A Quick Reference Guide to the Set of Natural Numbers

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Understanding sets and successors is easier than you think! Image by ddluong_

By singular request, if not by popular demand, today’s article presents the first in a series of guides to different types of numbers. It promises a roller coaster ride, with a slow uphill start and a mad rush to the finish. This article begins with “Sets and Successors: A Natural Foundation“.

All Set for Successors

It is tempting but premature to discuss numbers by saying, “Let’s start by counting 1, 2, 3…”. We need to cobble two shoes before we can walk on this path.

Set the Stage: What is a Set?

The first “shoe” is the “set”. A set is a collection of elements. The “empty set” has no elements; it is shown as “{}”. Let’s associate the number of elements in the empty set with the number zero.
A set’s elements could be numbers, groups of numbers, other sets, or even odder entities. Examples of the first two cases are:

  • A set consisting of numbers may look like this: {1, 2, 3}.
  • A set consisting of groups of numbers may look like this: {(1, 1), (2, 4), (3, 9)}
"Counting Change" by °Florian

Counting Change photo by °Florian

Normally we name sets with just a capital letter; for example, let A = {1, 2, 3}, or B = {a, b, c}. The empty set is shown as Ø = {}.
Suppose we have sets with just a few elements, such as ‘A’ and ‘B’ above. Because we can pair their elements in a one-to-one relationship, it is easy to see that ‘A’ and ‘B’ have the same number of elements. So far, we don’t have any official numbers other than zero, but we know that the “cardinality”, or number of elements, of ‘A’ and ‘B’ are the same. That is a hidden way of inserting an “equal” relationship into this discussion; it will remain hidden for a few paragraphs.
We show “the cardinality of set A” as “|A|”.
We also need to have some ways of building sets using rules that can be applied over and over again.

Ensure an Orderly Succession: What Are Successors?

The second “shoe” requires several pieces at the same time. We need to “order” things, so let’s say that a “less than” relationship means that there are fewer elements in one set than another. We use the ‘<‘ sign to depict “less than”. If we have new sets A = {99, 98} and B = {z, x, v}, we can pair (99, z) and (98, x). Since ‘B’ has the element ‘v’ left over, let’s say that |A| < |B|.

We also need a “successor” function, and will use the lower-case Greek letter “sigma”, ‘σ’, as the symbol. If an integer is ‘i’, then the successor of ‘i’ is shown as “σ(i)”. But what does the “successor” function do? The first definition is that a number is less than the successor of that number: “i < σ(i)”.

The Empty Set

Let’s build a series of sets, starting with the empty set. We had defined zero = 0 = |Ø| = |{}|.

Let’s make the next set A = {Ø}. Set ‘A’ contains one element, the empty set.  Let’s say that |A| = σ(0) = 1.
We then say that the successor of zero is ‘1’, written as σ(0) = 1.
We can also say that B = { A, Ø}, and that |B| = σ( σ(0) ) = 2.
Hopefully, no one will be surprised that σ(2) = 3, and so on. All these are being defined as successors, and do not yet have much reality.

We could just as easily say that σ(0) = ‘I’, σ(‘I’) = ‘II’, and continue counting using Roman numerals. But we won’t, since Arabic numerals are more convenient and familiar.
Perhaps this makes more sense if we try to make a one-to-one relationship between the elements of sets A and B. Both contain Ø, but B has the set ‘A’ as an extra element. So the cardinality of A must be less than the cardinality of B.
For this “orderly succession”, let’s use a different naming convention for the sets. Call “1 = {0}”. In other words, the set ‘1‘ contains only the number zero. We also set the cardinality |1| = 1.
The pattern will continue that 2 = {0, 1}; 3 = {0, 1, 2}; and so on. Likewise, |2| = 2 and |3| = 3.
The general rule is that, if n = {0, 1, 2… (n-1)}, then:

  • |n| = n
  • σ(n) defines the successor of the set n, which is the union of  the set n with the number ‘n’ , so σ(n) = {0, 1, 2… (n-1), n}
  • n+1 = the cardinality of the set σ(n)
  • σ(n) = n + 1; here we are back to the successor of a number, rather than a set

Now we can backtrack a bit and nail down the successor function in terms of sets: σ(n) = the union of n with the number ‘n’. The successor function for numbers, then, is the cardinality of the set created by the successor function for the corresponding set.

It is worth noting that there is no natural number between ‘n’ and σ(n). This can be seen by mapping their sets in a one-to-one relationship: only one element will be left without a partner.

Congratulations on Your Induction into the Set of Natural Numbers

Now we define the natural numbers, N, as the set containing zero, and also containing all numbers “n+1 = σ(n)” if ‘n’ is a member of N.

That tipped the roller coaster. It was one long sentence to define N = {0, 1, 2…}, but rooted in set theory and successor functions.

Warning of a Whole Other View of Natural Numbers

Some people define Natural numbers as N = {1, 2, 3…}, and say that Whole numbers are {0, 1, 2, 3…}. This article will continue using N = {0, 1, 2…}.

Students, please check with your math teacher as to what is correct in your class.

Looking to the Future

The next article in this series, “A Guide from Natural to Imaginary and Infinite Numbers”, begins with the questions, “What mathematical functions operate within Natural numbers?” and “Do any functions break away from Natural numbers?” The final article in this trio summarizes all of these numbers in “The Definitive Quick Reference Guide to All Types of Numbers.”

Guiseppe Peano: Photographer unknown

Who Set this Up? Guiseppe Peano

Born in Italy in 1858, Giuseppe Peano taught and researched in mathematics for nine years, with numerous publications in several disciplines, including calculus. In 1889 he published his famous “Peano Axioms” which “defined the natural numbers in terms of sets”.

References:

Hrbacek, K., Jech, T. Introduction to Set Theory. (Third Edition, 1999). Quoted in {Jech, Thomas. Basic Set Theory. Stanford Encyclopedia of Philosophy. (2002)}. Accessed Sept. 25, 2011.

O’Connor, J.J. and Robertson, E.F. “Giuseppe Peano“. University of St Andrews, Scotland. (Dec. 1997). Accessed Sept. 26, 2011.

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