The fifth assumption is the principle of complete induction.
M. Couturat considers these assumptions as disguised definitions; they constitute the definition by postulates of zero, of successor, and of integer.
But we have seen that for a definition by postulates to be acceptable we must be able to prove that it implies no contradiction.
Is this the case here? Not at all.
The demonstration can not be made by example. We can not take a part of the integers, for instance the first three, and prove they satisfy the definition.
If I take the series 0, 1, 2, I see it fulfils the assumptions 1, 2, 4 and 5; but to satisfy assumption 3 it still is necessary that 3 be an integer, and consequently that the series 0, 1, 2, 3, fulfil the assumptions; we might prove that it satisfies assumptions 1, 2, 4, 5, but assumption 3 requires besides that 4 be an integer and that the series 0, 1, 2, 3, 4 fulfil the assumptions, and so on.
It is therefore impossible to demonstrate the assumptions for certain integers without proving them for all; we must give up proof by example.
It is necessary then to take all the consequences of our assumptions and see if they contain no contradiction.
If these consequences were finite in number, this would be easy; but they are infinite in number; they are the whole of mathematics, or at least all arithmetic.
What then is to be done? Perhaps strictly we could repeat the reasoning of number III.