extensional

In others, type checking is undecidable (such as in Extensional Type Theory); does this mean that type checking cannot be done?

The former is extensional, the latter intensional.

This line of thought seems to rely on existence and verified existence being co-extensional (if not equivalent).

The other language features that get built on top of this (conditionals, tuples, etc.) only rely on functions' extensional features.

Associate this with a proof that the extensional side of it is pure (copying mechanics of substructural types) and you're done.

Because an extensional definition doesn't tell you how to add new elements to the set.

Interestingly, they don't look like mountains formed by collisional or extensional tectonic processes to my eye (a geologist who studies mountain formation).

And I think that's one of the problematic things about purely extensional logic advocated by philosophers like Quine.

It's not a function from (nat -> nat) of course since we're talking about something other than extensional equality...

The problem seems to stem from the tension between intensional and extensional definitions.

Paradoxically, this ambiguity allows the knowledgeable programmer to be more specific than would otherwise be possible, while retaining the extensional benefits of abstraction.

Can we substitute different steps with the same extensional effect that do the job more efficiently?

The point of reifying an effect in a monad is not because "effects are icky"; it is because we have written a non-extensional operation using effects, and we want it to be an extensional function.

Values have "extensional equality" which means that values A and B are equal exactly and only when they look the same and can be used in the same way.

It's not technically wrong (since after all O(n) and O(4 * n) each refer to sets of functions, and the sets they refer to are co-extensional), but it's a "code smell".

For example, in Programming Language Theory it's amazing how many different notions of "equal" there are (isomorphism, definitional, judgemental, propositional, extensional, etc.).

Extensional equality [1] in this context I think would mean that for any given situation, the human and the emulator would need to behave equivalently.