# Existence of a particular function that maps an arbitrary set of ordinals to a single ordinal

Does there exist a function $$f$$ that satisfies all of the following three properties?

1. The function converts an arbitrarily large (empty, finite, countably/uncountably infinite) set of ordinals to a single ordinal;
2. If two sets $$S$$ and $$T$$ of ordinals are not equal, then two ordinals $$f(S)$$ and $$f(T)$$ are not equal; if two ordinals $$\alpha$$ and $$\beta$$ are not equal, then two sets $$f^{-1}(\alpha)$$ and $$f^{-1}(\beta)$$ are not equal;
3. The function and its inverse are computable by Ordinal Turing Machines, i.e. there exists an Ordinal Turing Machine $$m$$ that (i) given an arbitrary set $$T$$ of ordinals as the input, outputs a single ordinal $$f(T)$$; (ii) given a single ordinal $$\alpha$$ as the input, outputs a set $$f^{-1}(\alpha)$$ of ordinals.
• This is kind of self-evident in the case of $V=L$ (unless I am misreading the question), because it would follow from equivalence of SO (sets of ordinals) in $L$ and those that are ordinal computable by (some suitable) ordinal parameters . But for the general scenario, it is unclear to me. Jun 28, 2022 at 9:44

The existence of a function $$f$$ as specified in the question cannot be proved in ZFC. This follows from the following theorem and the well-known independence of $$\mathrm{V = OD}$$ (equivalently: $$\mathrm{V = HOD}$$) from $$\mathrm{ZFC}$$. Recall that $$\mathrm{OD}$$ is the class of ordinal-definable sets.
Theorem. (ZFC) There is a (parameter-free) definable injective function $$f$$ from the class of subsets of ordinals to the class of ordinals (i.e., a definable function $$f$$ satisfying properties (1) and (2) of the question) if and only if $$\mathrm{V = OD}$$.
Proof. The right-to-left direction readily follows from the well-known fact that there is a (parameter-free) global well-ordering of the class of all sets $$\mathrm{V}$$ of order-type $$\mathrm{Ord}$$ (class of ordinals) in the presence of $$\mathrm{ZF + V = OD}$$.
For the other direction, recall that it is a theorem of $$\mathrm{ZFC}$$ that every set can be canonically coded by a subset of ordinals (more precisely, for every set $$x$$ there is a subset $$y$$ of ordinals such that $$x \in \mathrm{L}(y)$$). This makes it clear that if there is a definable injection $$f$$ of the class of subsets of ordinal into the class $$\mathrm{Ord}$$ of ordinals, then there is an injection $$F$$ of the class of all sets $$\mathrm{V}$$ to $$\mathrm{Ord}$$. More specifically, given a set $$x$$, let $$F(x)$$ be $$f(y_0)$$, where $$f(y_0)$$ is the minimum element of the set of all ordinals of the form $$f(y)$$, where $$y$$ canonically codes $$x$$. Since $$\mathrm{V=OD}$$ is equivalent to the existence of a parameter-free definable injection of $$\mathrm{V}$$ into $$\mathrm{Ord}$$, this completes the proof.