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Re: fom  01  preface
Posted:
Dec 9, 2012 11:34 PM


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 Mgeneric filter. On the > > other hand, in general one can at least prove > > something like: Given PeM, then it is consistent > > that P has an Mgeneric 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 selfdefining > > 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 secondorder 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 selfdefeating, because > > we then modify somewhat the notions > > of 'set,' 'wellordering,' '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 nonconstructible elements outside M, were > it not L and there are none. Given that combined with LS, isn't the > very method of forcing dependent on that V =/= L? Then, there is no > definable wellordering 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 > LS? > > 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 antifoundational or > illfounded ordinal, contain themselves, basically Russellizing the > prototypical "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 DinganSich, Kant's ThinginItself. > > No classes in set theory: no models in theory: settheoretic 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()_

