I should have said this more carefully. If one allows enough rules of inference so that all the logical consequences of the axioms can be proved, then there are no automated proof checkers. So you can have proof checkers, but only at the cost of restricting the system so that not all logical consequences (i.e. implications that are true in every model) can be proved.
Because then the problem is not "Does this non-axiomatized stuff obey that theorem ?" but "Does that theorem follow from these axioms ?". One is a pure logic problem, and proofs may be checked by automated proof-checkers. The other directly or indirectly relies on the mathematician's intuition of the non-axiomatized subject in question, and can't be checked by automated proof-checkers.
Except insofar as the mathematicians, unknown to each other, have different ideas of what constitutes a valid rule of inference.
A logical system is a...
Thanks for posting this. My intended comments got pretty long, so I converted them to a blog post here. The gist is that I don't think you've solved the problem, partly because second order logic is not logic (as explained in my post) and partly because you are relying on a theorem (that second order Peano arithmetic has a unique model) which relies on set theory, so you have "solved" the problem of what it means for numbers to be "out there" only by reducing it to the question of what it means for sets to be "out there", which is, if anything, a greater mystery.
JoshuaZ: No, I mean the former. The problem is that you have enough rules of inference to allow you to extract all logical consequences of your axioms, then that set of rules of inference is going to be too complicated to explain to any computer. (i.e. the rules of inference are non-recursive.)