Infeasible inconsistency

[Joe Shipman]

That’s a cute paper but you didn’t state the strongest version of the result that follows from your proof:

Let T be a consistent theory, and P be a subset of sentences of T closed under conjunction, and s be a statement independent of T and not equivalent to any member of P. Then adding all the P-consequences of s and all the P-consequences of not-s gives a consistent extension of T. This can be implemented by the axiom schemes

 (s->q)->q 
and 
(~s->q)->q  
for all sentences q in P.

If s is inconsistent but feasibly consistent, then adding those axiom schemes seems to give a feasibly consistent extension of T. So this sort of answers my question: given a statement s which is inconsistent but feasibly consistent, one just needs to find as broad as possible a class of statements closed under conjunction none of which are equivalent to s.

For large cardinal axioms and arithmetical statements, this doesn’t seem to help much (either in the consistent or the paraconsistent setting), because although large cardinal axioms have new arithmetical consequences in the form of consistency statements, the negation of a large cardinal axiom has no arithmetical consequences (because cutting down to a submodel with no inaccessibles gives the same arithmetical truths).

Does any previously studied independent statement give new arithmetical consequences whether you assume it or deny it? Or are all the known ones with arithmetical consequences one-sided?

— JS



Sent from my iPhone

> On Dec 14, 2020, at 6:41 PM, Kreinovich, Vladik <vladik at utep.edu> wrote:
> 
> 
> As a partial answer to this question, it is probably worth mentioning an easy-to-prove fact that for many ZF statements S, we can add, to arithmetic,
> ·         all arithmetic consequences of S AND
> ·         all arithmetic consequences of “not S”,
> and still get a consistent theory, see e.g.,
>  
> Olga Kosheleva and Vladik Kreinovich, "Contradictions do not
>   necessarily make a theory inconsistent", Journal of Innovative
>   Technology and Education, 2017, Vol. 4, No. 1, pp. 59-64.
> http://www.cs.utep.edu/vladik/2017/tr17-27.pdf
>  
>  
> From: FOM <fom-bounces at cs.nyu.edu> On Behalf Of Joe Shipman
> 
>  
> The reason I am concerned about the lack of examples is that paraconsistent logic doesn’t seem to be worth much unless there is such a system that it could be applied to.
>  
> — JS
-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20201215/c7f59786/attachment-0001.html>