Thanks for posting this. My intended comments got pretty long, so I converted them to a blog post <a href="http://www.thebigquestions.com/2012/11/14/accounting-for-numbers/">here</a>. 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.