What is an undefinable ordinal? How large are the least undefinable ordinal and supremum of all undefinable ordinals?

Question

What does it mean to say that a particular ordinal/cardinal number is "definable"? Where and how do we define this "definability"? If there are undefinable ordinals/cardinals, how large are the least undefinable ordinal/cardinal? How large is the limit of all definable ordinals/cardinals?

Answer 1

Given a formal language with a well-ordering, the property of being undefinable (for elements of the well-ordering) is not definable in that language unless every element of the well-ordering is already definable. However, by stepping outside of that language, we can talk about definability in a formal way.

On the one hand, every consistent extension $T$ of ZF has a model (called a definable ordinal model or Paris model) in which every ordinal is definable. If $T$ has a well-founded model, then it also has a well-founded Paris model (and if such $T$ is complete, then every Paris model of $T$ is well-founded).

On the other hand, whenever we switch to a more expressive language and add reasonable axioms, we can typically prove that some ordinals are not definable in the base language. In ZFC, we can quantify over arbitrary languages as long as their domain is a set, and given a formal language L₁ (with at most countably many symbols), the least ordinal not definable in L₁ is countable. However, it can be a very large countable ordinal, and it typically equals the supremum of ordinals $α$ such that a bijection $α→ω$ is parameter-free definable in L₁, equivalently (for typical languages) the supremum of L₁-definable ordinals that are countable in the analog of HOD for L₁.

Also, assuming zero sharp exists, the least ordinal not definable in the constructible universe $(L,∈)$ is countable in $L$, and the least ordinal not definable in $L$ from a finite subset of $\{ω_1^V, ω_2^V, ω_3^V, ...\}$ is the least Silver indiscernible.

Regarding the set theoretical universe $V$, by extending it with a $(V,∈)$ satisfaction relation Tr and adding separation and replacement axioms for formulas involving Tr (alternatively, using a reasonable theory such as KM that proves existence of Tr), we can prove that the least ordinal not definable in $(V,∈)$ is countable. The supremum of all $(V,∈)$ definable ordinals is the least $κ$ such that $(V_κ,∈)$ is an elementary substructure of $(V,∈)$.

Also, there are different philosophical views on whether "every ordinal is potentially definable by the human mind" is true, false, or meaningless.