[FOM] Comments on Feferman on Hellman

[Harvey Friedman]

On 1/7/06 11:56 PM, "Nik Weaver" <nweaver at math.wustl.edu> wrote:

> 
> A question for Harvey Friedman:
> 
> On what grounds are you certain of the consistency of ZFC plus
> the various large cardinal axioms used in your work?  And more
> generally of the truth of all number theoretic consequences of
> these axioms?
> 
I never claimed any such certainty. Some months ago I posted some research
on the issue of why one should believe in the consistency, and posted an
abstract on my website. (Relational system theory, I think, is the title).
More work needs to be done along these lines in order to make it fully
convincing. I am optimistic about this, but am not pursuing it now, due to
other matters.

What I tried to do there was to show how formal systems that have a real
coherence, but are not based on set theoretic ideas, are mutually
interpretable with set theory with various large cardinal axioms.

However, the systems there that I develop there for this purpose are still
closer to set theoretic ideas than I would like. I would like these systems
to be recast in a way that they do not in any way shape or form resemble set
theory - or even mathematics.

> You prove that various combinatorial statements are equivalent to
> assertions of this type and infer that this shows the "necessary
> use" of large cardinals.  But that begs the question of whether
> the large cardinal axioms you use are in fact consistent with ZFC
> and that all number theoretic consequences of these axioms are in
> fact true.  How do you know this?

The phrase "necessary use of large cardinals" has a well defined meaning -
but perhaps only to experts and not general FOM readers.

It means that the system ZFC + LCA is interpretable in the system consisting
of the statement in question together with a suitable base theory for the
statement in question. Also see the closely related explanation below that
is more "pragmatic".

Postings on the FOM are not full blown papers, and so I generally use
shortcuts that are very familiar to experts.
> 
> A related point: you falsely claim
> 
> "36. I proved that all of these 6561 statements are provable or
> refutable using large cardinals. I prove that this cannot be done
> without using these large cardinals."
> This is not true because obviously a different pattern of proofs
> and refutations is consistent with the non-existence of these large
> cardinals.  

It took me a while to see the point you are making.

First of all, you are not questioning that

"I proved that all of these 6561 statements are provable or refutable using
large cardinals".

The point you are making, presumably, is that, e.g., in the entirely
unnatural and silly formal system

ZFC + "ZFC is inconsistent"

one can prove or refute all 6561 statements. Or in the similarly entirely
unnatural and silly system

ZFC + "ZFC + Mahlo cardinals is inconsistent"

one can prove or refute all 6561 statements.

(I say unnatural and silly not simply on the grounds that it is false).

Yes, this is technically true. In a full blown paper (or book) I would
elaborate so as to avoid your minor point.

There are several ways to be more careful in a paper or book. Here is one of
several.

Over many decades, there has developed a robust hierarchy of formal systems
ranging from, say, EFA (exponential function arithmetic) all the way up to,
say, ZF + "there exists j:V into V".  The first place in this robust
hierarchy - some call the Godel hierarchy - where all of these 6561
statements become decided, is at some level of large cardinals well beyond
the place in the hierarchy that ZFC sits.

Although it is the case that these systems are not linearly ordered under
derivability, they are linearly ordered under logical strength. Furthermore,
they are also linearly ordered under arithmetical consequences. In any case,
where there is nonlinearity under derivability, this does not affect
statements such as these 6561.

Nobody has the slightest idea for upsetting this striking robust picture in
any substantial way, and so it is generally accepted. Of course, there are
different levels of conviction about where, if at all, the systems stop
being consistent.

Harvey Friedman