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    One of the questions I have to do for my most recent example sheet is: "Deduce the axiom of the empty set from the axioms of infinity and separation (and then do it just using infinity)'.

    The first bit, superficially, is fine. Take an infinite set A and the sentence p=¬(x=x), then {x in A: ¬(x=x)} is the empty set.

    But, the precise wording of the axiom of infinity presupposes the axiom of the empty set - our infinite set must contain the empty set to begin with. We could get around this problem by swapping the empty set for a some constant, but then this wouldn't technically be in the language we're working in.

    How do I get around this problem? Is there a formulation of the axiom of infinity that doesn't need a constant or the empty set, or does it not matter that infinity assumes empty set?
 
 
 
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