[xamuel]

Here's an open-ended programming project which, in a certain formal sense, spans the entire range of all difficulty levels: write an "intuitive ordinal notation" for as large of an ordinal number as you can.

What is an "intuitive ordinal notation"? Definition: The set of intuitive ordinal notations is the smallest set P of computer programs with the following property. For every computer program p, if, when p is run, all of p's outputs are elements of P, then p is in P.

So "End.", the program which immediately ends with no outputs, is vacuously in P (all of its outputs are in P, because it has no outputs). It notates the ordinal 0. Likewise, "Print(`End.')" is in P, because its sole output, "End.", is in P; it notates the ordinal 1. Likewise, "Print(`Print(End.')')" is in P, notating the ordinal 2. And so on.

The above can be short-circuited: "Let X=`End'; While(True){Print(X); X=`Print(\`'+X+`\')'}". This program outputs "End.", "Print(`End.')", "Print(`Print(`End.')')", and so on forever, all of which are in P, so this program itself is in P. It notates omega, the smallest infinite ordinal.

Here's a library of examples in Python, currently going up to a notation for the ordinal omega^omega: https://github.com/semitrivial/IONs

[carapace]

Oh that's cool.

[xamuel]

As an example to illustrate how this initially-simple-looking project spans the whole range of all possible difficulty levels: suppose you had a source-code for a superhuman flawless obedient honest AGI, call it HAL. From that source-code, you could derive a source code X for: "Print all the computer programs which HAL would list if HAL were commanded: 'Enumerate every computer program which you know to be an intuitive ordinal notation'". Since we're assuming HAL is obedient and honest, all of X's outputs will be intuitive ordinal notations, so X itself is an intuitive ordinal notation: you could say it notates "the ordinal of HAL". So this innocent-looking exercise in fact touches the most ambitious possible projects, even "Design a perfect superhuman AGI" :-)

[ianai]

They’re not sequential though so write a program which offers a way to check whether an output is in the solution set and stores the input and always returns true.

[xamuel]

Not sure what you mean exactly. Note, the set P of intuitive ordinal notations is not computably enumerable. Proof: if X were a computer program which enumerated all the intuitive ordinal notations and nothing else, then X itself would (by definition) be an intuitive ordinal notation. It would notate an ordinal number bigger than all ordinal numbers that can be notated by intuitive ordinal notations. Absurd. (Of course, to make this proof fully rigorous I'd have to give the formal definition of which ordinal an ION notates, which I left out above. But you can probably fill in those details yourself.)