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