Alternative axiom scheme for ZF(C)
Till Mossakowski
till at iks.cs.ovgu.de
Sun Aug 29 16:10:38 EDT 2021
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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