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]]>1. 0 is a natural number.

2. If n is a natural number, then σ(n), where σ is the successor function, is also a natural number. Furthermore, there does not exist a natural number whose successor is 0.

Axioms 1 corresponds with [∅∈N] since |∅| = 0, and axiom 2 corresponds with [∀x(x∈N⇒℘(x)∈N)], where σ is analogous to ℘, since |℘(∅)| = |{∅}| = 1 = σ(0), etc.

Of course, we can just as easily start natural numbers at 1 in the Peano axioms and still make a one-to-one correspondence between the natural numbers and the elements of the set N defined in the Axiom of Infinity (just shift the correspondence over by 1), but it doesn’t feel as natural since we lose the connection with the size of the elements of N.

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