Track / Stage 6 / 212 of 400 #1212 of 1964

Problem 1212

National Olympiad, first round

Number theory Difficulty 6.3 Prove it

Specifically illustrate with examples that among the four axioms (i), (ii), (iii), (iv) of the Peano axioms, if any one is arbitrarily removed, a set $M$ can be found that satisfies the remaining three axioms, and $M$ is essentially different from the set of natural numbers $N$ .

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.