Sto leggendo il libro "general topology" di John L. Kelley e non riesco a capire un passaggio logico che non mi sembrano ben giustificati nella parte sui numeri reali.
Prima definisce l'insieme dei numeri reali così: "An ordered field is a field $F$ and a subset $P$, called the set of positive elements, such that
a) if $x$ and $y$ are members of $P$, then $x+y$ and $xy$ are also members; and
b) if $x$ is a member of $F$, then precisely one of the following statements is true: $x in P$, $-x in P$, or $x=0$. One easily verifies that $<$ is a linear ordering of $F$, where, by definition, $x<y$ iff $y-x in P$. [...] The members of $F$ such that $-x in P$ are negative. It will be assumed that the real numbers are an ordered field which is order-complete, in the sense that every non-void subset which has an upper bound has a least upper bound, or supremum."
Poi definisce gli insiemi induttivi: "An inductive set is a set $A$ of real numbers such that $0 in A$, and whenever $x in A$, then $x+1 in A$. A real number $x$ is a non-negative integer iff $x$ belongs to every inductive set. In other words, the set $omega$ of non-negative integers is defined to be the intersection of the members of the family of all inductive sets."
E qua c'è l'affermazione che non mi pare dimostrata: "Each member of $omega$ is actually non-negative because the set of all non-negative numbers is inductive." Io per dimostrarlo ho pensato che basta dimostrare che se $x in P$ allora $x+1 in P$ ma per il punto a) basta dimostrare che anche $1 in P$ solo che non so come fare. Bisogna assumerlo?