Regarding the question of when to include zero amongst the natural numbers, I can refer readers to my previous comments on this question at https://terrytao.wordpress.com/books/analysis-i/#comment-439818 and https://terrytao.wordpress.com/2014/11/23/254a-notes-1-elementary-multiplicative-number-theory/#comment-442582 (and also Remark 2.1.2 of my “Analysis I” book). The short answer is that in some contexts it is slightly preferable to consider zero a natural number, and in other contexts it is preferable not to. (But it is ultimately worth bearing in mind that, the quote attributed to Kronecker notwithstanding, the choice of what we do or do not consider to be a natural number is ultimately a human convention.)

]]>*Axiom 2.1.* 0 is a natural number ]]>

There should not be any debate. See the middle of page 7 of the paper he links to (“sum-free sets in groups”, now named “sum-avoiding sets in groups”), which says “the natural numbers N = {1,2,…}”.

With this in mind, the responses to your points are the following:

1). Terry does not consider 0 a natural number.

2). I don’t see why 0 playing an important role should mean it is a natural number.

3). The example you provided seems more like an exception.

“Natural numbers have no zero divisors” should read “Positive natural numbers have no zero divisors”. ]]>

0 is not a natural number

]]>*[Corrected, thanks – T.]*

*[Corrected, thanks – T.]*