On Peano's 5 postulates

Question

I have been recently studying a C.G. Hempel's article on mathematical truth and pointed out his following quotation: "Every concept of mathematics can be defined by means of Peano's three primitives,and every proposition of mathematics can be deduced from the five postulates enriched by the definitions of the non-primitive terms".

I was wondering if it is possible for someone to make a scheme illustrating the sequence of the derivation of the whole theory of mathematics being derived by these postulates.Possibly starting from natural numbers?(notice that Hempel excludes geometry)

Answer 1

We have to put Hempel's article in context.

Carl Gustav Hempel (1905–97),

was born near Berlin, Germany, on January 8, 1905. He studied philosophy, physics and mathematics at the Universities of Gottingen and Heidelberg before coming to the University of Berlin in 1925, where he studied with Hans Reichenbach. [...] In 1929, at Reichenbach's suggestion, Hempel spent the fall semester at the University of Vienna, where he studied with Carnap, Moritz Schlick, and Frederick Waismann, who were advocates of logical positivism and members of “the Vienna Circle”.

Thus, his view on mathematics were influenced by the debate on Foundations of mathematics of the first decaded of 20th Century and by Logicism :

One of Hempel's early influential articles was a defense of logicism, according to which mathematics—with the notable exception of geometry, which he addressed separately—can be reduced to logic (for Hempel, including set theory) as its foundation (“On the Nature of Mathematical Truth”, American Mathematical Monthly, 52 (1945)).

See §6. PEANO'S AXIOM SYSTEM AS A BASIS FOR MATHEMATICS

Every concept of mathematics can be defined by means of Peano's three primitives, and every proposition of mathematics can be deduced from the five postulates enriched by the definitions of the non-primitive terms. These deductions can be carried out, in most cases, by means of nothing more than the principles of formal logic; the proof of some theorems concerning real numbers, however, requires one assumption which is not usually included among the latter. This is the so-called axiom of choice.

And see §8. DEFINITION OF THE CUSTOMARY MEANING OF THE CONCEPTS OF ARITHMETIC IN PURELY LOGICAL TERMS :

At first blush, it might seem a hopeless undertaking to try to define these basic arithmetical concepts without presupposing other terms of arithmetic, which would involve us in a circular procedure. However, quite rigorous definitions of the desired kind can indeed be formulated, and it can be shown that for the concepts so defined, all Peano postulates turn into true statements. This important result is due to the research of the German logician Gottlob Frege (1848-1925) and to the subsequent systematic and detailed work of the contemporary English logicians and philosophers B. Russell and A. N. Whitehead.

Thus, in a nutshell, what Hempel calls "formal logic" is the high-order logical system developed bt W&R in Principia Mathematica (and still used by Gödel into his 1931 article on incompleteness) encompassing our current set-theory.

Answer 2

Starting from Peano's 5 postulates for $N$, $S$ and $0$, plus logic and set theory, we can construct the usual addition, multiplication and exponentiation functions on $N$. From these, we can construct the set of integers $Z$ (as sets of ordered pairs of natural numbers) and the corresponding arithmetic operations on $Z$. Then the set of rational numbers $Q$ (as sets of ordered pairs of integers) with the corresponding arithmetic operations on $Q$. Then we can construct the set of real numbers $R$ (as sets of rational numbers) and the corresponding arithmetic operators. Then the set of complex numbers $C$ (as sets of ordered pairs of rational numbers) and its operators.

Other sets and operators (e.g. derivatives, integrals in calculus) can be constructed using logic and set theory, and their consequences worked out to derive the theorems of algebra, number theory and analysis.

As for geometry, if it is to include any notion of a real-valued distance between pairs of points, the construction of the real numbers must precede the introduction of any axioms of that geometry. In that sense, such a geometry could also be seen as a consequence of Peano's Axioms.