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:

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> > Define: Set(x) iff {x,..}

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> > Extensionality: x C y & y C x -> x=y

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> > Comprehension: {x| Set(x) & phi}

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> > Pairing: x C {a,b} -> Set(x)

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> > Generation: Set(x) & y C H(x) -> Set(y)

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> > where H(x)={z| m in TC({z}). |m| =< |x|}

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> > Size: |x| < |V| -> Set(U(x))

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> > 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.

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> > 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.

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> > 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, First-Order 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

well-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, 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 Post-Cantorian, Russell,

Burali-Forti, and Cantor himself noted that the set, or assemblage or

collection, of all the non-self-referential 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 e-minimal and no e-maximal element,

with an e-maximal 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 Neumann-Bernays-Goedel). 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/well-founded 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 unbounded-precision elements of the linear continuum

of du Bois-Reymond, that follows the exploration of infinitesimals of

Euler and Gauss. Working down from the set-theoretic 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 well-founded sets, ignoring by

calling it a class that the properties noted by Russell, Burali-Forti,

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