erm sounds as tho you may be one of the people who know the answer before they post it.
No answers on the back of a post-card,
BUT Boolos and Jeffrey 3 rd edn - Computablity and Logic,
and or (eek!) Cutland Computablity
are helpful in this
and I think the OU still run Computability and Logic - M483 ?
The only problem for you is that you have to wade through an awful lot of other stuff (Turing, Church, Oh God and all sort of bo==lox) before you get to the good bit which in your case is towards the end of the books and is
der daaah axiomatisation of arithmetic
I think the two books I have mentioned go after peano arithmetic (and axiomatise it)
and the problem with that is that the principle of induction is difficult to shoe-horn into the scheme of things, and turns out to be rather weak.
and having done that, all the books go onto the Biggie in this area, and that is Godel's first and second theorems whic is basically - there are some arithmetic truths which cant be proven even though they are true
OK and so make a better go at axiomatising the maths to take in the defect, and you find there are still some truths which can't be proven......
Godel Escher and Bach go into this in some detail -
whcih leads to the Real Biggie - "there is no axiomatisable extension of arithemtic wh is complete."
and so dear reader, it now doestn matter if there are five seven or eleven axioms, there will still be bits left out.
Oh and somewhere else in the book you will find a statement about Zermelo Franklin set theory whcih says that group theory is not axiomatisable competely either.
This completely (hahahhaa) answers your question, but most Abers will read a bit of this and pass on......