My first paper for my New Year's not a resolution is:
Wang, Hao. 1957. "The Axiomatization of Arithmetic", Journal of Symbolic Logic 22, no. 2: 145-158.
This was recommended to me as a recommendation for one of my students, but it's a topic I'm also interested in, especially as we're heading into the term where I teach PA! Maybe I can fill in a bit more history, this year.
The paper addresses a question that I've asked myself, and my own students have asked me, namely: Where do the axioms come from? This isn't just an abstract question, but a historical/conceptual one. Once you have an axiom set it's easy (well...) to see that they are the right ones; but how do you discover the axioms in the first place? Wang identifies one option:
- You start from typical proofs and results, and work backwards to determine what the underlying assumptions are.
Another option would be to pick some reasonable assumptions, and adopt them until they are shown to be inconsistent. A third would be to prove what you can, and when you get stuck, add what you need as an axiom.
The focus of Wang's paper is Peano's axiomatization of arithmetic, which is not wholly Peano's but is in fact a borrowing from Dedekind and Grassmann (p. 145). All three were, however, rooted in a desire to make "an explicit statement of some adequate group of natural rules and conventions which enables us to justify all the true numerical formulae containing 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, +, *, =, (, )" (p. 146)" (note the lack of exponentiation). Wang describes Grassmann's Lehrbuch der Arithmetik (1861) as "the first serious and rather successful attempt to put numbers on a more or less axiomatic basis" (p. 147); his scope covered not only the non-negative integers but also the negative ones. Wang gives Grassmann's axiomatization (which he calls L_2), and notes that from it, the system L_1 (which consists in the commutativity and associativity of + and *, the distribution of * over +, the fact that 0 is the identity for + and 1 for *, that a+(-a)=0, that if c is non-zero and ca=cb, then a=b, that sums and products of positive numbers are positive numbers, that every number is either positive, not positive, or 0, and a version of mathematical induction) can be derived. (L_1 is described as the contemporary -- i.e., in the 1950s -- characterisation of the integers in abstract algebra).
Wang points out a drawback of Grassmann's L_2, which is that it does not require distinct integers to have distinct successors, and hence L_2 has models consisting in only a single object (p. 149). This was made explicit in Peano, whose system contained the basic concepts of 1, number, and successor, and five axioms:
- 1 is a number
- The successor of any number is a number
- No two numbers have the same successor
- 1 is not the successor of any number
- Any property which belongs to 1, and also to the successor of every number which has the property, belongs to all numbers
(Nowadays, presentations of PA often start from 0, as opposed to 1; in this, Frege's account of numbers differs from Dedekind's in that Frege did begin with 0.) These axioms were taken from Dedekind's essay Was sind und was sollen die Zahlen? (1888) (p. 149), and Dedekind's source for these axioms is preserved in a two-page letter that Wang quotes (in translation) (pp. 150-151). What's important is that these are an axiomatisation of the concept of number only -- there's nothing here to cover the arithmetic operations. These (addition, multiplication, and exponentiation) Dedekind defines later in the essay.
Sadly, since Wang's interest is in how Dedekind got to the axioms, and not what the axioms were, he does not discuss the axioms for the arithmetic operations, which leaves me with two questions:
- How do you define the arithmetic operations if you're starting from 1 rather than from 0?
- Is Dedekind 1888 translated into English so I can read it and find out the answer to the previous questions myself?