[FOM] Inconsistent Systems

[Harvey Friedman]

On Fri, Sep 13, 2013 at 4:56 PM, Alan Weir <Alan.Weir at glasgow.ac.uk> wrote:

> Regarding Harvey Friedman's sketch of the triviality of a form of naive
> set theory- FOM Digest, Vol 129, Issue 17- it is well known that Curry's
> paradox trivialises naive set theory even in quite weak relevant logics
> without any appeal to the logic of negation. Those who seek to maintain
> that naive set theory is nonetheless still consistent or (this is the more
> usual line) at least non-trivial, often see the culprit as that principle
> of contraction which in rule form (sequent form) natural deduction goes:
>

I had a feeling that what I sketched in
http://www.cs.nyu.edu/pipermail/fom/2013-September/017596.html was known,
and I am glad I didn't make that one of my numbered postings!  Thanks also
to Avron, Arthan, and Marcos for pointing this out.

Of course producing a non-classical framework in which naive set theory is
> consistent (or at least non-trivial) is one thing, and certainly can be
> done. Producing a framework which is not so restricted that it cripples
> standard mathematics is another, much more challenging, thing.
>

I believe that "it cripples standard mathematics", and even "profoundly
cripples standard mathematics". But I would be fascinated if a good case
can be made that it does NOT cripple standard mathematics, and the
development of a suitable treatise in which mathematics is actually
developed with such "logics".

Of course, it would also be an interesting challenge to prove that such a
suitable treatise cannot exist.

So I will operate on the working assumption that CA(no) and other forms of
CA simply cannot be used for the foundations of mathematics.

Now I turn to a related matter which I also hesitate to put in my numbered
postings.

SUPPOSE THE USUAL ZFC IS INCONSISTENT. CAN IT STILL BE USED FOR THE
FOUNDATIONS OF MATHEMATICS?

I think I need to clarify a point here, that I didn't make with regard to
CA(no). Obviously, subsystems of CA(no), or even CA, can be used for f.o.m.
So what am I talking about?

What I mean is, can we use the entire set of axioms, but with a weakened
logic - where the weakened logic is sufficient to do ordinary mathematics?
It appears that the answer is no for CA(no).

Now, let's look at ZFC. Clearly we can complain about the use of classical
logic to derive its hypothetical inconsistency. And there is a perfectly
good intuitionistic ZF - see  The Consistency of Classical Set Theory
Relative to a Set Theory with Intuitionistic Logic, J. of Symbolic Logic,
Vol. 38, No. 2, (1973), pp. 315-319. I think I called this IZF (I forget).

THEOREM?? (finitary). Suppose ZFC is inconsistent. Then IZF without
absurdity and without negation proves every formula without absurdity and
without negation.

I leave this to the experts to complete. And perhaps discuss also whether
anything can be salvaged from an inconsistency in ZFC other than the
obvious move of going to subsystems in the usual sense.


> (I note also that Prof. Friedman's system CA(no) might perhaps be more
> naturally called a theory of naive properties, since there is no
> extensionality axiom. Hartry Field has given a consistency proof for such a
> system in quite a strong non-classical logic.)
>
> Did Field or others see just what happens if we try to use this "strong
non-classical logic" for actual mathematics?

Harvey Friedman
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