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From: kovsky@netcom.com (Bob_Kovsky)
Subject: Re: Re hierarchy typing, natural & artifical intelligence
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Marvin Minsky wrote:

>How about this as a serious question for what we might call metatheory of
>philosophy.
>     How can so many people who understand recursive proof
>     theory not see how to construct sequences of metalanguages.
>
>Surely this has something to do with the late John Myhill's theory of
>creative sets.


	At the risk of thread drift, I would like to focus on the 
notion of "sequences of metalanguages" because it relates to something 
else I've been thinking about.

	The place to start is Whitehead & Russell's Theory of Types.  This
is an external constraint that is imposed in set theory and logic to avoid
a class of paradoxes.  Perhaps the most paradox is "the set of all sets
that do not include themselves," which is not well-defined because its own
membership in itself is ambiguous. 

	Whitehead and Russell imposed a hierarchy of types:  sets (the 
lowest), classes (contains sets) and so forth.  A set cannot be a member 
of a set, but can be a member of a class, etc.    The theory of types 
"worked" like a good fudge in avoiding the problems posed, but it created 
new problems that (AFAIK) have not been resolved.

	In general, I speculate that any organized collection of
mathematical objects requires a hierarchy of internal references in the
nature of a theory of types.   Hence "meta" and "sequences."  These terms 
seem to me to imply such a hierarchy.  And such a hierarchy is global:  
the constraints imposed apply uniformly to the entire collection of 
mathematical objects.

	Folks using the brains they were born with do not require global 
hierarchies.  We get along ok without theories of types too.  Cretans 
notwithstanding.  In fact, global hierarchies often play us false.  (We 
posit the existence and consistency of "higher levels" without any 
assurance of validity.

	I looked at these issue many years ago (when my skills in logic 
were sharper) but found no satisfaction.  Curious if anyone else has 
something to say.
