FOM: a correction
holmes at catseye.idbsu.edu
Tue Mar 30 14:04:12 EST 1999
This is a clarification of part of my last posting on
foundations in NFU. The description of the intuitive motivation of
stratification is something I have written down many times, and this
time I was too elliptical. I didn't explain the use of the word "role"
which I use in this account.
I'll recapitulate and (I hope) make it clear what I meant (perhaps
it is clear anyway). One supposes that in defining a set theory one
starts with a class of "bare objects" with whose structure one is not
concerned. One then assigns to (some of) the objects in this class
extensions (which are classes of objects). (Note that in this picture
it is quite easy to postulate a universal set).
This may be regarded as an implementation of the abstract data type
"class". Certain objects are now used to represent classes. Implicitly,
one has also implemented "class of classes", "class of classes of classes",
and so forth.
The set definition "the set of all x such that x \in x" (which is not
paradoxical; there are set theories in which this set exists) can be
criticised on the grounds that the implementation of the abstract data
type "class" is being abused: which bare object x belongs to a particular
class A of bare objects is not a property of A; for the assignment of
extensions to bare objects might make A belong to itself or not (unless
A = V or A = \emptyset).
Here is where I should have explained my use of the word "role". In the
sentence "x \in x", the first x can be understood as referring to x as
a bare object, and the second as referring to the class associated with
x. The problem with the specification is that it depends on the correlation
between x and the class assigned to it, and this is not actually a property
either of x or of the extension associated with it. In more complex
sentences, the roles assigned to variables will include not only
"bare object" and "class", but also "class of classes" and so forth.
The criterion of security for this hierarchy of abstract data types
(corresponding to the familiar type hierarchy of the Theory of Types) is
equivalent to the stratification criterion for NFU. The motivation is
that the assignment of two different roles to the same variable in the
specification of a set involves an illicit appeal to the arbitrary assignment of extensions to bare objects (to irrelevant details of the implementation
of classes, classes of classes, etc.), rather than to the properties of
the classes, classes of classes, etc. which are being implemented.
The assignment of
roles corresponds to the assignment of types for purposes of stratification;
but here it may be clearer how the assignment of types can be a
in a theory which is actually one-sorted.
Sorry about the odd format; I am using an unaccustomed text editor.
And God posted an angel with a flaming sword at | Sincerely, M. Randall Holmes
the gates of Cantor's paradise, that the | Boise State U. (disavows all)
slow-witted and the deliberately obtuse might | holmes at math.idbsu.edu
not glimpse the wonders therein. | http://math.idbsu.edu/~holmes
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