Re: Skolem's 'Paradox'

Daryl McCullough wrote:
> David C. Ullrich says...
> >
> >On Fri, 16 Sep 2005 16:20:24 +0300, Aatu Koskensilta
> ><aatu.koskensilta@xxxxxxxxx> wrote:
>
> >>What I mean is simply that using perfectly ordinary mathematical
> >>principles formalizable in e.g. ZFC it's provable that the existence of
> >>an uncountable set is not equivalent to the Sigma_1 statement that there
> >>exists a derivation of this fact.
> >
> >I don't know what you mean by this...
>
> You don't know what it means to say that two statements
> are inequivalent? Well, if there is a model that makes
> one statement true and the other statement false, then
> they are inequivalent. That is certainly the case with
> "There exists an uncountable set" and "ZFC proves that
> there exists an uncountable set". There are models in
> which the first is false, but the second is true.
>
> >I can't imagine how a statement about sets could be inherently
> >meaningful unless we assume that there are real sets out there
> >and that statements about sets are statements about those
> >real sets. What other sort of inherent meaning could a statement
> >about sets have?
>
> Well, any *particular* theorem of set theory only uses a finite
> number of set-theoretic principles, so we can make sense of it
> in any structure for which those principles apply. We don't have
> to assume that there is one, unique, Platonic universe. The biggest
> example is Cantor's proof of the uncountability of the set of
> reals. It could be formalized in ZFC, but it actually
> has a much broader significance. For instance, if we restrict ourselves
> to computable functions, there is no computable enumeration of the
> computable reals.
>
> --
> Daryl McCullough
> Ithaca, NY

comp.lang.c_20050309.rtf:Heh, Not Con(ZF).
sci.logic_20041116.rtf:You say that there are no paradoxes with
Cantorian theory, and that I am supposed to show ZF inconsistent. It
is uncertain whether that will happen, I'm thinking about it. You
mention Russell's paradox, you probably didn't catch my recent twist on
it validating my zero vis-a-vis infinity dual representation. There
would be Burali-Forti, that's similar to what would be Cantor's
paradox, that the set of all sets would indeed be its own powerset.
With the above notion about ordinals, they are combined into one single
issue, where Ord is less than nothing, and it itself is its own
successor, which would again perhaps be the empty set, which is the
opposite and same as Ord. You might notice that that is the statement
of the singular proper class and exluded excluded middle. About
Skolem, in this thread again you see what there is about Skolem in ZF:
backpedalling and handwaving, insensate to the infinite character,
nature, of two infinite sets.
sci.logic_20050314.rtf:Bhupinder, what do you mean that ZF is
inconsistent? Are you basically saying "Not Con(ZF)", and all forcing
results based upon Con(ZF) are not necessarily thus co-consistent?
sci.logic_20050314.rtf:That's where "forcing" means statements along
the lines of "if ZFC is consistent, then IST is consistent." They're
not necessarily meaning that "if and only if ZFC is consistent, then
IST is consistent", just comparing their relative strengths. Instead,
the notion of the reverse mathematics is that given Not Con(ZF) not
every single statement ever attributed to Zermelo-Fraenkel Set Theory
is doomed. Many are.
sci.logic_20050314.rtf:Are you trying to tell me ZFC is inconsistent?
I think it is, because of regularity versus unrestricted comprehension,
etcetera. I think your stated justifications that ZF is inconsistent
are not very concrete, in the sense of non-abstractness,
antimetatheoreticalicity.
sci.logic_20050315.rtf:It seems then that you are not indicting ZF, but
any theory.
sci.logic_20050315.rtf:I have had what may be similar concerns, they
have led me to a theory with no non-logical axioms, where everything
resolves to a tautology, and the non-logical axioms that comprise ZFC
besides regularity are considered theorems instead of axioms.
sci.logic_20050315.rtf:As well, results in ZFC not dependent on
regularity would be results of the axiom-free theory. I say theory
instead of set theory, for the consideration of arbitrary primary
objects, where it is as well a set theory where sets are the primary
objects.
sci.logic_20050316.rtf:I think NBG, for von Neumann-Bernays-Goedel,
implicitly infinitely axiomatized ZFC, is to ZFC as hyperreal numbers
are to real numbers.
sci.logic_20050316.rtf:I enjoy reading your posts because you're
sincere and well-informed, and seem willing to consider
non-conventional viewpoints, which is necessary for progress. While
that is so, if you say ZF is inconsistent the immediate conclusion of
that statement is that ZF is inconsistent.
sci.logic_20050323_b.rtf:At some point I arrived at another notion: if
there were no axioms in the theory, then there would not need to be a
Goedel sentence for any axioms, as there were none. In distinguishing
the first-order predicate calculus tautologies which are called the
logical axioms, truth tables, de Morgan, from the what are called
non-logical or proper axioms, eg the seven axioms of ZF set theory, I
determined that Goedelian incompleteness only applies to non-logical
axioms.
sci.logic_20050323_b.rtf:Then, I consider that if I want to prove
results currently proven by ZF set theory, then it would be equivalent
to having some of the axioms of ZF being theorems of basically nothing.

sci.logic_20050323_b.rtf:I think that there's a theory with no
non-logical axioms, where the axioms of ZFC besides foundation are
theorems, a set theory with dually minimal and maximal ur-element that
self-referentially generates all elements, that has the expressive
strength of a first and higher order logic as a first order logic.
sci.logic_20050325_b.rtf:Please elaborate your informed opinions on
Bhupinder's assertion that ZF is inconsistent, or others at context in
this discussion thread, or don't. Thank you in advance.
sci.logic_20050326.rtf:In a theory with no non-logical axioms, how can
I say what is or isn't a set? It's simple, anything's a set,
everything's a set, there is no restriction on the set construction.
Thus, it follows that any set you define via, say, the Zermelo-Fraenkel
set theory's nine (not seven) axioms, exists, as well, so do others,
then the logical axioms still apply, and it's still exempt from
Goedelian incompleteness. Where you might use an axiom in ZF to assert
the existence of a given set, in the axiom free theory, the set already
exists, you just use a predicate on the domain of all sets that
represents that ZF "axiom", which is actually a low-level quantifier.
sci.logic_20050326.rtf:Does the generic extension of N contain any
elements not in N? (No, it doesn't, that's a pointed question about
the inconsistency of inequivalency of infinite sets, to prod
conformists to the conclusion that ZF is inadequate.) How about them
ordinals? Heh heh heh heh. "Class of all classes, set of all sets,
you know the rest." The numbers are the trees.
sci.logic_20050327.rtf:In Zermelo-Fraenkel set theory, all sets are
composites of the empty set, or rather, it is said that the empty set
is the minimal element of that theory. In a theory without the axiom
of foundation, the set in its construction might refer to itself, its
transitive closure is not regular (in ref. to trivial minimal element
regularity), via unrestricted comprehension those are as well sets.
Anyways, any set you can construct in ZF from the empty set as minimal
element is available as an element of the universal set of pure sets.
sci.logic_20050404.rtf:I'm interested in the concept that all sets (of
real numbers) are Lebesgue-measurable, but I think that can only happen
when the reals contain infinitesimals, because that leads to
reinterpretation of Vitali, that there exist non-measurable sets. I'm
trying to figure out some reasonings behind that and concrete
analytical results to do with discrete point-sets. I don't think AC is
falsifiable, but then again I think it's a theorem of the null-axiom
theory, along with the other non-logical/proper axioms of ZF besides
regularity.
sci.logic_20050406.rtf:A lot of people base their proofs upon
statements that ZF is consistent. Many of those theorems are still
correct when ZF is determined to be inconsistent.
sci.logic_20050410.rtf:The problem with your theory is you try to
define non-logical axioms. In my theory, there are no non-logical
axioms, and what are called the axioms of ZF except for regularity are
theorems, of the first-order axiom free theory, thus that's not a
problem. It is not necessary to axiomatize, and correspondingly
necessary to not axiomatize, to derive overall truth. Goedel shows you
that attempting to impose axioms on a logical theory leads to
incompleteness, and thus inconsistency.
sci.logic_20050420.rtf:That is to say, for each of the non-logical
axioms of ZF, for example, besides regularity, anything that could
possibly be a set exists, that the set has the properties ascribed to
them by those non-logical axioms is reasonable in that assuming them
so, to be sets, does not lead to contradiction.
sci.logic_20050424.rtf:Anyways, we don't have to worry about being
morons to some higher power, it doesn't matter, the null axiom theory,
and all that we can infer from it including ZF-R, is plenty for
mathematical statement.
sci.logic_20050426.rtf:To reach a point, if ZF is inconsistent with
regards to any truth, then it is not acceptable as the foundation of
mathematical logic. ZF, a set of axioms describing what a set is,
represents the combined, sincere, effort of many and is a powerful and
yes, even living monument. While that is so, it can only be seen as a
vehicle for exploring truth values as part of a more comprehensive
system.
sci.logic_20050427.rtf:In ZF, and I'm not a particular expert, in ZF,
many ask "where's my universal set", and so a standard extension of ZF
is "ZF with classes". I'm against ZF with classes, it illustrates in a
way the slippery slope fallacy, something like the "set of all sets"
brings in a systemic inconsistency to ZF, largely for reasons of the
powerset bijection result, so a new collection called a "class" is used
instead of "set". They might then ask: where's the class of all
classes? At that point, promoters of ZF with classes vary in respose
among catatonia, abject denial, barking, handwaving, and subtle and
not-so-subtle hints that a theory without the axiom of regularity might
better suit the learner's proclivity towards rationality and the
importance of reason within studies of mathematical logic in deep
foundations.
sci.logic_20050427.rtf:I prefer the Axiom-Free Set Theory, where
theorems of ZF not using regularity are theorems, where each infinite
set is irregular or with dually represented elements, as the ur-element
is the singular proper class, dually minimal and maximal often as the
empty set or universal set, and the computer is the anticomputer:
through the looking glass.
sci.logic_20050427.rtf:I hope not to be detracting from the specific
technical considerations of arithmetic in ZF. There is much to it.
sci.logic_20050501.rtf:This post doesn't have much to do with
arithmetic in ZF, except V = L. It seems that arithmetic as here
discussed is about, to some extent, bijecting the naturals or ordinals
to some set of formulae. I think ZF is inconsistent, because of
regularity, and that instead the universe of discourse is basically
comprised of naturals, or alternatively ordinals with the cumulative
hierarchy. That's a similar notion as to how a computer register, a
finite sequence of zeros and ones, can be interpreted as signed or
unsigned number, where for the infinite ordinals there is exactly one,
in a specified way, finite ordinal, and that that infinite ordinals is
basically the negation or additive inverse of that finite ordinal.
sci.logic_20050501.rtf:In ZF there is the Burali-Forti paradox, that
the order type of ordinals would be an ordinal, confoundingly, there's
a paradox in ZF with respect to, basically, arithmetic. That's the
same problem as that infinity+1 = infinity, and Cantor's paradox that
the set of all sets is its own powerset, and using flexible definitions
Russell, the Liar, etcetera, and a solution has to do with the dual
representation of the minimal and maximal.
sci.logic_20050501_b.rtf:What's the class of all classes? It would be
similar to the set of all sets. Maybe you think those are just
absurdities, if other theories address them and ZF is dumb, in the
sense of being mute, then ZF is missing some expressive power.
sci.logic_20050502.rtf:If that does not sit well with you, then I
encourage you to address the other points there that illustrate ZF's
inconsistency, basically because of irregularity.
sci.logic_20050502.rtf:If "Not Con(ZF)", that is, ZF is inconsistent,
then all the forcing results based upon "Con(ZF)" would reflect that.
sci.logic_20050504.rtf:On sci.logic we are discussing arithmetic in ZF
and got to discussion of the Domain Principle. Your book "Beyond the
Limits of Thought" was quoted, I wonder if you might have something to
say, on sci.logic on the thread "arithmetic in ZF".
sci.logic_20050504.rtf:Where I'm coming from, I'm an amateur logician
who advocates a theory free of non-logical axioms, and think that that
theory can thus be Goedelianly complete, and the axioms of set
structure of ZFC minus the regularity axiom are theorems of the Null
Axiom or Axiom-Free theory, which is a theory with sets, numbers, or
physical or geometric objects as primary objects, at once.
sci.logic_20050708.rtf:ZF is inconsistent.
sci.logic_2005071.rtf:ZF is inconsistent. I don't think separation
implies Russell, the paradox, but if so the objective of the ur-element
is Russell's resolution, not to mention those of Burali-Forti, Cantor,
and the Liar, where I have discussed resolutions to Russell before.
sci.logic_2005071.rtf:ZF's being inconsistent is a plain statement, and
where it might be seen true for variously many of these statements
about mechanical quantification, and qualified quantization, for which
I can probably manufacture an explanation, for where it might be seen
true, gather the true statements and infer from them.
sci.logic_20050710.rtf:ZF is inconsistent, because something like X:
for any x, x E X, is simple to define, and quantification involves a
choice function over U.
sci.logic_20050710.rtf:ZF is inconsistent. Can you get past that?
sci.logic_20050710_b.rtf:A model is to a theory as a class is to a set.
What is the set of all sets? What is the class of all classes or
group of all groups or collection of all collections ad infinitum? ZF
is inconsistent. The universe is infinite and infinite sets are
equivalent.

.

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