Here, the underlying system is the positive integers along with the conventional rules of arithmetic such as addition, subtraction, multiplication, division, etc. Call this system A.





There are true statements in Athat cannot be proven within the rules of A.

If a mathematical system containing A can be proven to be consistent by using its own rules, then it is inconsistent.

It is impossible to axiomatize A, the positive integers that we all intuitively understand.

All general computers are equivalent: There is no problem that one general computer can solve that another general computer cannot. Example of "general computer" is the one you are now using.