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Topic: Few questions on forcing, large cardinals
Replies: 2   Last Post: Mar 28, 2013 1:42 AM

 ross.finlayson@gmail.com Posts: 2,720 Registered: 2/15/09
Re: Few questions on forcing, large cardinals
Posted: Mar 28, 2013 1:42 AM

On Mar 26, 9:42 pm, fom <fomJ...@nyms.net> wrote:
> On 3/26/2013 9:35 PM, Ross A. Finlayson wrote:
>

> > On Mar 24, 10:20 am, "Ross A. Finlayson" <ross.finlay...@gmail.com>
> > wrote:

> >> On Mar 23, 11:42 pm, fom <fomJ...@nyms.net> wrote:
>
> >>> On 3/23/2013 10:23 PM, Ross A. Finlayson wrote:
>
> >>>> On Mar 23, 5:01 pm, fom <fomJ...@nyms.net> wrote:
> >>>>> On 3/23/2013 6:46 PM, Ross A. Finlayson wrote:
>
> >>>>>> On Mar 23, 3:34 pm, fom <fomJ...@nyms.net> wrote:
> >>>>>>> On 3/23/2013 5:09 PM, Ross A. Finlayson wrote:
>
> >>>>>>>> On Mar 23, 2:44 pm, fom <fomJ...@nyms.net> wrote:
> >>>>>>>>> On 3/23/2013 4:34 PM, Ross A. Finlayson wrote:
>
> >>>>>>>>>> In a sense, infinity _is_ the numbers.  Start from even more
> >>>>>>>>>> fundamental objects than natural numbers as elements.  Like the
> >>>>>>>>>> numbers, they are as different as they can be and as same as they can
> >>>>>>>>>> be, where they are each different in not being any other and each same
> >>>>>>>>>> in being defined by that difference.  There's no stop to that, it's
> >>>>>>>>>> gone on, forever.  Then, in a way like when you look into the void, it
> >>>>>>>>>> looks into you

>
> >>>>>>>>> Is this your way of saying that if you look
> >>>>>>>>> into the void, you and the void become one?

>
> >>>>>>>>>http://en.wikipedia.org/wiki/Cantor-Bernstein-Schroeder_theorem
>
> >>>>>>>>> Just kiddding....
>
> >>>>>>>>> I think Cantor would appreciate your sentiment that
> >>>>>>>>> the numbers of Cantor's paradise are more fundamental
> >>>>>>>>> than those of Kronecker's torment.

>
> >>>>>>>> I wouldn't say that infinity, even in the numbers, is either of those
> >>>>>>>> things.  In ZF, Infinity is _axiomatized_ to be an inductive set, and
> >>>>>>>> a well-founded/regular one, that's not a given.  Calling that the
> >>>>>>>> universe, Russell's comment is that it would contain itself.

>
> >>>>>>>> There's a case for induction, as it were, that each case exists.  Then
> >>>>>>>> it is to be of deduction, not fiat by axiomatization, from simple
> >>>>>>>> principles of constancy and variety, the continuum.

>
> >>>>>>>> In a theory with sets as primary objects, a set theory and a pure set
> >>>>>>>> theory, numbers would be very rich objects indeed, as not just
> >>>>>>>> individual elements by their elements, but all relations of numbers.
> >>>>>>>> Set theory (well-founded, as it were, regular or that objects are
> >>>>>>>> transitively closed) is at once over-simplification, to talk about
> >>>>>>>> anything besides sets, and over-complexification, to talk about itself
> >>>>>>>> when any universal statement is in the meta.

>
> >>>>>>>> There are no numbers in a pure set theory.  To call the natural
> >>>>>>>> integers a set, it contains only numbers, for the Platonists: elements
> >>>>>>>> of the structure, of numbers, as:  none exist in a void.

>
> >>>>>>> It is odd.  In some sense, modern mathematics actually
> >>>>>>> treats its objects as urelements relative to set theory.
> >>>>>>> Looking at Hilbert, he makes statements whereby his formalism
> >>>>>>> is intended to supersede the class-based constructions of
> >>>>>>> Dedekind.

>
> >>>>>>> Your frank statement that a set is not a number reflects
> >>>>>>> that sentiment.

>
> >>>>>> Particular finite sets are called ordinals, set-theoretic operations
> >>>>>> on them are defined that give the same results as Presburger/Peano
> >>>>>> arithmetic of the natural integers. The negative integers aren't
> >>>>>> simply the complement as in finite-word-width machine arithmetic, but
> >>>>>> again simple enough operations on sets (with the only ur-element being
> >>>>>> the empty set) give a "model" of the integers.  Rationals are defined
> >>>>>> simply enough as equivalence classes over any pairs of integers,
> >>>>>> besides zeros, the reals then see the Least Upper Bound as axiom.
> >>>>>> These are all to match number-theoretic features, and largely suffice
> >>>>>> for integers and rational numbers, but not so obviously do sets
> >>>>>> suffice to represent thusly elements (and all of) the continuum of
> >>>>>> real numbers.

>
> >>>>>> Then, though, to call the empty set the number zero:  wouldn't that be
> >>>>>> the number zero wherever there's an empty set?  Building upwards to
> >>>>>> have particular sets for each of of the finite integers:  then to
> >>>>>> build the numbers as sets, is to build all the relations of the
> >>>>>> numbers as sets, not just as to a set-theoretic model of only that set
> >>>>>> of numbers' operations:  but of all instances, besides the schema.
> >>>>>> Where the ur-element is any thing, it so implies all other  things,
> >>>>>> and is so implied.  The collection and aggregates of sets or
> >>>>>> categorization or refinement of types or partition or bounding of
> >>>>>> division, are all of the same corpus.

>
> >>>>>> Here back to the questions as above:
>
> >>>>>> 1) is not forcing simply transfinite Dirichlet box?
>
> >>>>> I am not sure what you mean by this.
>
> >>>>> However, forcing might be better thought of as comparable
> >>>>> to Euclid's proof that there is no greatest prime.

>
> >>>>>> 2) are there any results due transfinite cardinals, not of transfinite
> >>>>>> cardinals?

>
> >>>>> The Borel hierarchy is defined in terms of the first
> >>>>> uncountable ordinal.  Hence, results in descriptive
> >>>>> set theory that depend on that definition may count.

>
> >>>>> I do not have enough knowledge of that branch of
> >>>>> study to comment further.

>
> >>>>>> 3) is not an irregular model of ZF non-well-founded?
>
> >>>>> What is your definition of irregular?
>
> >>>>>> 4) does not a model of ZF contain itself?
>
> >>>>> There are relativizations of models.  So, one question
> >>>>> in set theory is whether

>
> >>>>> HOD=HOD^HOD
>
> >>>>> where HOD are the hereditarily ordinal-defined
> >>>>> sets and HOD^HOD is HOD relativized within itself.

>
> >>>>> In this sense, models may have representations
> >>>>> within themselves.  But, once again, expertise
> >>>>> is lacking here.

>
> >>>>>> 5) is ZF not a model of itself?
>
> >>>>> ZF is an axiomatization.  The question is not
> >>>>> well-construed.

>
> >>>> 1) Forcing might be better thought of as that there's an ordinal
> >>>> greater than all ordinals.

>
> >>> That is not quite what you should take from my statement.
>
> >>> What determines that a forcing model is "bigger" than its
> >>> ground model is that for the set of "names" in the forcing
> >>> language, there is one name for every object in the ground
> >>> model and one name for which there is no such object.

>
> >>> The ordinals are "special" as the "spine" of the model.  They
> >>> can be collapsed onto lower ordinals as given in the ground
> >>> model.  But, it is probably not correct to view the manipulations
> >>> in forcing as adding ordinals at the top of the hierarchy.

>
> >>>> 2) That may as well be stated as that the Borel hierarchy is in terms
> >>>> of ranks of countable ordinals.

>
> >>> Probably a better statement.
>
> >>>> 3) An irregular model is not well-founded.
>
> >>> I passed on this one as I recall.
>
> >>>> 4) There's a relativization of ZF down to the countable and even to
> >>>> omega.  Then that a model of HOD, hereditarily ordinally-definable,
> >>>> isn't itself HOD is again:  Russell's "paradox".

>
> >>> I am not certain that the nature and existence
> >>> of countable models should be considered as having
> >>> the same sense as relativization.

>
> >>> Simply put, relativization involves reinterpretation
> >>> of quantified formulas in the sense of

>
> >>> [phi(x)]^M for some class M
>
> >>> Ex(phi(x)) becomes Ex(xeM /\ phi(x))
>
> >>> Ax(phi(x)) becomes Ax(xeM -> phi(x))
>
> >>> Since classes are associated with the grammatical forms
> >>> of naive set theory,

>
> >>> M(z)={z|psi(z)}
>
> >>> One can also speak of a different sense of relativization.
> >>> Let k be a parameter.  Relative to the parameter k, let two
> >>> classes be given by

>
> >>> {z|M(z,k)}
>
> >>> {<p,q>|peM, qeM, /\ E(p,q,k)}
>
> >>> In this case, if E satisfies the axioms when interpreted
> >>> as the membership relation over M, then

>
> >>> <M,E>
>
> >>> is a model of set theory.  Relativization in this case
> >>> is denoted with

>
> >>> [phi(x)]^<M,E>
>
> >>> for a given formula.
>
> >>> Ex(phi(x)) becomes Ex(xeM /\ phi(x)^<M,E>)
>
> >>> Ax(phi(x)) becomes Ax(xeM -> phi(x)^<M,E>)
>
> >>> The additional complexity of the formulas indicates that
> >>> the membership relation is reinterpreted by a definite
> >>> class specification.

>
> >>>> 5) ZF as theory is all its theorems.  That as all the sets that don't
> >>>> contain themselves, again via Russell, does.  I'll agree it's a direct
> >>>> question as to the content of ZF, simply construed.

>
> >>> Pass.
>
> >> An analysis pointing out perceived deficiencies in forcing:
>
>
> >> Basically forcing "scales" the universe then as to where transfinite
> >> induction (over transfinite ordinals) is through all of them.  The
> >> difference between this and plain transfinite induction is as to the
> >> difference between induction and transfinite induction.  Then it is as
> >> to transfinite Dirichlet box.

>
> >> Borel's hierarchy in terms of finite languages and computability is as
> >> to countable ordinals and even more simply polynomials in omega.

>
> >> A model of ZF _is_ ill-founded.  Whether in "naive" set theory or not,
> >> with its concomitant paradoxes of Russell, Cantor, and Burali-Forti as
> >> are well known, it's in ZF:  or not, and as a model of ZF, includes
> >> all theorems of ZF, and then some, else ZF could model itself.  (Which
> >> it doesn't, directly.)

>
> >> So,
>
> >> a) model theory is in the meta, and in naive set theory
>
> >> b) forcing introduces elements that would exist in ZF that have
> >> properties of elements that wouldn't exist in ZF

>
> >> c) large cardinals presuppose a universe (and aren't regular sets nor
> >> cardinals)

>
> >> d) models of regular theories are irregular (as are large cardinals)
>
> >> e) transfinite ordinals and polynomials in w support transfinite
> >> Dirichlet box

>
> >> f) there are non-trivial elementary embeddings V -> V, else in pure
> >> sets no elements of structure with models (under isomorphism) are
> >> primary, and all are concrete/constructible

>
> >> g) there's a non-trivial elementary embedding V -> V, v -> V\v
>
> >> h) the paradoxes of naive set theory with regards to HOD transitive
> >> closure as regularity aren't resolved in their demurral

>
> > So:  what theory are large cardinals in, and, is it consistent, in
> > that theory, to force them into ZF?  Because, if they're independent
> > of ZF, the axioms establishing their definition, large cardinals,
> > isn't there a theory, with them?

>
> > One might aver to NFU (New Foundations with Universes, vis-a-vis, New
> > Foundations with Ur-Elements) and then having large "sets", compared
> > to the universe (or domain of discourse, as it were) of ZF(C), but
> > then they're classes, and there's only one proper class.  And, in a
> > pure set theory:  it's a set.

>
> For my part Ross, I do not want to misdirect
> you.  It has been a very long time since I
> had been studying set theory closely enough
> to talk about large cardinals.
>
> My primary investigations had been with respect
> to how the sign of equality is treated within the
> theory.  The model theory of forcing models is
> directly relevant to that understanding.  The
> specifics of large cardinals and their axioms
> are not.
>
> I hope I have been of some help to you.

Dichotomy is a basic reasoning. Everything or rather any thing is
defined by what it isn't. The "context", of a thing or assembly, is
basically the universe setminus the thing: v ~ V\v.

Western tradition has then Kant, Hegel, and Heidegger as the
philosophical leaders of the concepts of "Nothing", and "Being", or
existence, then for Heidegger also "Time". The Universal is often in
passage ascribed to the divinity, but it's as simple to start with

Then in terms of identity or tautology, equality, even the very
concept of equality is seen to vary: for all that n object evinces as
identity, and all the things that come together as it as tautology.

Then that is rather airy but the basic consideration of equality
consequent all structure has a variety of reasonings to so define it,
of basically sameness and convergence.

Forcing is a reflection that via properties of the universe, there is
as much tendency from greatest to smallest as least to most, here of
the definition of ordinals from zero, in the universe, here of
ordinals and as such: ordinal. Then, where technical philosophy
founds logic founds mathematics, forcing in logic is that there is a
universe with its self-similar and self-containing properties (a la
Kant's Ding-an-Sich or thing-in-itself) of here set theory: a
universe. Forcing is an axiomatization, of the maximal, for a theory
that the maximal would make inconsistent.

Then in the paraconsistent one can find simple enough that the
_direct_ results of ZF are still so, but that _eventual_ or _extended_
results are not, when _direct_ results of as intuitive a notion as
that there is an "All", and consequences that there is, leave the
happy fiction that ZF could be consistent when its own model, and so
modeling it, would invalidate it. Yet, specific to that notion is
that as well is justified at least as reasonable (given reason)
grounds that universe-like structures and deduction of their existence
and the import of all's existence has: not just an interminable
cascade of apologizing to the universe with ever greater and never
great non-set cardinals in set theory, but here theory, and set
theory, accommodating the dialetheic.

All and Nothing, Nothing and All: same.

So: forcing is an axiomatization: of a universe-like set. And, it's
a contrivance: for theory and its universe.

A theory: is the theory of everything.

Regards,

Ross Finlayson