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In the standard construction of natural numbers in axiomatic set theory (ZFC), zero is defined as being the empty set.

However, if we consider, for instance, the function $f:\mathbb N\rightarrow \mathbb N$ defined by $f(n)=n+1$, we have $f(0)=1$, but $f(\emptyset)=\emptyset$, because the image of the empty set is always empty.

Is this contradictory? What am I missing here?

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    This is (I think) essentially the same confusion as in http://math.stackexchange.com/questions/125209/an-unwanted-property-of-the-set-t-x-x.2012-04-13
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    You confusing the empty set as an argument of a function with the image of the empty set under the function. Unfortunately in the usual notation they are both denoted $f(\emptyset)$.2012-04-13
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    Remember that in set theory everything is a set. So the function is defined on sets. This is why in set theory we often use $f[A]$ or $f''A$ for the set $\{f(a)\mid a\in A\}$.2012-04-13

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