[FOM] The Gold Standard

[Harvey Friedman]

On 2/20/06 5:58 PM, "Arnon Avron" <aa at tau.ac.il> wrote:

> On Sun, Feb 19, 2006 at 05:47:26PM -0500, Harvey Friedman wrote:

Friedman wrote:

>> Just yet another argument against just one of the natural stopping
>> places in  the natural hierarchy (the level which essentially
>> represents not stopping).
> 
> According to what I understand about logic, if someone brings
> some argument as the justification  of A, and the same argument
> applied word by word to B, than the same person (if he is a logican)
> necessarily sees A and B as equally justified. ...

I was only giving a brief indication of some of the justifications I have
heard for the top of that hierarchy "large large cardinals", and not
surprised to see that you find an relatively easy and quick argument against
that justification. In fact, you are using one of the main techniques used
to argue against justifications.

> What is clear is that
> in your posting *you* found it necessary to add some arguments
> to distiguish between the cases of NF and ZFC + j:V into V.
> This *logically* means that *you* don't
> find your own  justification of ZFC + j:V into V
> as sufficient. 

I just made the standard moves to give an argument against your argument
against my original brief argument.

I don't find ANY justification of any place in the canonical hierarchy
"sufficient" - let alone any at this highest of positions.

As I said before, we don't know nearly enough about f.o.m. for us to give
"sufficient justifications" in the sense that I think you intend.

>Do you have a better one? (I hope it is not going
> to be: the original justification augmented with the thesis that
> there are no good reasons to reject ZFC + j:V into V, while
> there are such reasons for any other system to which
> the original justification applies, but you don't like them).

I have some optimism that I could do better, but I have no doubt that it
would be subject to serious criticism.

I think that many of the professional set theorists could do better than I
at this high level. Let's ask them. Hello, out there!?

Harvey Friedman