Re: Heap-Set Theory H-S

On Tue, 29 Jan 2008 16:17:31 -0800 (PST), Zaljohar@xxxxxxxxx wrote:


And right, imho it's natural (though not necessary from a logical point
of view - it seems) to rule out the "empty heap".

My personal argument (again a quote):

        "... of course there's no "empty heap", if there are no objects
         there is simply no heap, it "vanishes".


Indeed I agree with you totally on that aspect.

But I think in Mereology it is axiomatized that there is a heap that
is a part of every heap and they call it the Null heap.

No, IIRC there are different systems, some with and some without the
null "heap". (As I already mentioned: the null "heap" is _possible_
from a pure logical -or formal- point of view.)

"... few authors have gone so far as to postulate the existence of a
?null entity? that is part of everything:

(P.11) EzAxPzx Bottom"

(http://plato.stanford.edu/entries/mereology/)

My point is, that for a /theory of heaps/ (i.e. a _special_ mereological
theory) it is natural do exclude the "null heap", since "if there are no
objects, there is simply no heap, it "vanishes"."


I understand that the Null heap is not empty of course since this
would contradict the first axiom in Mereology that is Ax ( x c x ).

Imho your error here is to read/interpret "c" as "is an element". Note
that in set theory, we also have

0 c 0 (!).

- still 0 is the empty set (i.e. is empty).


But what is the intuitive basis for axiomatizing the existence of such
an object?

Well, the Stanford Encyclopedia of Philosophy again:

"(Two exceptions are Martin 1965 and Bunt 1985; see also Bunge 1966 for
a theory with several null individuals.) Without such an entity, which
one could hardly countenance except for good algebraic reasons, the
existence of a mereological product is not always guaranteed. Hence
(P.7) must remain in conditional form. Likewise, differences and
complements may not be defined -- e.g., relative to the universe U.
Hence, the corresponding closure principles (P.8) and (P.9) must also
remain in conditional form."


In my above theory I didn't axiomatize that, because I don't see the
intuitive basis behind the existence of such a Null object?

You are right in doing so. There is none. It contradicts the "natural
meaning" of /heap/.


F.

--

E-mail: info<at>simple-line<dot>de
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