There is no effective (computable) procedure for determining when a statement is an axiom. However, sometimes humans can make such a determination. Fundamentally a computer wouldn't be able to since there is no computable way of making such a determination. So the computer can't be programmed to work with, say, the second order Peano Axioms. Humans can and have worked with the second order Peano Axioms.
Basically, your argument is known to philosophers by name "Lucas'/Penrose argument", and has been discussed to death. Most of the philosophers and mathematicians consider it to be invalid. There are lots of references in Wikipedia article[1].
I would like to know the flaw in my reasoning.