Prev: Models of set theory vs. models of first order theories
Next: Cantor's Diagonal?
From: MoeBlee on 4 Feb 2010 13:27
On Feb 3, 9:41 pm, "Ross A. Finlayson" <ross.finlay...(a)gmail.com>
wrote:

> No, I'm explaining it.

No, as you usually do, you're just typing a bunch of words - a
nonsensical jumble of math terminology - that come to your mind, I
don't know what purpose you think that serves.

> 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.

Some of your finest nonsense. I especially admire the way you build,
getting yet more and more outlandishly ridiculous with each sentence.

MoeBlee

From: MoeBlee on 4 Feb 2010 13:36
On Feb 3, 2:48 pm, Marc Alcobé García <malc...(a)gmail.com> wrote:

> Kunen's formulation

As I looked that over, as well as one of Enderton's formulations, I
realized that an earlier claim I made was an overstatement. Yes, some
transfinite recursion formulations do each define a set of ZF
theorems, but the particular formulation you referred to is instead of
the form "Given a formula with property P, there exists a formula with
property Q" and I don't know how one would reformulate that as instead
a definition of a set of ZF formulas. In that sense, I do understand
your point: the signficance seems to syntactical, about formulas.
However, while still syntactical, let us not forget that from the
information about formulas we also go on to the information that such
situations allow us to introduce operation symbols such as 'rank' and
'aleph' in a way that preserves eliminability and non-creativity.

MoeBlee
From: MoeBlee on 5 Feb 2010 11:09
On Feb 4, 12:36 pm, MoeBlee <jazzm...(a)hotmail.com> wrote:

> I don't know how one would reformulate that as instead
> a definition of a set of ZF formulas.

But I'm working on it now. It's a bit difficult to fully unpack
Kunen's formulation as he's given it, but maybe in the next few days
I'll have some free time to do it. If I do show that the theorem
unpacks to a definition of a set of ZF theorems, then not only is the
"significance" of the theorem justfication of the tactical,
syntactical tool of adding operation symbols with transfinite
recursive definitions, but also the actual individual theorems taken
as a set.

MoeBlee


From: Jan Burse on 5 Feb 2010 11:26
MoeBlee schrieb:
> On Feb 4, 12:36 pm, MoeBlee <jazzm...(a)hotmail.com> wrote:
>
>> I don't know how one would reformulate that as instead a definition
>> of a set of ZF formulas.
>
> But I'm working on it now. It's a bit difficult to fully unpack
> Kunen's formulation as he's given it, but maybe in the next few days
> I'll have some free time to do it. If I do show that the theorem
> unpacks to a definition of a set of ZF theorems, then not only is the
> "significance" of the theorem justfication of the tactical,
> syntactical tool of adding operation symbols with transfinite
> recursive definitions, but also the actual individual theorems taken
> as a set.
>
> MoeBlee
>
>
There are papers on the net(*) that show that transfinite recursion
is part of ZFC. So it seems that weaker systems only make
a difference. Right?

Bye

(*)
A highly efficient ``transfinite recursive definitions'' axiom for set
theory.
Robert S. Wolf
Source: Notre Dame J. Formal Logic Volume 22, Number 1 (1981), 63-75.
From: MoeBlee on 5 Feb 2010 11:40
On Feb 5, 10:26 am, Jan Burse <janbu...(a)fastmail.fm> wrote:

> There are papers on the net(*) that show that transfinite recursion
> is part of ZFC.

What do you mean by "part of ZFC"?

> So it seems that weaker systems only make
> a difference. Right?

I don't understand your question. What difference are you referring
to?

MoeBlee

First  |  Prev  |  Next  |  Last
Pages: 1 2 3 4 5
Prev: Models of set theory vs. models of first order theories
Next: Cantor's Diagonal?