
Re: Just another exposition of MK.
Posted:
Mar 17, 2013 1:15 PM


On Mar 16, 1:29 am, Zuhair <zaljo...@gmail.com> wrote: > On Mar 16, 9:33 am, Zuhair <zaljo...@gmail.com> wrote: > > > > > > > > > > > Define: Set(x) iff {x,..} > > > Extensionality: x C y & y C x > x=y > > > Comprehension: {x Set(x) & phi} > > > Pairing: x C {a,b} > Set(x) > > > Generation: Set(x) & y C H(x) > Set(y) > > > where H(x)={z m in TC({z}). m =< x} > > > Size: x < V > Set(U(x)) > > > where TC, U stand for transitive closure, union respectively defined > > in the customary manner; C is subclass relation;   =<   and   < > >   relations are defined in the standard manner. > > > The theory above minus axiom of Size is sufficient to prove > > consistency of Z. With the axiom of Size it can prove the consistency > > of ZF+Global choice, and it is equiinterpretable with MK+Global > > choice. > > > Zuhair > > Of course V is the class of all sets
Defined by its elements: _why isn't it a set?_.
Modern mathematics builds on Peano/Presburger to define an inductive set, FirstOrder Predicate Logic FOL/FOPL, finite combinatorics in the finite are completely and consistently modeled. Then is added an axiomatization of infinity that the inductive set is of a particular wellfounded / regular form, and as well regularity is axiomatized (fiat) to restrict comprehension of other assemblages to the well founded / nonselfreferential: ZF (for Zermelo and Fraenkel, who himself averred that the foundations as ZF weren't "secure"). Then, the domain of discourse and mathematical universe, is all those things. A hundred years ago, in the nascent PostCantorian, Russell, BuraliForti, and Cantor himself noted that the set, or assemblage or collection, of all the nonselfreferential elements, would also contain this sputnik or follower of itself, that all the ordinals: iis yet another ordinal, and that the powerset of all the sets: would trivially embed into the set via identity.
Starting from the origin there is all that follows, then all that follows: does. The simple notion that All and None are indistinguishable under a simple reversal of all coding, is put aside for first establishing finite combinatorics. Yet, then when it comes to the actual infinite, and finite combinatorics is not finite as it is unbounded, the axiomatization of a particular structure of an inductive set or least set with an eminimal and no emaximal element, with an emaximal element that isn't itself an element, shoud have that that imposition carries through, and that the structure has other properties, besides that censoring. Then Goedel notes ZF is incomplete insofar as its consistent: there are true features of these objects: not ZF's theorems.
So, not wanting to be left out of discussing the domain of discourse and results about sets in the meta, another group noun, here the class, is ascribed to have some properties of what a set has (eg ZFC with Classes, NBG the infinite axiomatization in schema of ZF, with Classes, von NeumannBernaysGoedel). Sets are defined by their elements. Classes are defined by their elements. The set as duck has the class, as duck, then with the bill removed and its legs broken, so that the water falls off its back, as statements are made about its elements. ("If it walks like a duck and quacks like a duck, it's a duck", "... a rose by any other name would smell as sweet".)
Then, where the group noun as regular/wellfounded doesn't suffice for statements about it, it is on to the next from set to class to gaggle. Modern mathematics then tries to reach to the Absolute, as Cantor called it, the Universe or "Big" infinity, above that of all the sets and cardinals, and as well ordinals or other regular unbounded collections, and higher than the realm of all the asymptotics and unboundedprecision elements of the linear continuum of du BoisReymond, that follows the exploration of infinitesimals of Euler and Gauss. Working down from the settheoretic Universe V, and with regards to the "constructible" universe or universe of constructible (constructed) things L, then modern mathematics has that at once there is a model of all the wellfounded sets, ignoring by calling it a class that the properties noted by Russell, BuraliForti, and Cantor would still apply. This is somewhat unsatisfactory (for conscientious formalists).
With the foundations as to being a rigorous formalization and language for all results in mathematics, then the course of the analysts in their realms of the continuous and real still sees results and systems in development, that are basically mute on the foundations, above measure theory with countable additivity in forming the needed foundation for (standard) real analysis, ignorant and furthermore uncaring of the foundations beneath, as modern mathematics is ignorant of properties of the linear continuum a la the Infinitarcalcul.
That is somewhat unsatisfactory. As an exercise in the pure mathematics, the bridge to potential application is only through the countable, in additivity, in measure theory. Then, a consideration of a more direct and applicable route from primary objects and first principles as final cause, sees justification.
Regards,
Ross Finlayson

