Re: Skolem's 'Paradox'

sci.math_20050214.rtf:I suggest discarding all the non-logical axioms
and just theorizing from tautology the axioms of ZFC besides
regularity. That's about a vague notion that infinite sets can not be
regular, well-founded.
sci.math_20050228.rtf:Piotr, I just suggest you reread the posts on
this very thread about that kind of question. I'm working from an
axiom-free set theory, or null-axiom set theory, "NAST", heh, an AFT,
where in general theorems of ZFC minus regularity are theorems. A set
theory is comprised of only sets and if you care they are all ordinals
or even naturals and there is dual representation via the dually
minimal and maximal ur-element, which is eternity and void and
variously infinity and nothing and perhaps one or negative one,
confoundingly. If you have a set theory with proper classes I say
there can only be one proper class, then I intone "set of all sets,
class of all classes, half a shadow play" and damn you to
Skolemization.
sci.math_20050308.rtf:Basically that's the same thing as talking about
the set of all sets, which would evermore be its own powerset. I think
if you are rational then you ascribe to an axiom-free set theory with
ubiquitous ordinals, which has the option of being consistent,
complete, and concrete. At the very least, you should admit that ZF
has some problems, in terms of its existence. Otherwise I think you're
an ignoramus.
sci.math_20050409.rtf:In ZFC with classes, how you do quantify over
classes?
sci.math_20050415.rtf:Quantification implies a universal set. ZF,
Zermelo-Fraenkel Set Theory, is inconsistent, because of regularity.
sci.math_20050418_d.rtf:Well, this is good, I'm pleased with how this
discussion was progressing. Will, I'm somewhat different than you in
that I attack Cantorian and Goedelian results, and ZF itself, the
foundations of the status quo's mathematical logic, quite directly,
because Goedel calls you a liar. I'm glad that you agree that
considering alternatives to transfinite cardinals is not wrong.
sci.math_20050501_b.rtf:f0fs24 cf0 Consider the universal quantifier,
and the axioms, the non-logical axioms of ZF set theory. If "for each"
set, it satisfies regularity, what is it "for each" of? Is it for each
set? Does that mean for each set,of all sets?
sci.math_20050501_b.rtf:Is there a class of all classes in "ZF with
classes"? No, there's not. ... If you pick another group noun, for a
different type of collection that is actually the same thing and
fooling yourself, "ZF theory of sets and classes and collections", then
"ZF theory of sets, classes, collections, and groups", then there is no
group of all groups.
sci.math_20050501_b.rtf:As alluded to in mention of a theory of
everything, that's about a theory of everything, and where all the
truths of ZF can be inferred from another theory, then, "they're not
done yet, doc."
sci.math_20050519_b.rtf:In set theory, people generally use a set of
axioms, axiomatic set theory with axioms. One group of those is called
Zermelo-Fraenkel, or ZF, set theory, the list of what are called
non-logical or proper axioms that clarify what a set is, or can be.
sci.math_20050519_b.rtf:One implication of accepting the Domain
Principle, which I understand is an extension of some theories of
Immanuel Kant, a Western technical philosopher, which I just heard of
but already said, is that ZF is inconsistent because of regularity.
sci.math_20050523.rtf:That might seem obvious, those results might seem
obvious, but there are some basically contradictions to Cantorian set
theory as it is often practiced. For example, the cardinality of the
reals, or c, has been shown equivalent, that a bijection exists, to
Aleph_1, Aleph_2, Aleph_3, .... (As well, where infinite sets are
equivalent, for example as in my null-axiom theory, Aleph_0.)
Hausdorff says: a countable union of coutable sets might be
uncountable. ZF is inconsistent because of the Domain Principle, and
the reals are equivalent to the natural integers because of the
Well-Ordering Principle.
sci.math_20050524.rtf:Back to set theory, in the current most
estabilshed set theory of the foundations of mathematical logic ZF, as
it is enshrined in text and taught, there is no universe of sets.
There are problems with that. One is that quantification over sets
implies a universal set, or as Cantor himself said, according to the
"domain principle", there must exist a set of all sets. There are a
varieties of amendments to set theory to address that and other issues
having to do with otherwise inescapable ramifications of those simple
facts about mathematical logic, for example type theory, and later to
the notion of a non-set container, or collection, the class, and then
back again to anti- or non-foundational set theories that have no axiom
of regularity.
sci.math_20050524_b.rtf:ZF is inconsistent.
sci.math_20050525_d.rtf:About set theory, and its consistency, there's
a difference between saying quantification over sets implies a
universal set and thus the regularity axiom of ZF is inconsistent and
saying any set theory is inconsistent.
sci.math_20050526.rtf:On the contrary, perhaps you might consider why
all these people, for example A.A. Fraenkel, the F in your ZF set
theory, think that reliance on transfinite cardinals is a mistake.
sci.math_20050526_b.rtf:The set of all sets is its own powerset.
Quantification over sets implies a universal set exists. A variety of
modern theories have non-well-founded sets for which the powerset
result does not hold. Where quantification over sets implies the
existence of a universal set, ZF is inconsistent, as is its cousin with
the same axioms NBG.
sci.math_20050526_b.rtf:In NSA, there may be the hyperintegers,
infinitely long strings of finite radix, finite base, that represent
integers. Where here we have been talking about how the tree is the
same for the integers and those of the unit interval of reals, that it
has no leaf nodes, in NSA it is the same, except the hyperintegers (or
"p-adic integers") may well be infinite in precision and extent. The
transfer principle is that what is provable in NSA, IST, is provable in
ZFC, where IST and ZFC are coconsistent.
sci.math_20050528_b.rtf:ZF is inconsistent in a theory with a universal
set, because regularity would be one of what you've been calling a
"false axiom".
sci.math_20050529.rtf:ZF is inconsistent.
sci.math_20050530.rtf:Skolemize, your model is countable. Sets of
numbers are measurable. The universe is infinite. Infinite sets are
equivalent. ZF is inconsistent.
sci.math_20050530_b.rtf:Where functions between physical object and
physical objects are physical objects, the physical universe, which
exists as a totality or collection for anything to exist, are physical
objects, then the universe is infinite and infinite sets are
equivalent, because you and I each exist. So, that fact of existence
is an empirical example or physical proof that infinite sets are
equivalent. Replacing the word "physical" with "set-theoretical" leads
to a parallel deduction, but I repeat myself. ZF is inconsistent.
With a variety of types of objects to consider, eg, sets, numbers, sets
of numbers, numbers of sets, and physical and geometrical objects, some
notions that affect any theory, or theory of anything, affect those.
sci.math_20050601.rtf:The set of all sets is its own powerset. In a
set theory, there are only sets and quantification over sets implies a
universal set or set of all sets. ZF is inconsistent. That aside, a
set of all sets is a counterexample to that there exists no bijection
between a set and its powerset because the set of all sets is its own
powerset. By assigning basically an index or ordinal to each set, via
well-ordering, that set is as well orderable by what is generally
called the set of natural integers, via transfer.
sci.math_20050601_d.rtf:ZF is inconsistent.
sci.math_20050603.rtf:ZF is an inconsistent set of axioms. Please
present any well-ordering of the real numbers.
sci.math_20050604_d.rtf:There can be, only one, theory, the null axiom
theory. ZF is inconsistent.
sci.math_20050605.rtf:ZF is inconsistent because quantification over
sets implies that there exists a universal set, thus regularity is
inconsistent with a set theory where a formula is valid. Doing away
with that non-logical axiom, do away with the rest of them towards an
axiom-free theory.
sci.math_20050605_b.rtf:Currently in particle physics there are
quantities that as they are measured more precisely, their value
appears to diverge. The universe is infinite and infinite sets are
equivalent. Infinitesimals in and as reals invalidate Vitali's
non-measurability criteria. ZF is inconsistent. Somewhat more
extendedly: an incomplete theory is inconsistent, and only the null
axiom theory can be complete, and the axioms of ZF besides regularity
are theorems of the null axiom theory's axiom-free set theory.
sci.math_20050605_c.rtf:In your opinion, is ZF consistent?
sci.math_20050605_d.rtf:All the axioms of ZF besides regularity are
theorems instead. It's basically just a semantic difference.
sci.math_20050606_b.rtf:ZF is inconsistent. The axioms of ZFC besides
regularity are theorems of the null axiom theory. Infinite sets are
equivalent.
sci.math_20050609.rtf:Infinite sets are equivalent. ZF is
inconsistent. There's a universal set. In the monadic universe
functions between monadic objects and monadic objects are monadic
objects, and so the universe is infinite and infinite sets are
equivalent. Here the monad is not the Lebniz monad, nor the pervert
gonad, it's a variously physical, set-theoretical, number-theoretical,
or geometrical primary object of the theory, and quite similar to the
monad of Leibniz' monadic theory.
sci.math_20050609_c.rtf:ZF is inconsistent because there exists a
universal set. Cantor acknowledged that, he called it the domain
principle, and after it is named Cantor's paradox. In your set theory,
do you quantify over... sets? How? Is it from a choice function on
all sets? Why can't you acknowledge that? Is it because... that
invalidates what you have so vociferously said?
sci.math_20050610.rtf:f0fs24 cf0 Why? I don't use ZF.
sci.math_20050610.rtf:ZF is inconsistent, because those axioms purport
to describe all sets, and do not. If you want to discuss only regular
sets, it's within some broader theory, and you're only discussing
regular or well-founded sets. You'll notice that there are a variety
of anti-foundational theories. The non-logical axiom, or rule, of
regularity is inconsistent with the domain principle, comprehension.
sci.math_20050610_b.rtf:You might notice that I just said regularity
and comprehension are at odds with each other. I use an axiom-free
theory, the non-logical axioms of ZF besides reglarity of which are
theorems.
sci.math_20050620.rtf:ZF is not consistent with a universal set, which
must exist for it to be a set theory. Drop regularity, that's a very
simple way to achieve the consistency, of ZF-R, except for that it's
incomplete.
sci.math_20050620.rtf:About the _utility_ of transfinite cardinals,
there is not much to say. The search has been on for a _use_ of
transfinite cardinals for quite some time. There's more to logic than
ZF's hamster maze, and it's not even utterly fundamental.
sci.math_20050620_b.rtf:Basically ZF says that the empty set is a set,
and that the set containing the empty set, its powerset, is a set, as
is its and theirs, and via union, and subsets, all of those.
sci.math_20050620_b.rtf:If c = Aleph_1 is a "model" of ZF, what that
means is that c = Aleph_1 is a "theorem" of ZF. If c = Aleph_1 is a
theorem, and c = Aleph_2 is a theorem, then Aleph_1 = Aleph_2 is a
theorem.
sci.math_20050620_b.rtf:About the metatheory and the countability of
any set it seems that Skolem showed that it is not possible to deny in
ZF the countability of any set in a given n'th order theory, because
via induction there exists another. I simply don't find entertaining a
puppet show in this case.
sci.math_20050624.rtf:ZF is inconsistent.
sci.math_20050625.rtf:I don't know about NBG, which I think is just
infinitely axiomatized ZF(C), but in ZF with Classes there are those
notions of the class and proper class. Now, if a proper class can not
be contained in any other container, is it not obvious that there is
only one proper class? A conundrum arises that consideration of the
class of all classes leads to the same or similar problem as the set of
all sets. In having a universal container, or Ding-an-Sich, thing in
itself, ZF does not have one, having a universal container should be
done thus that in a set theory, it's a set. Otherwise, it's not in the
domain of discourse, and mentioning it kicks sand on theory.
sci.math_20050625.rtf:David, I think ZF is inconsistent, and that
instead the theory is the null axiom theory, with no non-logical
axioms, only the logical axioms or rules of inference of first-order
predicate logic.
sci.math_20050628.rtf:ZF is inconsistent and passe.
sci.math_20050702.rtf:ZF is inconsistent.
sci.math_20050704_d.rtf:ZF is inconsistent. For example, the order
type of ordinals would be an ordinal, or the set of all sets would be a
set. You might want to consider why in ubiquitous ordinals the
powerset is order type is successor, and why that leads to resolution
of these and other self-same set-theoretical paradoxes, which you
appear to have in _your_ professed theory. Confront your
misperceptions.
sci.math_20050707.rtf:ZF is inconsistent, because the set-theoretical
universe is infinite, among other reasons that resolve to the same
thing.
sci.math_20050719.rtf:Obviously I suggest the null axiom theory. ZF is
inconsistent.
sci.math_20050731_c.rtf:ZF is inconsistent, because there is a
universal set in a universal set theory.
sci.math_20050825.rtf:The set of all sets is inclusive. If
quantification implies a universal set, ZF is inconsistent.
sci.math_20050827.rtf:So, are the reals a set in ZFC? Some people
accept the well-ordering principle as self-evident. Biject any set to
an ordinal, it's well-ordered.
sci.math_20050828_b.rtf:If there is an ordinal for each "transfinite
cardinal" in ZF, is not each set in ZF well-orderable via bijection to
an appropriately equipollent well-ordered ordinal, and thus the
well-ordering principle a trivial corolarry of the well-ordering of
each ordinal, and ZF is ZFC?
sci.math_20050828_b.rtf:What we've been discussing is why the reals are
not a set, or else for any other set to biject to them there is
generated via an extension of Cantor's first and transfinite induction
certain reasonings why the rationals are uncountable, ie contradiction,
or any well-ordering of the reals contains two points with no points
between them in the natural ordering, or else... the reals are not a
set in ZF.
sci.math_20050829.rtf:I had not heard of these "sets in ZF with no
cardinality." There's an ordinal for each cardinal, a well-ordered
ordinal, so if there were no sets in ZF with no cardinality then
well-ordering would be a trivial corrolary of a set's definition as a
set. Please explain why am infinite set could not map to one of N,
P(N), P(P(N)), etcetera. Perhaps that gets into the inaccessible
cardinals?
sci.math_20050830.rtf:If there's an equipollent ordinal for each
cardinal, then the well-ordering principle is a trivial corolarry and
ZF is the same as ZFC. That is not so?
sci.math_20050830_b.rtf:f0fs24 cf0 Yes, that's what I thought. So, in
ZF does every set have a cardinal? Is each ordinal well-ordered?
Where that is so, Choice is a theorem of ZF. No?
sci.math_20050831.rtf:So anyways, besides the discussion about
ramifications of well-ordering the reals, which are a well-orderable
set in ZFC for everybody, does the existence of an ordinal for each
cardinal, even if it is not simple to say what it is, does the
existence of said ordinal imply a well-ordering exists for that set,
any set? Similarly, there is the existence proof of a well-ordering of
the reals in ZFC, and if the reals are a set, then they are
well-orderable, and the well-ordering implies the extension of Cantor's
first to that there must be adjacent points, regardless of whether it
implies that there are more than countably many disjoint intervals.
sci.math_20050831_b.rtf:With regularity in ZF, is the existence if not
identity of an ordinal for each cardinal certain? All there needs be
is an ordinal for any greater cardinal.
sci.math_20050901.rtf:f0fs24 cf0 ZF is inconsistent. I think that
because I think there needs to be a universal set in a set theory, and
regularity, the axiom of regularity or axiom of foundation, precludes
the existence of a universal set. So, ZF: inconsistent, ZF - R: less
necessarily inconsistent.
sci.math_20050901.rtf:ZF being inconsistent doesn't mean the rework of
all other mathematics, that's ridiculous. It just would mean the
rework of a variety of intermediate results, retrofitting
coconsistential schemata.
sci.math_20050904.rtf:f0fs24 cf0 ZF is inconsistent.
sci.math_20050911.rtf:ZF is inconsistent. Try explaining again the
extension of Cantor's first to any well-ordered set and the reals in
their normal ordering.
sci.math_interview.rtf:I theorize that some of the axioms of ZF are
decided by existence and excluded middle and tautology, or not. Thus I
aim to strike at the root without killing the tree.
sci.philosophy.meta_20050618.rtf:ZF is inconsistent. The universe is
infinite, and infinite sets are equivalent.
sci.physics_20050512.rtf:There should be some note of a difference
between the logical and non-logical (or proper) axioms. When there is
discussed ZF, for Zermelo-Fraenkel, Set Theory, in general reference to
its axioms are references to its non-logical axioms, eg powerset,
union, pairing, ..., where those are basically qualifications of what a
set is in terms of construction from, basically, the empty set.
sci.physics_20050512_b.rtf:Here's something to consider, the
non-logical axioms of ZFC, except for regularity or well-foundedness,
plus what is called "inverse", are theorems of the null axiom theory.
Thus, many proven results simply remain true, soundly.
sci.physics_20050521.rtf:The universe is infinite and infinite sets are
equivalent. Any theory of everything is necessarily a logical theory
of everything, and ZF is inconsistent. The electron and photon are
point particles, but so are the other particles. There is not a
theoretical smallest particle, except for any of those. That's a
similar consideration to that the reals, nonstandardly, have a smallest
positive real number.
sci.physics_20050526.rtf:One thing we were talking about was whether an
incomplete theory was inconsistent. Perhaps that was a different
discussion. As a consequence of an axiomatization is Goedelianly
incomplete, the notion with that was that a theory with a null
axiomatization is not subject to that dilem-na. The idea is to show
that as a perspective of the theory as, for example, a set theory as
set theory is used in a variety of foundational treatments that from
first principles and tautology and excluded middle the axioms of ZFC
besides regularity are actually theorems of the null axiom theory. In
that way, the results provable in ZFC - R are results provable in the
null axiom theory, with the added semantic that the theory is complete.
sci.physics_20050531.rtf:In real numbers with a scalar infinitesimal,
Vitali's result does not hold and all sets of numbers are measurable.
ZF is inconsistent.
sci.physics_20050608.rtf:The universe is infinite where functions
between physical objects are physical objects, and infinite sets are
equivalent. ZF is inconsistent, etcetera.
sci.physics_20050618_b.rtf:ZF is inconsistent. The universe is
infinite and infinite sets are equivalent.

.

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