In several places I have come across what seems to be a standard proof by contradiction that there is no greatest natural number. As follows:- Assume there is a greatest natural number (+ve integer). Call it n. Add 1 to it to get n+1. n+1 is an integer greater than n. Therefore n cannot be the largest +ve integer. This proof seems to fall down because we are assuming in the proof that there is a larger +ve integer than n which we can obtain by adding 1 to n. If n were indeed the largest integer then we could not do this. I know it is intuitively obvious that there is no greatest integer but is there another proof. The Peano construction, I believe, have as an axiom that every integer has a sucessor and soit follows that there can be no greatest integer by definition. But that is a definition and so we would not need to prove it. Where is my logic, or lack of it, failing me. Matheinste.