"A relation exists between two terms," etc., etc.
Sometimes this inconvenience would not be unavoidable, but sometimes also it is essential. A relation is incomprehensible without two terms; it is impossible to have the intuition of the relation, without having at the same time that of its two terms, and without noticing they are two, because, if the relation is to be conceivable, it is necessary that there be two and only two.
V
Arithmetic
I reach what M. Couturat calls the ordinal theory which is the foundation of arithmetic properly so called. M. Couturat begins by stating Peano's five assumptions, which are independent, as has been proved by Peano and Padoa.
- Zero is an integer.
- Zero is not the successor of any integer.
- The successor of an integer is an integer.
To this it would be proper to add,
Every integer has a successor.
- Two integers are equal if their successors are.
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.