Proper or custom JSON serialization of non-finite float values

Feature or enhancement

The goal of this feature is to allow the JSON serializer in stdlib to serialize non-finite values (NaN, Inf, -Inf) according to the JSON specification. Going beyond just conforming to the spec, we could allow for custom serialization behavior.

Previous discussion

This problem was previously discussed in #84813, and a related PR was submitted in 2019, however, @mdickinson suggested I open a fresh issue where we can discuss the implementation in-depth.

Pitch

Currently, python's default JSON serializer encodes values like NaN as-is, with the explanation being that many JS-based JSON libraries also do this, and that the corresponding parsers can handle such non-conforming input.
In reality, most major browsers do not support this type of encoding and even NodeJS(v14.16.0) acts according to the previously-linked JSON spec.
The keyword argument allow_nan makes the serializer throw when encountering non-finite values when set to true, but I'd argue it is paramount to ensure compatibility with the spec and modern browsers. Changing the default behavior is of course not needed or possible at this point.

When implementing this feature, there are two main decisions to make.

Firstly, it has to be decided if allow_nan should be extended to take more datatypes like strings and callables, or if we should create a separate argument for this functionality.
Re-purposing allow_nan would make the control over such behavior centralized, however the name is very limiting.
It doesn't say anything about other non-finite values, and without looking at the docs, one would think it only takes bool values.

Secondly, it has to be decided how far we want to take this feature.
Do we want to have pre-defined cases like as_is, throw, and to_null, or do we want to allow the user to pass their own callable? The latter is implemented by the linked PR. Having both options is also a possibility.

Overall, each combination of decisions has its advantages and drawbacks. Since I wasn't a part of such discussions before, I don't have a preference.
All I want is to see this feature get implemented, and I can create a PR once consensus is reached.

Linked PRs

• gh-115246

Start by reviewing the previous discussion in #84813, the related PR 13233, and linked PR gh-115246. The work is ready when the project reaches consensus on the API and behavior for serializing NaN, Inf, and -Inf, including whether custom callables are supported.