[FOM] Consistent logics with non-well-founded definitions?
Andrew Boucher
Helene.Boucher at wanadoo.fr
Sun Sep 5 05:59:24 EDT 2004
On Sep 5, 2004, at 3:02, Bryan Ford wrote:
> Hi folks,
> On a similar note, can anyone give me pointers to any studies of
> formal logic
> systems (however weird or contrived) that can prove themselves
> consistent,
> but are nevertheless consistent (e.g., their consistency is provable
> in ZFC)?
> I'm pretty sure I've heard of such things being studied somewhere, but
> can't
> remember where and can't find the references...
There are some systems mentioned in my paper "'True' Arithmetic Can
Prove Its Own Consistency". A pdf version can be downloaded at
http://www.andrewboucher.com/papers/consistency.pdf.
Basically the system is second-order arithmetic without the successor
axiom, so the system has as models all initial segments of the natural
numbers - {0}, {0,1}, {0,1,2}, ... - as well as the natural numbers
themselves.
Recently Dan Willlard and Yves Gauthier have worked on such systems.
Willard had a couple of papers which appeared in the Journal of
Symbolic Logic, "Self-Verifying Axiom Systems, The Incompleteness
Theorem and the Tangibility Reflection Principle" (2001), pp. 536-596.
Gauthier has a book "Internal Logic" (Synthese Library / Volume 310,
Kluwer Academic Publishers) as well as a paper, "The Internal
Consistency of Arithmetic With Infinite Descent" in Modern Logic, vol.
8 no 1/2 (Jan 1998-April 2000), pp. 47-86.
I am unfortunately not knowledgeable about less recent work (e.g. that
of Kreisler and Jeroslow?).
Rdgs
Andrew Boucher
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