Re: Heap-Set Theory with Null heap.

On Sun, 10 Feb 2008 07:09:11 -0800 (PST), Zaljohar@xxxxxxxxx wrote:


But Frege, the Null heap is empty of proper parts but it is not really
empty in the sense you are speaking about.

Nonsense. First of all: there is no empty heap (intuitively). Didn't you
read my argument?

Second: Heaps have _constituents_ (my term). If /a/ is a constituent of
the heap H, I write

a c H.

A very simple case occurs if the heap H _just_ consists of the
constituent /a/ (where it doesn't matter if /a/ is atomic or not). Then
we have

H = a.

Hence

a c a.

Or the other way round. Given any heap H, H is the only constituent of
H, hence

H c H

(for any heap H).


In this theory (H-S-Null) we do have ~Ex Ay( y!@x) as a theorem.
i.e there do not exist heap that has no parts.

That's fine. Since we have just shown that for every heap H

H c H.

Now let H_0 be an arbitrary heap. Hence (from FOPL):

Ey(y c x)

and hence

AxEy(y c x)

since H_0 was arbitrary.

With other words,

~ExAy(y !c y).

So this is in complete agreement with your axiom.


The [formal] proof of this theorem is axiom of reflexiveness Ax(x@x)

Right. Again

Ax(x c x)

is in complete agreement with my (intuitive) approach. (Just stated
above.)


So by negation if we assume that ExAy(y!@x) then we will have
Ex( x!@x) which contradict reflexiveness clearly, so there do not
exist a heap that is empty of parts.

Right. Still from this we do not get the existence of "the empty heap"
(or "bottom").


The definition of the Null heap which symbolized as [ ] is

Definition: x = [ ] <-> Ay(y @ x -> y = x)

To do this (i.e. to get a _correct_ definition) you first have to
show/prove that such a heap exists (in your theory). And -as explained
already several times- *I* reject any axiom to this virtue (in my
approach of a /theory of heaps/).

But this way you do not get just the "Null heap" but any heap consisting
of an atom! (If there's no empty heap, of course.)

After all, if a is an atom, we have

a c a

and for any heap b =/= a, b !c a.

Hence

Ay(y e a -> y = a)

holds. Hence from your "definition"

a = [a]

- where "[a]" denotes the heap just consisting of the atom a.


So as you see from reflexiveness we do have [ ] @ [ ] , i.e x is a
part of itself , but it is not constituted of any parts other than
itself.

The same holds for

[a]

if a is an atom (and there's no empty heap, of course).


So [ ] is empty of PROPER parts.

The same holds for

[a]

if a is an atom (and there's no empty heap, of course).


Also since I defined atom in the following way:

x is atomic <-> (x !=[ ] and Ay( (y@x & ~y=[ ]) -> y=x ))

So an atom is a heap that has one proper part that is [ ].

Now this is _counterintuitive_. A grain of sand certainly is not a grain
of sand + some unseen (undetectable) "empty heap". That's exactly why I
reject an "empty heap" in my approach - i.e. the theory of ("real")
heaps.


What I want to say is that the intuitive line that you presented has
nothing to do with axiomatizing a heap that has no atoms, ...

Idiot.

I SAID (quote):

"[...] though an empty heap may be acceptable from a _formal_ point of
view on may argue that from an _intuitive_ point of view there's no
empty heap - hence we have good reasons to reject it in the theory _of
heaps_. (Though it may be allowed in other mereological theories.)"

With other words,

[...] the importance of having a Null heap is to have a Boolean algebra,
were W has a complementary heap that is Null, and the intersection
(Product) of any two heaps is defined by a heap, so it is only axiomatized
for formal reasons.
~~~~~~~~~~~~~~~~~~~


BUT if you are interested in a THEORY _OF HEAPS_ (i.e. those things we
intuitively would call a /heap/), then a "empty heap" seems not to be
appropriate. Hence it has to be rejected in such a theory.

HENCE, you claim that

"Leniewski which is a potent Mereologist has rejected bottom"

does not surprise me the least (though I don't know his reasons for
doing so. but I suspect the may be similar to mine).


EOD


F.

--

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

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