
Re: fom  01  preface
Posted:
Dec 9, 2012 11:28 PM


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=221287 http://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+model+&as_ugroup=sci.*&as_uauthors=Ross+Finlayson
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

