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 <> 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.
> 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.
> 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.
> Theory is its theorems.

In theory...