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I am reading "Topology 2nd Edition" by James R. Munkres.

In Ch. 1.4, Munkres defines the set of real numbers $\mathbb{R}$ with the field axioms (including completeness), and then defines $\mathbb{Z}_{+}$ as the smallest inductive set in $\mathbb{R}$, as follows:

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 $\mathcal{A}$ be the collection of all inductive subsets of $\mathbb{R}$. Then the set $\mathbb{Z}_{+}$ of positive integers is defined by the equation $$\mathbb{Z}_{+} = \bigcap_{A\in \mathcal{A}} A.$$

Munkres didn't define $\mathbb{Z}_{+} := \{1, 1 + 1, 1 + 1 + 1, \dots\}$.

Why?

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    $\begingroup$ Topologists love unions and intersections. $\endgroup$ – Théophile Jul 31 '20 at 11:32
  • $\begingroup$ How does he go about defining the real numbers? It's been a while since I've read it and don't have a copy anymore. $\endgroup$ – Cameron Williams Jul 31 '20 at 11:36
  • $\begingroup$ Munkres defines $\mathbb{R}$ as a field with the linear order relation which has the least upper bound property. $\endgroup$ – tchappy ha Jul 31 '20 at 11:48
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    $\begingroup$ Munkres' definition is precise and rigorous. $\{1, 1+1, 1+1+1, \dotsc \}$ is clear (needs intent to be misunderstood) but not rigorous. $\endgroup$ – Daniel Fischer Jul 31 '20 at 11:54
  • $\begingroup$ @DanielFischer Thank you very much for your comment. $\endgroup$ – tchappy ha Jul 31 '20 at 12:56
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In general, definitions containing "$\dots$" means that everybody can intuitively understand them, but they don't provide details for a formal and rigorous definition of the object.

What is the problem? If you want to prove some properties about an object defined by means of "$\dots$", since you don't have a rigorous definition, you don't know exactly how to formally prove it, even though you intuitively understand what you have to prove: your proof will be inevitably hand-waved, and this could be a way to overlook some important and unexpected details (for instance, the fact that you need some further hypothesis to prove the desired property).

So, your definition of the set of positive integers as $\mathbb{Z}_+= \{1, 1+ 1, 1+ 1 + 1, \dots\}$ is perfectly understandable but then if you want to prove something about $\mathbb{Z}_+$, what do you do? Thanks to the rigorous and formal definition of $\mathbb{Z}_+$ as the smallest inductive subset of $\mathbb{R}$, it is clear which are the elements of $\mathbb{Z}_+$ and how you can use and refer to them.


By the way, there are many different but equivalent ways to define $\mathbb{Z}_+$ formally and rigorously. As @Théophile pointed out in his witty comment,

topologists love unions and intersections

so this is a possible reason why, in his topology handbook, Munkres defined $\mathbb{Z}_+$ as the intersection of some subsets of $\mathbb{R}$. This is not only a joke, but also due to the fact that this definition (among all the possible ones) is maybe the handiest one to deal with positive integers in a topological context, where you usually cope with intersections and unions.

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  • $\begingroup$ Taroccoesbrocco, Thank you very much for your answer. $\endgroup$ – tchappy ha Jul 31 '20 at 12:55
  • $\begingroup$ @tchappyha - You're welcome! $\endgroup$ – Taroccoesbrocco Jul 31 '20 at 12:59

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