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