The axioms of natural numbers can't be determined because they are axioms. If that's not true, "derive 0 is a natural number" and "1 is the succesor of 0" without any notion of numbers.

Again, they cannot be derived within the formal system where they are axioms. They can be determined in a different system which uses distinct axioms or derivation rules. This is, more or less, how you could interpret the parent comment.

The axioms of arithmetic are irreducible simple - to simple to be derived.

Your argument seems to be

1. Humans have derived arithmetics.
2. Arithmetics can't be algorithmically derived from a simpler system.
3. Therefore, humans are not algorithmic.

It seems that you are equivocating in your demands. Your original assertion is that an algorithm can't derive (in this case, meaning invent) formal arithmetics, but the quoted argument supports another claim, namely that the formalisation of arithmetics is the most austere possible. But this claim is not (at least not obviously) relevant to the original question whether intelligence is algorithmic or not. Humans haven't derived formal arithmetics from a simpler formal system. Removing the equivocation, the argument is a clear non-sequitur:

1. Humans have invented arithmetics.
2. Arithmetics can't be simplified.
3. Therefore, humans are not algorithmic.

What else could describe, for example, the emergence of the patterns out of cellular automata rules? It seems to me nature is inherently magical.

What do you mean by magical? Saying "emergence is magical" doesn't look like a description.

I think this statement is ridiculous

I would suggest you to be more careful with such statements. It comes across as confrontational.