Re: fuzzy relations between non-well founded sets?

Stephen Paul King (stephenk1@home.com)
Mon, 11 Jan 1999 12:40:18 +0100 (MET)

Hi all,

I have found the following:

>7.2.1.2. Fuzzy Hypersets (*)

> Although fuzzy sets are now commonplace in artificial intelligence, so far as I know fuzzy hypersets have never before been
>discussed. Fortunately, therewould appear to be no particular problems involved with this useful idea: the basic mathematics of
>fuzzy hypersets, at least as far as I have worked it out, is completely straightforward.

> The simplest example of a fuzzy hyperset is the set x defined by:

> dx(x) = c,

> dx(y)=0 for y not equal to x.

>Here, if c=0, one has an ordinary well-founded set, namely the empty set. If c=1, one has the set x={x}. Otherwise, one has
>something inbetween the empty set and x={x}.

> Each fuzzy hyperset is characterized by a fuzzy apg, which is exactly like an apg except that each link of the graph has a
>certain number in [0,1] associated with it. The Fuzzy AFA then states that each fuzzy apg corresponds to a unique fuzzy set. It is
>easy to see that the natural analogue of the Solution Lemma holds for fuzzy hypersets. And, of course, the consistency of fuzzy
>hypersets with the axioms of set theory (besides the axiom of reducibility) follows trivially from the fact that each fuzzy hyperset
>is, in fact, a hyperset under AFA.

in:
http://goertzel.org/ben/chlog3.html

Now I am wondering if we could use Kosko's mutual entropy formalism to
think about relations between streams. Any ideas?

Later,

Stephen Paul King

https://members-central.home.net/stephenk1/Outlaw/Outlaw.html

spking1@mindspring.com (Stephen Paul King) wrote:

>Has any work been done on fuzzy subsethood relations between (and/or
>within) non-wellfounded sets?
>please reply via e-mail. :)
>Thanks,
>
>Stephen Paul King

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