There are N useful AI tools that can be made.

Where N is less than infinity.

Is it known that there are fewer than infinity tools?

I would assume that for any given tool you could make a "tool maker" tool.

You make a tool, then a tool factory, then a tool factory factory, ad infinitum.

Sprinkle in minimization and virtualization and it's extremely cool!

There is no ASML toolmaker maker.

Not yet, but could there be?

For any given time period N, if it takes > 0 time or effort to make a tool, then there are provably less possible tools than infinity for sure.

If we consider time period of length infinity, then it is less clear (I don’t have room in the margins to write out my proof), but since near as we can tell we don’t have infinity time, does it matter?