On 2/15/2013 7:44 AM, Shmuel (Seymour J.) Metz wrote: > In <oPydnQ9MUcW1vIPMnZ2dnUVZ_s6dnZ2d@giganews.com>, on 02/15/2013 > at 05:50 AM, fom <fomJUNK@nyms.net> said: > >> As this part of the construction had been >> motivated by the axiom of regularity in >> set theory, it is intuitively reasonable >> to think of a letter as "a collection >> of letters" although I suspect many will >> find that objectionable. > > Use of naive set theory leads to paradoxes. If you are using any > mainstream set theory, e.g., ZFC, then no set is an element of itself. > There are theories where you could get away with that, e.g., NF, but I > suspect that you would find them awkward. More to the point, having a > letter be a collection of letters would violate the very axiom of > regularity that you cite as your motivation.
No problem here. No difference set contains the letter with which it is associated.
If one looks at the statement of regularity closely, the class of sets that do not contain the empty set all act to separate at least one set-as-object from that same set in the sense of set-as-collection. That is what I had been thinking of when I made the statement.
Probably a bad choice.
> > Things like that are more reasonable in Mereology, about which I don't > know much, but if you're addressing issues related to set theory that > doesn't help. >
I replied to someone a while ago when Zuhair was writing about mereology. The stuff on the web is the "rediscovery" of mereology via Husserl and Brentano. I have read Husserl and do not like what I obtain second-hand from Brentano.
Mathematically, Lesniewski's mereology had been far ahead of what phenomenology and the Vienna Circle produced. In case a quick synopsis would be of any interest to you, here is what I prepared before:
> > Arguably, mereology as an investigation into foundational > mathematics is from Lesniewski > > Lesniewski wrote several papers criticizing Russell's > Principia as basically being incoherent > > He did his own investigations along the lines of logical > structure of sentences having existential import that > are different from both Frege and Russell > > Subsequently, he characterized a notion of class that was > different from Russell's > > At first, he tried to characterize his ideas in traditional > logical formats but ultimately began to pursue it > using formal syntax > > This actually gets dense and I have not really > examined it. But, for example he begins a system > he calls protothetic with > > A1. ((p <-> r) <-> (q <-> p)) <-> (r <-> q) > > A2. (p <-> (q <-> r)) <-> ((p <-> q) <-> r) > > and then lists 79 theorems about logical equivalence. > > He then switches notation to quantify over propositional > variables > > A1. ApAqAr(((p <-> r) <-> (q <-> p)) <-> (r <-> q)) > > A2. ApAqAr((p <-> (q <-> r)) <-> ((p <-> q) <-> r)) > > So that he can extend the system using variables ranging > over truth-functions > > A3. AGAp(AF(G(p,p) <-> > ((Ar(F(r,r) <-> G(p,p)) > <-> > (Ar(F(r,r) <-> G(p <-> Aq(q),p)))) > <-> Aq(G(q,p))) > > He then lists 422 theorems in order to obtain the > three logical axioms of Lukasiewicz grounding > the usual theory of deduction based on implication > and negation. His primary stated goal at the > outset for extending the original system was to > obtain > > ApAqAF((p <-> q) <-> (F(p) <-> F(q))) > > and the equivalent was obtained at step 381 > > Next, he turns to his system of ontology > grounded upon the logical foundation of his > protothetic. His only axiom is > > A0. AZAz((Z class of z) <-> ( > (-(AY(-(Y class of Z))) > /\ > AYAX(((Y class of Z) /\ (X class of Z)) -> (Y class of X))) > /\ > AY((Y class of Z) -> (Y class of z)) > )) > > > In his exposition he comments on Russell's > paradox: > > "... can be strengthened in ontology by > means of the easily proved sentence which > says that: > > AZAz((Z class of z) -> (Z class of Z)) > > > (I call this the 'ontological identity sentence'; > it should be noticed that the yet stronger thesis > > AZ(Z class of Z) > > is not provable in ontology -- indeed, its > negation is provable.) In connection with this > sentence, I want to emphasize expressly that in > ontology there is always a very good possibility > of proving theses having a single component of > the type (Z class of Z) or (what is indifferently > the same in ontology) (z class of z). This does > not, however, lead to a contradiction via the > well-known schema of Principia Mathematica > because the definition directives of ontology > have been appropriately formulated so that > no thesis of the type > > AZ((Z class of x) <-> -(Z class of Z)) > > can be obtained." > > > This assertion might best be viewed much like > the situation with general relativity. Philosophers > who have been looking at Lesniewski's systems > have not run into any contradictions such as > Russell's (at least, in so far as Peter Simons > has reported accurately) > > The character of his predicate in ontology > allows him to formulate terminological > explanations about ontology within the language > of ontology. > > The axiom of ontology, and the equivalent axioms > he discusses in his exposition derive from > his analysis of Russell's paradox. > > The Lesniewskian notion of class is based upon > a part relation and this is the formal mereology > associated with his investigations: > > > A1: > If P is a part of object Q, then Q is not a > part of object P > > A2: > If P is a part of object Q, and Q is a part of object R, > then P is a part of object R > > D1: > P is an ingredient of an object Q when and only when, > P is the same object as Q or is a part of object Q > > D2: > P is the class of objects p, when and only when the > following conditions are fulfilled: > > a) > P is an object > > b) > every p is an ingredient of object P > > c) > for any Q, if Q is an ingredient of object P, then > some ingredient of object Q is an ingredient of > some p > > > > These conditions formalize a statement Lesniewski > made in his analysis of Russell's paradox: > > "I use the expressions "the set of all objects m" > and "the class of all objects m" to denote every > object P which fulfills the two following > conditions: > > 1) every m is an ingredient of the object P > > 2) if I is an ingredient of object P, then > some ingredient of object I is an ingredient > of some m" > > > > Since ingredient is effectively the reflexive > subset relation in set theory by Zermelo's > 1908 language, you can see why Zuhair chose the > language he did to describe atoms. In an > atomistic theory, I and m must at least share > some atom of P > > I expressed this idea in a formal sense long > ago only to be flamed (singed by you and firebombed > by someone else) > > What actually makes mereology work is something that > is associated with the constructible universe. It is > called almost universality. Of course, that is not > how Lesniewski referred to it: > > "Lukasiewicz writes in his book as follows: 'we say > of objects belonging to a particular class, that > they are subordinated to that class' > > "It most often happens that a class is not subordinated > to itself, as being a collection of elements, it > generally possesses different features from each of > its elements separately. A collection of men is not > a man, a collection of triangles is not a triangle, > etc. In some cases, it happens in fact to be otherwise. > Let us consider e.g., the conception of a 'full class', > i.e., a class to which belong, in general, some > individuals. For not all classes are full, some > being empty; e.g., the classes: "mountain of pure > gold', 'perpetual motion machine', 'square circle', > are empty, because there are no individuals which > belong to those classes. One can then distinguish > among them those classes to which belong some > individuals, and form the conception of a 'full > class'. Under this conception fall, as individuals, > whose classes, e.g., the class of men, the class > of triangles, the class of first even number (which > contains only one element, the number 2), etc. > A collection of all those classes constitutes > a new class, namely 'the class of full classes'. > So that the class of full classes is also a full > class and therefore is subordinated to itself." > > In almost universal models of set theory, every subclass of > the universe is an element of the universe. Thus, > "is an element or is equal to" is the same as > "is a proper part or is equal to". So, the satisfaction > predicate can be reflexive containment... except for > one thing. The identity predicate in the theory can > not be based on extensionality. It must be based on > first-order object identity as described by Frege and > this cannot arise just because one invokes the > ontological position of a "theory of identity". The > reason that it must be based on object identity is that > reference to the universe can only be made if, as > Lesniewski has observed > > P is an object > > In an almost universal model, every proper part > of the universe is an element of a class of which > the universe is not an element. Thus every proper > part of the universe is distinguished from the > universe on the basis of object identity. Since > to be a 'full class' the universe can have no > other parts, it is unique and may be denoted > by a singular term. > > I have a very strong suspicion that no one will > ever derive a Russellian paradox in Lesniewskian > mereology. This is especially true if one considers > George Greene's explanation that the paradox arises > from grammatical form. As we have seen, Lesniewski > specifically devised his mereology to circumvent the > grammatical forms he thought would be problematic.