Gödel's work proved this statement false.
To give an example, what if the algorithm is to generate the correct algorithm? What if the correct algorithm ends up being the original algorithm?
That is not deterministic.
That's not a description of an algorithm. You've stated a "problem". An algorithm, by definition, is an actual deterministic procedure.
I wanted to say the same thing, but you managed to find the words I couldn't.
> An algorithm, by definition, is an actual deterministic procedure
That's not necessarily true. There are non-deterministic algorithms: https://en.wikipedia.org/wiki/Nondeterministic_algorithm
Well they can. Quantum computers are inherently non-deterministic.
Even for a classical computer, access to a true randomness source (such as a sufficiently good hardware RNG) is enough to make a classical computer non-deterministic, and hence programs that rely on that true randomness source are classified as non-deterministic.
In practice, we sometimes classify a program as nondeterministic even if it only has access to pseudorandomness, provided that pseudorandomness is "random enough". If a program uses a high quality PRNG seeded with the current time, that might be practically considered non-deterministic, even though strictly speaking the program's behaviour is a deterministic function of the current time (and other inputs).
Are you tautologically saying “determinism is determinism” because you define an algorithm as a “deterministic procedure”.
Biological neural networks are non-deterministic.
(You can debate philosophically whether apparent non-determinism is actually fundamentally non-deterministic or ultimately reduces to some hidden determinism. It depends on one's choice of interpretation of quantum theory – many worlds and hidden variables both claim that reality is fundamentally deterministic, other interpretations do not. But, whether or not non-determinism ultimately exists, it certainly apparently exists, both in biological and technological systems.)