Michael, Ali, et al., (01)
In my criticisms of your paper, I want to emphasize that I was *not*
criticizing your competence or mathematical prowess. Please note that
my most scathing criticism is that you are *too industrious* -- namely
you started from scratch with very complicated assumptions. (02)
I apologize for using the word 'muck' but that is exactly how I felt.
When you talk about lattices, all the proofs are clean and simple.
But when you take finite hierarchies, you don't have a closed system.
You never know whether an operator applied to elements of the hierarchy
will produce a result that is in the hierarchy. (03)
But with the lattice, all the functions and operators map some theory(s)
in the lattice to another theory that is also in the lattice. For
any two or more theories, both the supremum and infimum are guaranteed
to exist in the lattice and to be unique. No new proofs needed. (04)
I agree that work on interpreting one theory in another was done by
very competent logicians. But they were just looking at two or three
theories at a time. And they did a lot of good work on analyzing
the conditions for the interpretation to be possible. (05)
But when we're talking about the OOR, we're faced with a large number
of theories. What we usually do is to *generate* theories that aren't
*yet* in the hierarchy -- but they are *already* in the lattice. (06)
The most common kind of generation is to combine a collection of
modules and find their supremum (minimal common generalization)
or their infimum (maximal common specialization). (07)
We always *know* that the infimum and supremum exist. But we need
to check whether they are trivial -- a trivial infimum is the
absurd theory at the bottom when the theories are contradictory,
and a trivial supremum is the universal theory at the top when
the modules have nothing in common. (08)
If the infimum and supremum are not trivial, then the methods for
finding them and testing them have been discussed and explored
for years. Most of the difficult work has been done long ago. (09)
In summary, I'd suggest that a paper that's been accepted should be
published as is. But for the next edition, I would recommend a much
shorter and simpler paper that starts with the Lindenbaum lattice
and AGM. Anything else is defined in those terms, and the finite
hierarchies are treated as starting subsets that can be enlarged
as needed. (010)
John (011)
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