On 5/26/2013 2:06 PM, Ross A. Finlayson wrote: > On May 26, 11:25 am, fom <fomJ...@nyms.net> wrote: >> On 5/25/2013 8:36 PM, William Elliot wrote: >> >>> On Sat, 25 May 2013, fom wrote: >> >>>> On 5/24/2013 10:57 PM, William Elliot wrote: >>>>> V is the ZFC universe. >>>>> L is the constructible universe. >>>>> L_omega0 is the omega_0th level of the constructible universe. >> >>>>> Correct or needing correcting? >> >>>> Your apparently simple question seems to have generated >>>> some interesting replies. >> >>> Seemingly off the mark because I'm asking about notation and not about >>> theory. >> >> Just to clarify something concerning your distinction here, >> let me observe that the sign of equality in "V=L" is >> metamathematical or metalogical or metalinguistic or >> however some purist would like to take it as not being >> part of the "object language". >> >> In other words, part of the reason your question received >> "theory" replies is because of the class/set distinction >> and its model-theoretic implications. >> >> These particular notations correspond precisely with >> the point at which philosophy and mathematics (prior >> to intuitionism and categories) meet. Enough said. > > What part of philosophy has non-sets...
To be honest, Ross, most of the mathematical words involved with these topics have become meaningless to me.
"The priority monist will hold that the one basic concrete object is the world (the maximal concrete whole). She will allow that the world has proper parts, but hold that the whole is basic and the parts are derivative."
In Chapter 6 of "Logic, Logic, Logic" by Boolos you will find the statement,
"A different conception of set, to be examined below, is the doctrine of 'limitation of size'. The doctrine comes in at least two version: On a stronger version of limitation of size, objects for a set if and only if the are not in one-one correspondence with all the objects there are. On a weaker, [...]"
So, the question becomes how to differentiate "all the objects that are" from sets. Well, that happens to be the same question as how to interpret the universal quantifier in set theory as an *ontologically foundational* theory.
"Ontological priority: there is an irreflexive, asymmetric, and transitive relation of ontological priority between entities."
So, it is ludicrous to accept the standard account of identity based on logical atomism. In relation algebra, there are 4 necessary relations: full, empty, identity, and diversity. Combining the roles of "diversity" and "ontological priority" I can grammatically define a relation,
AxAy(xcy <-> (Az(ycz -> xcz) /\ Ez(xcz /\ -ycz)))
that is provably irreflexive, asymmetric, and transitive within the standard deductive calculus and without any interpretative philosophy beyond the fact that the transformation rules of the calculus "mean what they do".
"Ontological foundationalism: the ontological priority relation is lower-bounded."
"Concreta foundationalism: there are lower-bounded ontological priority relations within the domain of concrete objects."
The grammatical definition,
AxAy(xey <-> (Az(ycz -> xez) /\ Ez(xez /\ -ycz)))
provides for the directionality to establish these principles. But, it is the complexity associated with the identity relation that must be addressed before these two aspects of priority monism can be properly implemented.
I have written a set theory in which the only proper class is the universe (Boolos' strong limitation of size). The null class is the ontological lower bound. But, the singletons are the concreta. It may be compared with Euclid. On the one hand, "a point is that which has no part" describes the empty set. On the other hand, "a unit is that by which what exists is one" describes the relation of singletons to singular terms in the universe.
The subtle complexity of "identity" and "diversity" come into play because singular reference to the universal class follows from the role of description theory in the history of foundations. But, Hilbert's formalism obscured this relationship and the influence of Russell and Tarski divorced "names" from foundations as an extra-logical concept.
So, a simple beginning becomes an almost impenetrable argument.
In the end, the description theory trail leads to David Boersema's "Pragmatism and Reference" in which he points out that "the given" (for pragmatic philosophers) is a given whole. Prior to individuation, the given whole must be interpreted as decomposed into parts. Only then may the identity presuppositions of description theories apply.
You may contrast that idea with the distinction between "priority monism" and "priority pluralism" in the often-referenced link,
That is, when faced with an independent question, one must come up with reasonable philosophical positions for the introduction of new axioms. It appears, however, that my efforts are in vain. Half of mathematicians believe that God gave Kronecker the natural numbers. Half of mathematicians believe that God gave Russell an intractable paradox. And Hilbert ensured that skeptics could make any argument meaningless in order to save Cantor's paradise.