The “standard model” of arithmetic is the idea you probably have when you think about natural numbers (0, 1, 2, 3, …) and what you can do with them. So, for instance, you can keep counting as far you like and will never run out of numbers. You won’t get a struck in a loop anywhere when counting: the numbers don’t suddenly go 89,90, 91, 80, 81, 82, … Also 2 + 2 = 4, x + y = y + x, etc.
One of the things mathematicians do is take structures like this standard model of arithmetic and devise lists of properties describing how it works and constraining what it could be. You could think of this as playing mathematical charades. Suppose I’m thinking of the natural numbers. How do I go about telling you what I’m thinking without just saying, “natural numbers” or counting 0, 1, 2, 3, … at you? What’s the most precise, unambiguous, and concise way I could do this, using principles that are more basic or general?
Of the people who gave this a go for the natural numbers, the most famous are Richard Dedekind (1888, What are numbers and what should they be?) and Giuseppe Peano (1889, The principles of arithmetic, presented by a new method). The result is called Peano Arithmetic or Dedekind-Peano Arithmetic. What I find interesting about this is where the ideas came from. Dedekind helpfully explained his thinking in an 1890 letter to Hans Keferstein. A chunk of it is quoted verbatim by Hao Wang, (1957, p. 150). Here’s part:
“How did my essay come to be written? Certainly not in one day, but rather it is the result of a synthesis which has been constructed after protracted labour. The synthesis is preceded by and based upon an analysis of the sequence of natural numbers, just as it presents itself, in practice so to speak, to the mind. Which are the mutually independent fundamental properties of this sequence [of natural numbers], i.e. those properties which are not deducible from one another and from which all others follow? How should we divest these properties of their specifically arithmetical character so that they are subsumed under more general concepts and such activities of the understanding, which are necessary for all thinking, but at the same time sufficient, to secure reliability and completeness of the proofs, and to permit the construction of consistent concepts and definitions?”
Dedekind spelt out his list of properties of what he called a “system” of N. Key properties are as follows (this is my paraphrase except where there is quoted text; also I’m pretending Dedekind started the numbers at zero when he actually started at one):
- N consists of “individuals or elements” called numbers.
- Each element of N is related to others by a relation (now called the successor), intuitively, “the number which succeeds or is next after” a number. But remember that we don’t have “next after” in this game. The successor of an element of N is another element of N. This captures part of the idea of counting along the numbers.
- If two numbers are distinct, then their successors are also distinct. So you can’t have say, the successor of 2 as 3 and also the successor as 4 as 3.
- Not all elements of N are a successor of any element.
- In particular, zero isn’t a successor of any element.
Dedekind notes that there are many systems that satisfy these properties and have N as a subset but also have arbitrary “alien intruders” which aren’t the natural numbers:
“What must we now add to the facts above in order to cleanse our system […] from such alien intruders […] which disturb every vestige of order, and to restrict ourselves to the system N? […] If one assumes knowledge of the sequence N of natural numbers to begin with and accordingly permits himself an arithmetic terminology, then he has of course an easy time of it. […]”
But we aren’t allowed to use arithmetic to define arithmetic. Dedekind explains again the intuitive idea of a number being in N if and only if you can get to it by starting at 0 and working along successors until you reach that number. This he formalises as follows:
- An element n belongs to N if and only if n is an element of every system K such that (i) the element zero belongs to K and (ii) the successor of any element of K also belongs to K.
So, we get the number 0 by 6(i), the number 1 by 6(ii) since it’s the successor of 0, the number 2 by applying successor to 1, and so on until an infinite set of natural numbers is formed. This approach is what we now call mathematical induction.
There are a few issues with Dedekind-Peano Arithmetic, though – for another time…