Shipman's schema for ZF(C)

[Arnon Avron]

What is obvious is that foundation follows from \in-induction, even if the latter
is limited to bounded formulas. As for the converse - the proof I know of it uses
the existence of the transitive closure of sets, and at least the standard proof
of that existence uses the axioms of infinity, replacement and union. Are you sure
that full separation suffices?

Arnon



________________________________
From: FOM <fom-bounces at cs.nyu.edu> on behalf of Anton Freund <freund at mathematik.tu-darmstadt.de>
Sent: Tuesday, August 31, 2021 10:15 AM
To: fom at cs.nyu.edu <fom at cs.nyu.edu>
Subject: Shipman's schema for ZF(C)

Just to begin with the obvious: \in-induction for arbitrary formulas
follows from foundation in the presence of full separation.

Results about \in-induction in the absence of full separation are known,
e.g., in the context of Kripke-Platek set theory:

Gerhard Jäger, A Version of Kripke-Platek Set Theory which is Conservative
over Peano Arithmetic, Mathematical Logic Quarterly 30 (1984) 3-9,
https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.19840300102

Michael Rathjen, Fragments of Kripke-Platek set theory with infinity,
in: Peter Aczel, Harold Simmons and Stanley Wainer (eds.), Proof Theory: A
selection of papers from the Leeds Proof Theory Programme 1990,
https://www1.maths.leeds.ac.uk/~rathjen/FRAGMENT.pdf

An over-simplified summary is: With very little induction, set theories
can really become quite weak. Adding full induction yields theories of
medium strength -- which are, however, nowhere near as strong as ZF.

Best,
Anton


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