ordinal definability—find them in the ordinal-definability tag. I shall discuss two puzzles arising with Gödel’s notion of ordinal definability. These essays are adapted from a longer, much more technical work joint with myself and Bokai Yao on the topic of what we call the Contingent HOD Dichotomy. That paper is still currently in progress and will be released soon.
There is something about allowing ordinal parameters, it turns out, that enables us to accommodate what would otherwise be a purely philosophical, metatheoretic notion in the underlying theory—amazingly, we are able to bring this notion from the metatheory into the object theory. In this manner, a philosophical notion becomes mathematical, becoming part of the subject matter.
Let me tell you all about it—I shall explain everything.
