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Extra info for Advanced courses of mathematical analysis II: proceedings of the 2nd international school, Granada, Spain, 20-24 September 2004
Thus we should have two different numbers with the same successor. This failure of the third axiom cannot arise, however, if the number of individuals in the world is not finite. 2 Assuming that the number of individuals in the universe is not finite, we have now succeeded not only in defining Peano’s [page 25] three primitive ideas, but in seeing how to prove his five primitive propositions, by means of primitive ideas and propositions belonging to logic. It follows that all pure mathematics, in so far as it is deducible from the theory of the natural numbers, is only a prolongation of logic.
In the case of an assigned number, such as 30,000, the proof that we can reach it by proceeding step by step in this fashion may be made, if we have the patience, by actual experiment: we can go on until we actually arrive at 30,000. e. by proceeding from 0 step by step from each number to its successor. Is there any other way by which this can be proved? Let us consider the question the other way round. What are the numbers that can be reached, given the terms “0” and [page 21] “successor”? Is there any way by which we can define the whole class of such numbers?
We have thus reduced Peano’s three primitive ideas to ideas of logic: we have given definitions of them which make them definite, no longer capable of an infinity of different meanings, as they were when they were only determinate to the extent of obeying Peano’s five axioms. We have removed them from the fundamental apparatus of terms that must be merely apprehended, and have thus increased the deductive articulation of mathematics. ” How stands it with the remaining three? It is very easy to prove that 0 is not the successor of any number, and that the successor of any number is a number.