To know is one thing, to understand is another ...

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LoginThis is the axiomatic definition of the counting numbers (starting with zero).

The definition is both logical and implicit. We usually know them explicitly, thru counting marbles etc (addition and subtraction) and derive zero from subtraction problems (and the negative integers as well.

This is a good brain-up to get your head thinking logical, before tackling something else, like a warmup before playing sports ;-)

The axiom of induction may be necessary to clear up "next", to avoid loops in counting, and give entry to using countably infinite sets. But it looks a lot different from the first four axioms, just like the parallel postulate in Euclidean geometry seems out of place.

So removing the induction axiom, is like removing the parallel postulate. It opens up to more general data structures (ones that allow indirect self reference (loops) just as ignoring the parallel postulate opens us up to non-Euclidean geometry.