Existence of the empty set in ZFC, not using the axiom of specification

Asked Mar 16 '22 at 11:48

Active Mar 16 '22 at 11:48

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Some sources include the existence of the empty set as one of ZFC's axioms. Some don't, but they include the axiom of specification and prove the existence of the empty set using this axiom and first-order logic rules.

However, in the hebrew wikipedia I found the following list of axioms in ZFC: extentionality, union, infinity, replacement, power, regularity, chocie.

Is it possible to prove the existence of the empty (or the axiom of specification, which can be used to prove the existence of the empty set) set using only these axioms?

asked Mar 16 '22 at 11:48

Ben

N.B.: This may depend on the exact formulation of these axioms. – Mees de Vries Mar 16 '22 at 12:00
You can prove seperation using replacement. – Vivaan Daga Mar 16 '22 at 12:06
If you want to avoid the Empty Set axiom and use Specification to prove its existence, you have to assume that some set exists, in order to apply Spec to it... – Mauro ALLEGRANZA Mar 16 '22 at 12:55
@MauroALLEGRANZA The fact that some set exists follows from the axioms of FOL; no special axiom is required. – Vivaan Daga Mar 16 '22 at 13:14
The existence of an empty set follows from the axiom of infinity, or at least it can be formulated as saying that $\omega$ has an ampty set as an element. – Arvid Samuelsson Mar 16 '22 at 18:00
@ArvidSamuelsson It does follow from infinity, but infinity is not needed. – Vivaan Daga Mar 16 '22 at 18:23
But I don't like the proof from pure logic. – Arvid Samuelsson Mar 16 '22 at 18:26
@ArvidSamuelsson would you mind sending me a link to a proof not using infinity (nor seperation)? – Ben Mar 17 '22 at 19:47
@Ben Why are you asking me? It was Voilet Flame who brough up that proof. – Arvid Samuelsson Mar 18 '22 at 19:12
My bad. @VioletFlame would you mind sending me a link to a proof not using infinity (nor seperation)? – Ben Mar 19 '22 at 20:08

Existence of the empty set in ZFC, not using the axiom of specification

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