[FOM] Consistent logics with non-well-founded definitions?

[Andrew Boucher]

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