# Defining $\mathbb{Z}_+$

While reading Munkres' Topology section about integers and reals (Chapter 1, Section 4), he defines the set $\mathbb{Z}_+$ as:

Definition: A subset $A$ of the real numbers is said to be inductive if it contains the number 1, and if for every $x$ in $A$, the number $x+1$ is also in $A$. Let $\cal{A}$ be the collection of all inductive subsets of $\mathbb{R}$. Then the set $\mathbb{Z}_+$ of positive integers is defined by the equation $\bigcap _{A \in \cal{A}} A$

Isn't it possible to define $\mathbb{Z}_+$ as the set defined this way:

$1\in B \wedge (j \in B, n-1 = j) \Rightarrow n\in B$

Then $B = \mathbb{Z}_+$

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This doesn't determine B as the set of positive integers. Basically you're just saying that B is an inductive set. – Adrián Barquero Oct 29 '10 at 2:34
In fact, if you look at the definition you gave from Munkres's book, the set of positive integers being defined as the intersection of all inductive sets is just saying that the positive integers constitute a minimal inductive set. – Adrián Barquero Oct 29 '10 at 2:38
Ohh, you're right. I thought the second definition excluded other non integers from appearing but actually B could be any inductive set. Thanks a lot! – F M Oct 29 '10 at 2:47
fmartin: could you write that up as a community wiki answer, so that you can mark it as accepted? – Carl Mummert Oct 29 '10 at 4:33
Taking the intersection of all inductive sets is like taking the smallest inductive set. It's also like taking the minimal fixpoint of your other definition. – Yuval Filmus Oct 29 '10 at 5:52

This doesn't determine B as the set of positive integers. Basically you're just saying that B is an inductive set. In fact, if you look at the definition you gave from Munkres's book, the set of positive integers being defined as the intersection of all inductive sets is just saying that the positive integers constitute a minimal inductive set.

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