Pitfalls of Peano.

The following is an interesting example that I came across in the article "Whither Mathematics?" by Brian Davies in the December issue of Notices of the American Mathematical Society.

Consider the following list A1 of axioms.

(1) There is a natural number 0.
(2) Every natural number a has a successor, denoted by S(a).
(3) There is no natural number whose successor is 0.
(4) Distinct natural numbers have distinct successors: a = b if and only if S(a) = S(b).
(5) If a property is possessed by 0 and also by the successor of every natural number which possesses it, then it is possessed by all the natural numbers.

Now consider the following list A2 of axioms.

(1) G is a set of elements and these elements obey the group axioms.
(2) G is finite but not isomorphic to any known list of finite simple groups.
(3) G is simple, in other words, if N is a subset of G satisfying certain properties then N=G.

We can roughly compare A2 with A1. The second axiom in A2 can be thought of as analogous to the third axiom of A1. Also the third axiom of A2 is analogous to the fifth axiom of A1, insofar as it refers to an unspecified set with cetain properties and concludes that it is equal to G.

Now, as is generally believed by most group theorists, the system A2 is internally inconsistent and the proof its inconsistency runs for more than 10000 pages.

So who is to deny that the system A1 is also probably internally inconsistent! Particularly since Godel proved that you can not prove it is consistent (staying inside the system). May be the shortest proof of its inconsistency is one hundred million pages long!