Talk:Mereology
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[edit] M8
Are we sure that M8 corresponds to the ZF axiom of replacement and not the axiom of separation? --NoizHed 22:35, 12 March 2006 (UTC)
- M8 says that all individuals satisfying some property φ can be fused together. In standard mereology, one attaches a nonemptiness proviso; there exists at least one individual satisfying φ. The set theory analogue to M8 then is the principle of unrestricted comprehension of naive set theory, not Separation of Replacement. Separation is an axiom only in Zermelo set theory. In ZF, the presence of Replacement makes Separation provable.202.36.179.65 20:04, 10 August 2006 (UTC)
[edit] M2
- "M2 rules out closed loops formed using Parthood; so that the part relation is well-founded."
Well-foundedness means something else, and the subset relation of ZF fulfills M1-M3, but is not well-founded. So in general I see no reason why parthood should be. --129.187.111.178 12:43, 28 June 2006 (UTC)
- By "well-founded" I mean "circularity, i.e., closed loops, is impossible." And I emphatically expect Parthood and subsethood to be well-founded in this sense. This follows from the antisymmetry of Parthood. See Casati & Varzi (1999: 35).202.36.179.65 19:57, 10 August 2006 (UTC)
- I don't see why mereology should be dogmatic about forbidding weakening of Foundation to Aczel's AntiFoundation Axiom AFA. Insisting on Foundation pointlessly circumscribes the applicability of mereology, just as insisting that multiplication is commutative pointlessly restricts the scope of ring theory to a world in which there are neither matrices nor quaternions. Likewise the existence or nonexistence of atoms is not something one should dogmatically commit to, both points of view are useful in mereologies of both the real world and ideal or abstract or theoretical worlds. And the antisymmetry of partial orders is useful on some occasions but an inconvenient superstition on others, witness the many examples of preorders. Vaughan Pratt 20:03, 16 August 2006 (UTC)
[edit] Bottom vs. Atoms
I don't know if it's a confusion of the article or the subject itself, but there seems to be a confusion here between bottom and atoms in the axiom saying that bottom is an atom. Done right, part-whole theory should allow the possibility of absence of stuff, e.g. an empty glass, without any commitment to the atomicity of the stuff constituting a full glass, e.g. water. Water is atomless when it is infinitely subdividable, i.e. you can always pour out half of what remains in the glass without emptying it (unless it was empty to begin with). But that's not to say that you can't pour out all the water! Water is atomic when, with only one atom of water left in the glass, pouring out any nonzero amount of water is guaranteed to empty the glass. The open intervals of the real line are atomless in that sense, as are the elements of any infinite free Boolean algebra (since for any formula f there is a smaller element formed by conjunction with a variable not appearing as a variable of f). The closed intervals on the other hand are atomic, the atoms being of the form [x,x]. Finite free Boolean algebras are atomic, the atoms being the conjunction of any set of literals (possibly negated variables) in which every variable appears. All these situations allow bottom or empty, and none of them confuse bottom with atoms. There are many atoms but only one bottom. Vaughan Pratt 20:03, 16 August 2006 (UTC)
- I don't think this is a confusion at all, merely a difference in terminology. Mereologists simply define an atom to be something with no proper parts. Thus by definition bottom is an atom. However since most mereologists reject the existence of bottom for philosophical reasons, this definition of atom turns out to be equivalent to the notion of atoms you are describing.--NoizHed 10:03, 17 October 2006 (UTC)
The bottom element is a trivial object that is a proper part of every object in the domain of discourse. But it only have a heuristic function. But when we apply mereology with a modeltheory of logic we have to think if we should accept a null individual. If we use free logic we can use the null individual. If we use ordinary logic I would shunn the null individual.
The issue is really if we have mereological simples in our mereology or gunk (a substance where everything has a proper part). And one of the advantages of the mereological axioms is that we don't have to answer that question. We can leave it to the philosophers.
The issue of gunk is, for instance, important in how we should understand objects located in euclidean threedimensional space and how we should understand a biological organism.(Sir.R - 15:16, 23 October 2006 (GMT))
[edit] Mereology and Set Theory
"In set theory, "a is a member of b" and "a is a subset of b" cannot be both true. If the first is true, then only "{a} is a subset of b" can also be true. {a} is the singleton corresponding to a."
this is incorrect:
let b be the set of the set Ø and its singleton (v. Neumann's Number 2)
b = { Ø, { Ø } }. The singleton of the empty set { Ø } is both subset and member of b.
Even more, if the above statement were correct, the fact that the empty set is subset of every set would prohibit the empty set to be member of any set. Already in { Ø }, and in fact in every natural number the empty set is both subset and member.
But even without the empty set this is false: replace all occurences of Ø by the set of your choice and the singleton of this set will still be both subset and member of b.
Atoll 20:10, 10 October 2006 (UTC)
I've removed this part now:
In set theory, "a is a member of b" and "a is a subset of b" cannot be both true. If the first is true, then only "{a} is a subset of b" can also be true. {a} is the singleton corresponding to a.
(and also : ... Why not? Try A={x}, B={{x}, x}. Works also with sets only: A={}, B={{}} ...)
Atoll 11:23, 22 November 2006 (UTC)
[edit] History before the 20th century
Since the issues of Mereology were raised by Boethius and his medieval successors, and before them, by Plato [1][2], it seems strange that the historical part of this article begins with Husserl in 1901. --SteveMcCluskey 20:25, 29 November 2006 (UTC)
