Re: Skolem's 'Paradox'

sci.math_20041006.rtf:The "set of all sets" does not exist, per se, in
ZF, it is disqualified from existence primarily by the axiom of
regularity. Basically its existence is denied because there are a
variety of antinomies or paradoxes associated with it. One of these is
called Cantor's paradox, that the set of all sets would be its own
powerset. Another is the Burali-Forti paradox, that the order type of
all ordinals would be itself an ordinal. These are clearly associated
with basic problems of considering any infinite number, and thus their
consideration leads to either: a)ultrafinitism, or b) combining each
of them into basically a superclass of conundrums, and trying to solve
the problem at once, instead of just restating it.
sci.math_20041115.rtf:You might have heard of "proper classes", where
someone says "in ZF, the set of all sets does not exist, there is only
the proper class of all sets." Under some naive and correct
definitions of the proper class, there can only be one or none of them.
The empty set is the proper class or ur-element, as is infinity. It's
a _set_ theory, founded on nothing and everything.
sci.math_20041117.rtf:I'm not talking about ZF, although you could call
it ZFF, simply removing the axiom of foundation and adding perhaps an
inversion axiom.
sci.math_20041117.rtf:For example, I read about Nam Duc Nguyen's new
axiom he wants that he hasn't stated, and I've already determined it
might be a theorem of the null-axiom theory, the axiom-free theory, as
the axioms of ZFC are theorems. That enables me to validate much of
logic with my non-standard theory, in much the same way standard
analysis is validated in the regular non-standard analysis, where the
powerset result is not a theorem.
sci.math_20041117.rtf:Instead of pushing off the resolution of ZF's
problems into ever-higher order logics, with thus no resolution, they
are resolved in what can be a finitist logic, it is not, there is the
completed infinity.
sci.math_20041120.rtf:Consider a construction in "ZF with classes" that
cannot be a set, in that supposed "set theory", and cannot be a class.
English readily runs out of unique "group nouns." Does not that seem
to demand infinitely many different group nouns? Where that is so, it
is a good idea to get rid of that notion.
sci.math_20041124.rtf:I probably don't recall the axioms of ZF without
research, let's see:
sci.math_20041124.rtf:If ZF were consistent, and the axioms of ZF
theorems of NAT, or AFT, the Axiom Free Theory, or FNAT, the Finlayson
Null Axiom Theory, except for perhaps regularity, foundation, where
every set contains a minimal element anyways, besides that, then
theorems in ZF not using foundation would be theorems of FNAT.
sci.math_20041203.rtf:By and large those have little to do with
foundations. Anyways, today there are theories that do not necessarily
use ZF, for example, an excellent set theory with a somewhat restricted
notion of a set, and calling everything else a class, except the class
can not contain other classes, leading to no class of all classes, and
no reolution of Burali-Forti.
sci.math_20041206.rtf:Do you use ZF with classes? What's the class of
all classes? If your answer is no, can there be more than one proper
class?
sci.math_20041208.rtf:If you use the notion of proper classes in ZF
because you accept the powerset theorem that infinite sets aren't
equivalent then there is the regularity axiom which pre-vents the set
of all sets, Kant's Ding an sich and Kant's noumena, that is its own
powerset, then you're stuck, because there are collections of those
things in our discourse. If you don't allow classes, then if you want
a set theory perhaps you should have an anti-foundation axiom, or
instead no (non-logical or "proper") axioms.
sci.math_20041208.rtf:Do you use ZF with classes? What's the class of
all classes? If your answer is no, can there be more than one proper
class?
sci.math_20041219.rtf:That notion also conveniently fits well with
other set-theoretical notions attempting to explain the very real
nature of mathematical objects, where a set theory needs an ur-element
that should be as well a set. It can not be only a set, and the only
other things that can not be a set must share a literal value of sorts,
because there can only be one proper class, or none. There are a
handful of what they call paradoxes or antinomies that have been
plaguing set theory since its inception and coagulation upon ZF that
are neatly resolved in that way.
sci.math_20041231_b.rtf:VNB is said to be a synonymous acronym of
NBG/GBN, von Neumann - Bernays - Goedel, an infinitely axiomatized ZFC.
sci.math_20050110.rtf:Not much use can be made of a presupposed fact
that the rationals and irrationals alternate in the well-ordered reals,
because those sets of numbers are fields. It is only that that fact
can be presupposed for applications that do not conflict with their
characteristics as NCD sets for the reals in the well-ordering of the
reals, via fiat in ZFC, where any set can be well-ordered.
sci.math_20050110.rtf:Under what conditions can it be said the reals
are well-orderable? In ZFC, any set is well-orderable. If any set of
reals dense is well-orderable then so is the complete superset and any
subset. Is not that obvious to you?
sci.math_20050117_b.rtf:We have by fiat (a consequence of
axiomatization) that the reals are well-orderable, in ZFC. Where the
reals are well-orderable, then it stands to reason that there is a
well-ordering on the set. The most "obvious" ordering on the reals is
the normal, linear, total ordering. Do you know any well-orderings of
the reals?
sci.math_20050117_b.rtf:I read somebody say the reals' well-orderings
have complexity Sigma-1-2 in Goedel's constructible universe and thus
the well-ordering is not Lebesgue measurable, and do not understand
that. I don't yet know what some few of those phrases mean and if they
can't be explained completely in a few paragraphs then I suspect them
meaningless. If something is independent of ZFC, then it's of little
use to a Platonist. Please describe Sigma and Pi of Goedel's
constructible hierarchy. Those are closely held.
sci.math_20050120.rtf:f0fs24 cf0 In ZFC, there is at least one
well-ordering of the reals.
sci.math_20050120.rtf:A theorem of ZFC set theory is that there is a
well-ordering of any set.

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