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I recently landed on a book written for elementary school teachers which introduced the concept of whole numbers in the following manner:

We have a set $\{\alpha, \beta, \gamma\}$. There are other sets we can identify that share a common property with this set by virtue of which a one-to-one correspondence can be established between the elements of the set $\{\alpha, \beta, \gamma\}$ and all other sets identified. This common property abstracted away from all these sets including $\{\alpha, \beta, \gamma\}$ and considered as a thing in and of itself is named '3'.

I've also come across some books on set theory that define 3 as literally the set {0,1,2} or {{}, {{}}, {{}, {{}}}}.

So now I'm confused what is the correct way of thinking about whole numbers? A property abstracted from sets and considered to be a thing in itself or literally as sets themselves (as in 3={0,1,2}?

The first approach appeals to me better because it is in line with what I think of 3. However, the second one, says 3 is a set, that's not what my intuition tells me. 3 and the set {0,1,2} are clearly different things.

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    $\begingroup$ Are you asking a philosophical question, "what is the nature of whole numbers?" Or a pedagogical question, "how should we teach about whole numbers?" $\endgroup$
    – Justin Hancock
    Jun 12 at 16:37
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    $\begingroup$ @JustinHancock My confusion lies in why at more advanced levels we are treating whole numbers as literal sets, even though at elementary levels we have established that they are abstracted properties of sets? I'm kind of scared of learning what is in the elementary book because things in more advanced texts are different. I'm confused as to what is the correct way to think about whole numbers among these two, the one that the elementary book says or the one that more advanced texts on set theory say. $\endgroup$
    – Harshit Rajput
    Jun 12 at 17:38
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    $\begingroup$ What's interesting is that mathematicians who disagree wildly on what numbers "are" (I'm personally fond of numbers just being numbers, not secretly being something else like sets or equivalence classes) can still collaborate and agree on their arithmetic. $\endgroup$
    – TomKern
    Jun 12 at 19:06
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    $\begingroup$ In the U.S., just the fact of that definition using Greek letters would scare many students planning to teach elementary school. And I see no need for doing that. If you're teaching a class, I'd start by asking the students how they would define the number 3. After discussing those, I'd then (stuck with this textbook) change that definition to use {a,b,c}. $\endgroup$
    – Sue VanHattum ♦
    Jun 12 at 20:04
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    $\begingroup$ @Sue VanHattum: The Greek letters issue is something I completely missed, probably because for me it was buried beneath what seems to be excessive abstract "math-speak". In fact, my earlier comment was intended to convey that if you [= textbook author(s)] are going to do this, at least do it correctly. I think it would be better to talk about prehistoric people keeping track of how many chickens (or spears, or whatever) they have, who also wish to compare how many they have vs how many someone far away has (using pebbles or notches on a stick for the one-to-one correspondence) (continued) $\endgroup$
    – Dave L Renfro
    Jun 13 at 7:26

3 Answers 3

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Disregarding research on the foundations of mathematics, what we really want to have are the usual numbers with the usual properties. The usual properties are really nice, because they correspond to how the world works: I can have three apples in one hand and four in the other, and no matter which order I count them in, I always get seven apples total. And so I want $3+4 = 4+3 = 7$. Likewise for the other properties.

  • You can declare a set of axioms that are minimal and capture this behaviour. (As a first approach, check Peano axioms.
  • You can declare axioms for how set works and then build up natural numbers from them.
  • You can declare how categories, in category theory, work, and build up natural numbers from there.
  • I presume there are other ways of going about this stuff, too.

However, the main point is that you always get the same natural numbers (again, disregarding subtleties in the foundations of mathematics) which have all the familiar properties, because those are the ones we want to model our reality.

I suggest not seeing the different ways to construct natural numbers as somehow fundamental, unless you are in fact interested in looking into the foundations of mathematics. The properties of the natural numbers are the most important part, for almost everyone, including almost every mathematician.

In the context of teaching mathematics you also want to have the properties make sense; you want people to understand that $a+b = b+a$ corresponds to first counting the apples in one hand and then the other, or the other way around, and getting an equal amount of apples both ways. Being able to draw this connection is important; that is, more generally, giving meaning to the formulae.

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    $\begingroup$ Thank you. Yes. @Harshit Rajput, this is so much more helpful than anything else. When a textbook is focusing on the wrong thing, you as the teacher can ignore it. $\endgroup$
    – Sue VanHattum ♦
    Jun 13 at 15:29
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The first approach is an attempt to define an equivalence relation on sets, so that two sets are equivalent if and only if they have the same cardinality. An Achilles heel of this approach is that the set of all sets is not a proper set. In fact the collection of all things with cardinality three is not a proper set. If we are trying to contain everything in the theory of sets, we have come to a problem.

Axiomatic set theory such as Zermelo-Frankel begins with an axiom that gives the empty set {} and then proceeds to construct "one" which is {{}}, etc. Then there is the axiom of infinity which begins the construction of a new generation of sets that are not in bijective correspondence with those constructed in our first generation, which consists of finite sets.

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    $\begingroup$ However, Zermelo-Fraenkel is far too advance for elementary school (or even most elementary school teachers). $\endgroup$
    – Gerald Edgar
    Jun 12 at 17:08
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    $\begingroup$ So despite the shortcomings of the first approach, it can be used to build up to calculus and linear algebra without facing any issues (as Axiomatic set theory is not taught in schools)? $\endgroup$
    – Harshit Rajput
    Jun 12 at 18:02
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    $\begingroup$ @HarshitRajput yes I think it is a very good approach! In fact I recall this from my primary school workbook from the 1960s...drawing lines between pictures showing people and objects like balls and jacks to show the sets were "equinumerous." I think Piaget would approve. $\endgroup$
    – user52817
    Jun 12 at 21:45
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    $\begingroup$ @user52817: My primary school workbook was similar (may have been the identical book), and I describe it a bit (and give a precise author/title cite) in my comments to this answer. I realize you've seen these comments before (one of the comments is by you), but maybe others here would be interested, especially since the description and author/title cite stuff was added 5-6 weeks after the question appeared, and thus likely missed by others. $\endgroup$
    – Dave L Renfro
    Jun 13 at 7:40
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What you seem to be referring to is the cardinality of a set i.e. the number of elements in a set. If you want take the cardinality of set A you should write |A|.

e.g., A = {1, 2, 3} => |A| = 3

Often you will see the cardinality symbol is not used for convenience. However this is not technically correct as a set is different from a number.

i.e. {1, 2, 3} != 3

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