Date: Dec 10, 2012 12:35 AM Author: fom Subject: Re: fom - 01 - preface 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.

> No classes in set theory: no models in theory: set-theoretic theory.

>

> http://en.wikipedia.org/wiki/Theory

>

> Theory is its theorems.

In theory...