Shipman's schema for ZF(C)

[JOSEPH SHIPMAN]

Yes, I said “set induction” and meant to compare it with Peano induction (without AxInf it is equivalent because of the bi-interpretability between PA and the theory of hereditarily finite sets using the well-known map f(x)=the sum of 2^f(y) for the elements y of x).

— JS

Sent from my iPhone

> On Aug 29, 2021, at 4:48 AM, Zvonimir Sikic <zvonimir.sikic at gmail.com> wrote:
> 
> 
> As noted by Carnerio and, in one direction, by Kepke, Shipman's schema is equivalent to Ay Ex (y in x). Hence, the schema is more akin to Peano's axiom  Ay Ex (x succeeds y) then to the induction. Is it possible that Shipman meant
> 
> Ax (Ay (y in x -> S(y)) -> S(x)) -> Az S(z) 
> 
> instead of
> 
> Ax (Ay (y in x -> S(y))            ) -> Az S(z)?
> 
> On Fri, Aug 27, 2021 at 11:26 AM JOSEPH SHIPMAN <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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