If this seems like a bad idea, it's because it is.
(In mathematics you don't technically need very many axioms or inference rules, although you do need a large body of heuristics and hints if you want a proof system which confines itself to proving inferences that are actually useful in some sense of the word, rather than proving a combinatorial explosion of trivial theorems. Dealing with the body of elementary arithmetic problems, however, wouldn't necessarily be particularly intractable -- last time I checked software proof systems can already deal with proving theorems in areas of mathematics like this and a fair bit further.)