```Date: Mar 17, 2013 1:15 PM
Author: ross.finlayson@gmail.com
Subject: Re: Just another exposition of MK.

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 equi-interpretable with MK+Global> > choice.>> > Zuhair>> Of course V is the class of all setsDefined by its elements:  _why isn't it a set?_.Modern mathematics builds on Peano/Presburger to define an inductiveset, First-Order Predicate Logic FOL/FOPL, finite combinatorics in thefinite are completely and consistently modeled.  Then is added anaxiomatization of infinity that the inductive set is of a particularwell-founded / regular form, and as well regularity is axiomatized(fiat) to restrict comprehension of other assemblages to the well-founded / non-self-referential: ZF (for Zermelo and Fraenkel, whohimself averred that the foundations as ZF weren't "secure").  Then,the domain of discourse and mathematical universe, is all thosethings.  A hundred years ago, in the nascent Post-Cantorian, Russell,Burali-Forti, and Cantor himself noted that the set, or assemblage orcollection, of all the non-self-referential elements, would alsocontain this sputnik or follower of itself, that all the ordinals:iis yet another ordinal, and that the powerset of all the sets:  wouldtrivially embed into the set via identity.Starting from the origin there is all that follows, then all thatfollows:  does.  The simple notion that All and None areindistinguishable under a simple reversal of all coding, is put asidefor first establishing finite combinatorics.  Yet, then when it comesto the actual infinite, and finite combinatorics is not finite as itis unbounded, the axiomatization of a particular structure of aninductive set or least set with an e-minimal and no e-maximal element,with an e-maximal element that isn't itself an element, shoud havethat that imposition carries through, and that the structure has otherproperties, besides that censoring.  Then Goedel notes ZF isincomplete insofar as its consistent:  there are true features ofthese objects:  not ZF's theorems.So, not wanting to be left out of discussing the domain of discourseand results about sets in the meta, another group noun, here theclass, is ascribed to have some properties of what a set has (eg ZFCwith Classes, NBG the infinite axiomatization in schema of ZF, withClasses, von Neumann-Bernays-Goedel).  Sets are defined by theirelements.  Classes are defined by their elements.  The set as duck hasthe class, as duck, then with the bill removed and its legs broken, sothat the water falls off its back, as statements are made about itselements.  ("If it walks like a duck and quacks like a duck, it's aduck", "... a rose by any other name would smell as sweet".)Then, where the group noun as regular/well-founded doesn't suffice forstatements about it, it is on to the next from set to class togaggle.  Modern mathematics then tries to reach to the Absolute, asCantor called it, the Universe or "Big" infinity, above that of allthe sets and cardinals, and as well ordinals or other regularunbounded collections, and higher than the realm of all theasymptotics and unbounded-precision elements of the linear continuumof du Bois-Reymond, that follows the exploration of infinitesimals ofEuler and Gauss.  Working down from the set-theoretic Universe V, andwith regards to the "constructible" universe or universe ofconstructible (constructed) things L, then modern mathematics has thatat once there is a model of all the well-founded sets, ignoring bycalling it a class that the properties noted by Russell, Burali-Forti,and Cantor would still apply.  This is somewhat unsatisfactory (forconscientious formalists).With the foundations as to being a rigorous formalization and languagefor all results in mathematics, then the course of the analysts intheir realms of the continuous and real still sees results and systemsin development, that are basically mute on the foundations, abovemeasure theory with countable additivity in forming the neededfoundation for (standard) real analysis, ignorant and furthermoreuncaring of the foundations beneath, as modern mathematics is ignorantof properties of the linear continuum a la the Infinitarcalcul.That is somewhat unsatisfactory.  As an exercise in the puremathematics, the bridge to potential application is only through thecountable, in additivity, in measure theory.  Then, a consideration ofa more direct and applicable route from primary objects and firstprinciples as final cause, sees justification.Regards,Ross Finlayson
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