Some days ago, I posted a question about <a href="http://mathoverflow.net/questions/26676/incompleteness-and-nonstandard-models-of-arithmetic">models of arithmetic and incompleteness</a>.  I then made a mixture of too many scattered ideas. Thinking again about such matters, I realize that what really annoyed me was the assertion by Ken Kunen that the circularity in the informal definition of natural number (what one gets starting from 0 by iterating the successor operation a finite number of times) is broken “by formalizing the properties of the order relation on &omega;” ( page 23 of his “The Foundations of Mathematics”). What does actually “breaking the circularity” mean? Is there a precise model theoretic statement that expresses this meaning? And what about proving that statement? Is that possible?