[koalala]

Do any mathematicians around here know if this is a special case of Gödel's incompleteness theorem [1]?

[1] https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_...

[maweki]

I'd say yes insofar, that it's clearly a case of Rice's theorem. Whether such a backdoor is contained in a program is a non-trivial property and therefore generally undecidable.

All the other theorems (Gödel, Church, Tarski, Turing) are basically the same if you squint your eyes hard enough.