Only some versions of the ZF axioms include an axiom stating that an empty set exists. According to mathworld, the Axiom of the Empty Set (AES) follows from the Axiom of Infinity (AI) and Axiom of Separation (AS), via [itex]\exists x (x = x)[/itex] and [itex]\emptyset = \{y : y \not= y\}[/itex]. I guess they think the AI states that a set exists? But the AI is defined inductively, and the empty set serves as its only basis. Here's the AI:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\exists x (\emptyset \in x \ \wedge \ \forall y (y \in x \rightarrow \cup \{y, \{y\}\} \in x))[/tex]

The AI doesn't say the empty set exists. It seems to me to be, well, like a proof by weak induction without a basis. It only says that an infinite set exists provided that an empty set exists. No? I'm confused.

Edit: haha, though I guess without the empty set, the set that AI says exists has no extension, so... ouch, my head.

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# Axiom of Empty Set superfluous in ZF?

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