Re: Heap-Set Theory H-S

On Jan 29, 3:34 pm, G. Frege <nomail@invalid> wrote:
On Tue, 29 Jan 2008 15:03:58 -0800 (PST), Zaljo...@xxxxxxxxx wrote:

but Frege I also had [ ] in my mind , but I see it too heavy for that
purpose.

I wanted the lightest symbol,

I see. While on the other hand, ' seems to "light" to me. :-)

Moreover, one might mistake them for quotation marks. ;-)



Ideally we would use spaces to do the job

I don't think so. :-)



so for example we say z = xy

Uhhh... :-)



so xy here is actually the heap of xy

Well ... But this way you would not even be able to _formulate_:

        [[x, y], z] = [x, y, z].



but I feel that a lot of objections will be raised against that

Indeed! (Yo may count on me here! :-)



so I preferred to use the lightest symbol like ' ' , so 'xy'
would be the nearest to xy, I guess?

Well... Another point, I guess: [ ] resembles the usually braces { }
used for sets.

Good point.

But one might object that [ , ] are used in mathematics to represent
bounded interevals, so it can cause some confusion with binary heaps,
don't you think so.

I think you should omit the coma inside the [ ]

[xy] makes a better notation than [x,y]

However I still feel that we should use a lighter symbole than [ ].

What about the symbole ` ` , I think this will not be confused with
the quotation mark.

so ` x y ` is the heap of which x and y are its only atomic parts.

what do you think?







I like your notation  c  to represent 'is a constituent of' or what I
call ' is part of' .

Yes. Parthood has indeed some "features" in common with the usual set
theoretic "c", I guess.

Yes, there are some common features no doubt.

Especially

        [a, b] c [a, b]

etc., of course.



But what is interesting to me is that if we stipulate that all sets
are atomic (NOTE: this is not the standard, that standard is to
stipulate that all singleton sets are atomic: Review David Lewis in
Parts of Classes) However in this theory I stipulated that all sets
are atomic,

Good idea. After all they ARE, from the viewpoint of heaps. (Since a set
is not a "molecular" heap - but just a heap consisting of that very set.
It's an atom.)



and from comprehension in this theory we can actually get the heap of
any sets that we fulfill P(x) for any P(x) provided that there should
exist at least one set for which P holds.

Very nice approach. Yes, it's certainly a nice idea to combine set
theory with "heap theory". Or with other words, to do set theory in the
context of a heap framework. (This way we might get the best from both
theories, I guess.)

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".

Beautiful!

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.

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 ).

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

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?



Zuhair





So in this way we can have the heap of all ordinals, the heap of all
sets, the heap of all well founded sets, the heap of all sets
bijective to a set, the heap of all sets bigger than a set, etc...,
the heap of all sets other than a set, the heap of all sets that are
not members of a set etc....

Yes. It's a nice approach, indeed.

F.

--

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

.

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