Alternative axiom scheme for ZF(C)

[Till Mossakowski]

I think a proper set induction scheme should look like this:

(Forall x ((Forall y  (y in x implies Phi(y))) implies Phi(x))) implies 
(Forall z Phi(z))

Informally: if for every set x, Phi-ness of all its members implies 
Phi-ness of x itself, then Phi holds for all sets.

Till Mossakowski

Am 27.08.21 um 23:49 schrieb Mario Carneiro:
> That axiom is equivalent to
>
> forall y exists x (y in x)
>
> from which one can derive the existence of singletons (via the subset 
> axiom) and not much else. Considering that singletons are usually 
> derived from the axiom of pairing, I don't think it will eliminate any 
> normal ZFC axioms.
>
> Mario Carneiro
>
> On Fri, Aug 27, 2021 at 11:26 AM JOSEPH SHIPMAN <joeshipman at aol.com 
> <mailto:joeshipman at aol.com>> wrote:
>
>     Consider the set induction scheme:
>
>     (Forall x Forall y (y in x implies Phi(y))) implies (Forall z Phi(z))
>
>     With this included, which other axioms of ZFC may be dispensed with?
>
>     — JS
>
>

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