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I think the title summarises the question well. In most treatments of ZFC, why is an axiom postulating the existence of an empty set included? Is the Axiom of Infinity ("There exists a set $\mathbb{N}$ such that there exists $\emptyset \in \mathbb{N}$ such that $\forall x, x \notin \emptyset$ and $\forall n, n \in \mathbb{N} \rightarrow n^+ \in \mathbb{N}$") for some reason insufficient?

carfog
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  • See this post as well as this one. – Mauro ALLEGRANZA Dec 01 '22 at 10:38
  • If we assume that the "universe" of sets is not empty, i.e. there is a set according to formula $\exists x (x=x)$, we may apply Separation to it with property $x \ne x$ and the result - by Extensionality - is the empty set. – Mauro ALLEGRANZA Dec 01 '22 at 10:39
  • Thus, the Empty Set assumption is only a convenient choice: we have to assume something to start with. – Mauro ALLEGRANZA Dec 01 '22 at 10:45
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    That is exactly my question: why do we need the empty set if we have postulated the natural numbers? Does the latter not imply the former? – carfog Dec 01 '22 at 10:53
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    If you write the Axiom of Infinity in the way: $\exists \mathbf {I} ,(\emptyset \in \mathbf {I} ,\land ,\forall x\in \mathbf {I} ,(,(x\cup {x})\in \mathbf {I} ))$ you need to have it already. If you assume your proposed form, this is only a way to pack two assumptions in one... – Mauro ALLEGRANZA Dec 01 '22 at 10:56
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    The usual approach IMO is only convenience: we start step by step introducing the axioms: the first one are more simple: Empty set, Extensionality, Pair, Separation. – Mauro ALLEGRANZA Dec 01 '22 at 10:57

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There isn't any deep reason; as you say, the empty set axiom follows from the other axioms, and some people omit it. Probably it is mainly done out of historical tradition or for expository purposes (if you are introducing the axioms one by one, with the axiom of infinity being one of the last ones). One reason that it can be convenient to include it is if you are interested in studying weakened versions of ZFC where some of the axioms are omitted or changed. For instance, if you want to study ZFC without the axiom of infinity, then you do need to include the empty set axiom (or some other axiom that implies a set exists (which is sometimes taken to be built into the underlying first-order logic, though this is really not a good idea)), since otherwise there could exist no sets at all (no axiom besides the empty set and infinity implies a set exists without presupposing some other set is already known to exist).

Eric Wofsey
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