On 12/9/2012 10:28 PM, Ross A. Finlayson wrote: > On Dec 9, 6:36 pm, fom <fomJ...@nyms.net> wrote: > >> 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 > > 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 non-constructible elements outside M, were > it not L and there are none.
I think that is put correctly.
How does one see non-constructible elements?
The transitive classes built from Goedel operations generate all of the finite sets seen in V_omega. One must understand that the cumulative hierarchy using the power set operation is merely specifying a topological cover. Beyond the finite sets there is no means to know the definite structure of a power set. One must think of the language acting at V_alpha as second-order until V_(alpha+1) is iterated.
To assume non-constructible elements is to assume partiality. It is a good counterfactual, but acting on it ought to require justification.
> Given that combined with L-S, isn't the > very method of forcing dependent on that V =/= L?
That was what the quote from Jech conveyed. However, that is only with respect to class models. Set models reside within the universe.
> Then, there is no > definable well-ordering of the reals as a result of Feferman, with V = > L.
Is there a definable well-ordering anywhere?
Yet, with V = L, wouldn't one not even need forcing, with up/down > L-S?
One needs forcing for independence proofs. And some of the models are just cool.
> > 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. >
I actually have Cohen's book on order. I thought it time to look at the original development closely for myself.
> 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).
Good for some math.
> 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).
One of my recent posts answering William Eliot summarizes Lesniewski's formalization of ontology. And, Zuhair is working out a mereological theory for himself. That is a better place to look for these issues.
> 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. >
Yeah... mereology is a much, much better place for ontology.