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From: Ross A. Finlayson on 4 Mar 2010 22:19
On Feb 3, 7:41 pm, "Ross A. Finlayson" <ross.finlay...(a)gmail.com>
wrote:
> On Feb 3, 9:56 am, MoeBlee <jazzm...(a)hotmail.com> wrote:
>
> > On Feb 2, 9:00 pm, "Ross A. Finlayson" <ross.finlay...(a)gmail.com>
> > wrote:
>
> > > Its set-theoretic content, the content of the transfinite recursion
> > > theorem that transfinite recursion schema define induction, is that
> > > there is induction over limit ordinals, defined via transfinite
> > > recursion schema(ta).
>
> > You don't know what you're talking about. You're completely confused
> > about the difference and relationship between transfinite induction
> > and transfinite recursion.
>
> > MoeBlee
>
> No, I'm explaining it.  Yet, thanks for the comic relief.
>
> Feel free to go over again the general structure.
>
> Marc, yes, I think that I know what I wrote and it is very clear and
> direct, you seem clear to me and my writing is no notice of an error,
> basically the only usage is that the function space of a into d is in
> the machinery for proof by induction or with the recursion and so on,
> with definitions in terms of ordinal successorship the limit ordinal
> induction schema, for the transfinite induction over limit ordinals.
> Of course, finite induction just needs the addition with the inference
> rule:  induction, that what's true for f(0) is true, and that for
> f(n), that f(n+1) is true, because the numbers go zero, one, two,
> etcetera.  Recursion is defined by the expansion of the function,
> their inductive rules in forward inference apply.
>
> In set theoretic structures these generally use ordinals.  With
> cardinals there would still be maintained the space.  Yet, that is
> where the measurement in keeping the variables transparent to
> cardinals would expand the regular space, adjusting the refinement.
> Ordinals, maintain cross-discretely, they're ordinal-valued.  The
> refinement, here that addresses when maintaining the cardinals in the
> set theoretic framework reduces to ordinals through refinement.
> Seriously, this makes sense to me, the space that the cardinals
> maintain is where the functions of inter-related variables are scaled
> together so they're analyzed together, maintaining analog function
> spaces.
>
> It's just not that much definition.
>

Yes, this has that the quantizing works out while the discrete is
maintained so besides that the constructive universe is good for
exhaustion of approximation, in effects, furthermore the results of
exhaustion are available in the constructible universe, under real
analysis.

This was where otherwise it is a general discussion of a result that
is known as a fundamental concept of transfinite ordinal induction,
basically definition of the induction schema over the limit ordinals.
Here the class of ordinals in set theory isn't a set, in regular set
theories. It's all of them so it would be one of them, an ordinal.
Obviously then this is an ubiquitous ordinal. It is a well-covered
situation, Cesare Burali-Forti's set of all ordinals that couldn't
exist in the naive set theory with the regularity in consistency.
It's contents are well-quantified over all of the ordinals, well-
ordered etcetera, basically with ordinal allusion to limit ordinals
and processes in different scale etcetera, for much for what they are
useful, intuitive, obvious, etcetera.

That, and the above holds true, it's very simple in definition and
where it maintains general accumulation of products, it's exhaustive
in the finite and standard.

Still, it's a simple model, obviously I think in a simple model of the
null axiom theory. That sees the polydimensional in the phase
boundary between the finite and infinitesimal or finite and infinite.
At the point it starts with the small dimensions, at the space the
large ones. This is where simple mathematical concepts derivative
from the notice that a particularly useful axiomatized function called
the unit line segment has that it is still well-defined by geometry,
to geometry of points and spaces instead of points and lines, besides
that obviously lines, planes, etcetera, are still the same, continua.
The space is all the points. Standardly, a vector space with a real
basis vector is all the linear combinations of the basis vector.
Still, a vector can be defined by a point besides zero.

So, yeah, I get it.

Regards,

Ross Finlayson
From: MoeBlee on 5 Mar 2010 12:58
On Mar 4, 9:19 pm, "Ross A. Finlayson" <ross.finlay...(a)gmail.com>
wrote:

> this has that the quantizing works out while the discrete is
> maintained so besides that the constructive universe is good for
> exhaustion of approximation, in effects, furthermore the results of
> exhaustion are available in the constructible universe, under real
> analysis.
>
> This was where otherwise it is a general discussion of a result that
> is known as a fundamental concept of transfinite ordinal induction,
> basically definition of the induction schema over the limit ordinals.
> Here the class of ordinals in set theory isn't a set, in regular set
> theories.  It's all of them so it would be one of them, an ordinal.
> Obviously then this is an ubiquitous ordinal.  It is a well-covered
> situation, Cesare Burali-Forti's set of all ordinals that couldn't
> exist in the naive set theory with the regularity in consistency.
> It's contents are well-quantified over all of the ordinals, well-
> ordered etcetera, basically with ordinal allusion to limit ordinals
> and processes in different scale etcetera, for much for what they are
> useful, intuitive, obvious, etcetera.
>
> That, and the above holds true, it's very simple in definition and
> where it maintains general accumulation of products, it's exhaustive
> in the finite and standard.
>
> Still, it's a simple model, obviously I think in a simple model of the
> null axiom theory.  That sees the polydimensional in the phase
> boundary between the finite and infinitesimal or finite and infinite.
> At the point it starts with the small dimensions, at the space the
> large ones.  This is where simple mathematical concepts derivative
> from the notice that a particularly useful axiomatized function called
> the unit line segment has that it is still well-defined by geometry,
> to geometry of points and spaces instead of points and lines, besides
> that obviously lines, planes, etcetera, are still the same, continua.
> The space is all the points.  Standardly, a vector space with a real
> basis vector is all the linear combinations of the basis vector.
> Still, a vector can be defined by a point besides zero.

Ah, now I understand perfectly.

MoeBlee
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