Date: Dec 9, 2012 11:34 PM
Author: 
Subject: Re: fom - 01 - preface

On Dec 9, 8:28 pm, "Ross A. Finlayson" <ross.finlay...@gmail.com>
wrote:
> On Dec 9, 6:36 pm, fom <fomJ...@nyms.net> wrote:
>
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> > On 12/9/2012 2:05 PM, Ross A. Finlayson wrote:
>
> > > On Dec 8, 11:21 pm, fom <fomJ...@nyms.net> wrote:
> > >> On 12/8/2012 10:50 PM, Ross A. Finlayson wrote:
>
> > <snip>
>
> > >    http://en.wikipedia.org/wiki/Forcing_(mathematics)
>
> > Thank you for the web pages.  I have not had
> > an internet account for some time and never
> > thought to see what was written on wikipedia

>
> > So, this is from that webpage,
>
> > CONSISTENCY
>
> > The above can be summarized by saying the
> > fundamental consistency result is that given a
> > forcing poset P, we may assume that there exists
> > a generic filter G, not in the universe V, such
> > that V[G] is again a set theoretic universe,
> > modelling ZFC. Furthermore, all truths in V[G] can
> > be reduced to truths in V regarding the
> > forcing relation.

>
> > ====
>
> > This is completely legitimate.  Relative to the deductive
> > calculus, the consistency spoken about here justifies
> > investigating the independence of the axioms from one
> > another.  These questions are wrapped in the completeness
> > and soundness of the deductive system.

>
> > Just before Jech makes precisely the same observation,
>
> > he writes:
>
> > "To conclude this section, I shall briefly
> > touch the question of existence of generic
> > sets.  Of course, if M is countable,  then
> > every PeM has an M-generic filter.  On the
> > other hand, in general one can at least prove
> > something like: Given PeM, then it is consistent
> > that P has an M-generic filter.  Even when
> > formulated correctly, one has to be careful
> > to exclude the obvious false cases (like
> > when M=the universal class).  In fact, it
> > seems necessary to formulate the theorem only
> > for M having an absolute definition (like
> > M=the consructible universe.)

>
> > In other words, if V=L is true, generic class
> > model that take L as the ground model are
> > illegitimate candidates for consideration
> > of foundational questions.

>
> > Suppose one asks, why might it be
> > consistent to assume the existence of a
> > filter converging outside of the model?
> > My answer was:

>
> > Definition of proper part:
> > AxAy(xcy <-> (Az(ycz -> xcz) /\ Ez(xcz /\ -ycz)))

>
> > Definition of membership:
> > AxAy(xey <-> (Az(ycz -> xez) /\ Ez(xez /\ -ycz)))

>
> > In other words, membership is essential in that
> > its own occurrence is necessary to its own
> > definition, but, membership is not the primitive
> > of the language.  The proper subset relation
> > can be characterized with a self-defining
> > syntax.  Thus, forcing methods are possible
> > because the part relation can be manipulated.

>
> > This is exactly how Jech gives his simplest
> > description:

>
> > "... to introduce "imaginary" sets and
> > adjoin an imaginary set G to the universe
> > to obtain an extension V[G] that is a
> > model of ZFC."

>
> > As for those definitions, while Russell's mistake
> > was becoming the ontology of mathematical logic,
> > there had been others thinking differently.  In
> > "Cantorian Set Theory and Limitation of Size"
> > Michael Hallet discusses the 1906 position taken
> > by Young and Young:

>
> > "After giving Cantor's 1882 definition
> > of a set the Youngs [1906], p.146,
> > comment: 'Here Cantor emphasizes the
> > fact that in any logical and more
> > especially mathematical thinking, we must
> > confine our ideas to some particular
> > field ...'  They then underline (though
> > they do not refer explicitly to the
> > paradoxes) the danger associated with
> > arbitrary 'conceptual spheres' or
> > 'fundamental regions' as they call
> > them: namely, that there is a temptation
> > to allow the fundamental region itself
> > to be a set. Certainly, this would be
> > logically convenient and many of the
> > natural 'fundamental regions' are treated
> > as sets in other contexts.  But, say
> > the Youngs:

>
> > '... it should be postulated that the
> > fundamental region itself is not to be
> > regarded as a set unless there is
> > another fundamental region containing
> > the first fundamental region together
> > with other elements.'

>
> > This certainly characterizes the
> > situation for any hierarchy of set
> > models.  But, with regard to the universe,
> > it means something like this:

>
> > Definition of top:
> > Ax(x=V() <-> Ay(-(ycx <-> y=x)))

>
> > Assumption of top:
> > ExAy(-(ycx <-> y=x))

>
> > where the fundamental region has proper
> > parts, and, only proper parts can be
> > sets.  One can still speak of classes
> > as grammatical forms associated with
> > formulas (which is the formal way of
> > speaking as "a ZFCer" on Sunday when
> > every mathematician is a "formalist").
> > But, the universe is not defined in
> > terms of membership and is not technically
> > a class in terms of second-order identity
> > by extensions.

>
> > The real problem with forcing is that
> > it is somewhat dishonest.  The partial
> > order in the explanation given in the
> > web page,

>
> > A forcing poset is an ordered triple
> >   (P, ?, 1)

>
> > where "?" is a preorder on P that satisfies following
> > splitting condition:

>
> > For all p ? P, there are q, r ? P such that q, r ? p with no s ? P such
> > that s ? q, r

>
> > and 1 is a largest element, that is:
>
> >   p ? 1 for all p ? P,.
>
> > Members of P are called conditions. One reads
>
> >   p ? q
>
> > as
>
> >   p is stronger than q.
>
> > seems innocent.  But, look at how it is applied,
>
> > The simplest nontrivial forcing poset is ( Fin(?,2), ?, 0 ),
> > the finite partial functions from ? to 2={0,1} under
> > reverse inclusion

>
> > Now, suppose I assume (for definiteness) -(V=L).
>
> > Next, I take a partial order bounded below.  It either
> > has a bottom or it is like a forest of trees.  That is,
> > when I reverse the inclusion relation, I have a
> > direction that is guaranteed to converge.  Moreover,
> > I have stipulated a further condition matching the
> > definition of convergent filterbases that
> > characterize the neighborhood base definitions
> > of a topology.  In other words, I have taken the
> > Cantor nested set theorem with vanishing diameter
> > and pointed it to heaven.

>
> > Is it any surprise that I suddenly find something
> > outside of what I have assumed to be partial?

>
> > You cannot do that with
>
> > Definition of top:
> > Ax(x=V() <-> Ay(-(ycx <-> y=x)))

>
> > Assumption of top:
> > ExAy(-(ycx <-> y=x))

>
> > Just like with the coatoms one element away from the
> > whole set in the ordering of its power set, all
> > notions of convergence end at the universe.

>
> > As far as the consistency aspects of forcing, the
> > best explanation I have found is in Rosser.  He
> > compares it to something that has nothing to do
> > with partial orders.  He compares it with Cayley's
> > demonstration that the axiom of parallels is
> > independent:

>
> > "... That is, he began by assuming
> > the truth of postulate 5.   As we
> > mentioned in Section B, we begin
> > similarly by assuming AC and GCH,
> > preparatory to proving their independence.
> > This is not self-defeating, because
> > we then modify somewhat the notions
> > of 'set,' 'well-ordering,' 'cardinal
> > number,' etc. just as Cayley introduced
> > modified 'points,' 'lines,' 'distances,'
> > etc.  Indeed, we need the presence of
> > powerful principles such as AC and
> > GCH to be able to prove that our
> > unorthodox 'sets,' etc. have the
> > desired properties, just as Cayley needed
> > Postulate 5 to prove the properties of
> > his 'points,' 'lines,' 'distances,' etc.

>
> > "Note further that Cayley did not
> > modify his 'points,' 'lines,' etc. very
> > much.  After all, he wished to invalidate
> > only Postulate 5.  Thus his 'points' were
> > actual points;  he just didn't use all
> > possible points."

>
> > The last sentence strongly suggests that
> > partiality is a fundamental requirement
> > for independence proofs in order to manipulate
> > an existing system of definitions to show that
> > the new definitions yield a system satisfying
> > the assumptions that are held invariant.

>
> Yes, there is quite the bit more on the Internet these days with
> regards to the foundational than there was even ten years ago, and
> Wikipedia has grown to be a central resource, I'm a supporter.
>
> http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=221287http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=300611
>
> These references to Cohen's invention of forcing to show the
> independence of the Continuum Hypothesis from the theory of ZF
> complement, for example, his book "Set Theory and the Continuum
> Hypothesis", of which I don't have the resources to send everyone a
> copy.
>
> http://groups.google.com/groups/search?hl=en&as_q=generic+standard+mo...
>
> You describe that if V=L that it would be false to force an extension
> to the model as there's nothing outside L, but M having an absolute
> definition as L would see non-constructible elements outside M, were
> it not L and there are none.  Given that combined with L-S, isn't the
> very method of forcing dependent on that V =/= L?  Then, there is no
> definable well-ordering of the reals as a result of Feferman, with V =
> L.  Yet, with V = L, wouldn't one not even need forcing, with up/down
> L-S?
>
> I think it's relevant to examine the ordering relations on Cohen's
> development, for where M is and isn't maximal, here for that it
> preserves its properties, in the inverse, in the inversion.
>
> Then, with quite a shift and to the consideration of the fundamental,
> primary, or ur- elements of our theories (of sets, numbers, aspects of
> geometry, theorems, and etc.), there is a strong underpinning for the
> foundations from Kant and Hegel, Frege, and to an extent Wittgenstein,
> then to Heidegger, in what would be the genera or noumena, with
> Euclid's geometry and Cantor's sets and Archimedes' numbers, and, say,
> Goedel's theorems.  The technical philosophy offers a strong
> counterpoint of these "conceptual spheres" or "fundamental regions" as
> a conceptual sphere and fundamental region.
>
> I'm all for the construction of von Neumann's ordinals for regular
> ordinals, and there are various equivalent constructions for general
> purposes, they're regular and always have something outside  (ZF
> perfectly models all finite bounded combinatorics completely).  But, a
> universe or the universe we inhabit has all things, with Ax x=x => x e
> U, including U e U, that there's a universe or there isn't generally
> identity (nor would there be for that matter distinctness).  Then,
> some would go so far as that, Skolemizing that to the countable, N e
> N, the natural integers themselves, simply as an anti-foundational or
> ill-founded ordinal, contain themselves, basically Russellizing the
> proto-typical "least" infinite (that it has, not that it hasn't, the
> Russell element).  Of course I don't say this is so in ZF except in as
> to where Goedel proves in ZF there are true facts about the objects in
> ZF, not in ZF, then getting in as to where the theory is its theorems,
> to the Ding-an-Sich, Kant's Thing-in-Itself.
>
> No classes in set theory: no models in theory:  set-theoretic theory.
>
> http://en.wikipedia.org/wiki/Theory
>
> Theory is its theorems.
>
> I hope you find that apropos.  Basically the notion of "foundation of
> mathematics" encompasses the technical philosophy, the logic, the
> definitions, the constructible in the symbolically abstract, for among
> its goals:  the applied.
>
> Regards,
>
> Ross Finlayson

Now the man is the idiot on earth. Forgive me Ross.Musatov0-()_