YOUR FRIDAY MATH with Mathematician KP Hart: NATURAL NUMBERS Part 1

The question has been posed explicitly here: “What are numbers?”. The answer that was given there may not feel completely satisfactory: “We do not know what numbers are but we do know how they behave”.

Practically everyone will say: “we use them every day, so they are there!” That is, taken very strictly, not true; what we use is a system of notations and abbreviations that represent and describe something that we all agree upon. However, whomever observed that thing that we denote by `the number 3’: let me know where that was. The idea that we can stick a label like `3’ on something and that almost everyone in the world will associate that with the same notion is quite exceptional.

Mathematics has functioned quite well without a proper definition of, say, `natural number’. That nobody noticed this was probably due to the aforementioned collective act of abstraction (or conspiracy if you will). One of the first who realized that mathematics did not have a `standard’ set of natural numbers was Richard Dedekind. In his Wass sind und was sollen die Zahlen? he took matters in hand and created a set of natural numbers.

That `creation’ went as follows. He took a (Dedekind-)infinite set, S say. That means that there is a map, φ, from S to S that is injective but not surjective. Thus we have a point a in S that is not of the form φ(s) with s in S. What Dedekind showed was that in this situation there is a subset N of S that satisfies

1. if s in N then also φ(s) in N
2. N is the smallest set that satisfies 1. and that contains a
3. a≠φ(s) for all s in N
4. φ is injective

There are a few things in this list that were said before but what Dedekind showed in the rest of his booklet was that these four properties suffice to have this N serve as the/a set of natural numbers. The point a plays the role of 1, we can define addition and multiplication in such a way that all requirements we have of the natural numbers are met (and φ(s) is just s+1).This then means that every Dedekind-infinite set S carries such an N within itself and that all those N’s can serve as `set of natural numbers’. This appears to lead to a huge variety of natural numbers but we have a solution for that: those N’s are isomorphic. That means: if N and M are both sets of natural numbers, with special elements a and b respectively, and their respective maps φ and ψ, then there is a bijective map f from N to M that satisfies f(a)=b and f(φ(s))=ψ(f(s)). Everything we can do with N we can do with M and vice versa. Or, as a mathematician would express it: up to isomorphism there is only one set of natural numbers.

This leaves us with the question where to find such a set S. That we will discuss later.

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Read all of KP Harts math questions here!

About Dutch Mathematician KP Hart: In the beginning of this year the Dutch government opened a website, The Dutch Science Agenda, where everyone could post questions that they thought were of scientific interest. This was an attempt to involve the whole country in determining what the Dutch science agenda should be in the coming years.

I looked through the questions and searched for terms like `mathematics’, `infinity’ … to see what mathematical questions there were and I noticed various questions that already have answers (and have had for a long time). On a whim I decided to post answers to those questions, in Dutch. For your edification I will translate these posts into English.

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