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:

>

>

>

>

>

>

>

>

>

> > 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-()_