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From: Marc Alcobé García on 3 Feb 2010 15:38
On 3 feb, 04:00, "Ross A. Finlayson" <ross.finlay...(a)gmail.com> wrote:
> On Feb 2, 6:08 am, Marc Alcobé García <malc...(a)gmail.com> wrote:
>
> > As usually stated the transfinite recursion theorem on ON seems more a
> > syntactic statement than a truly set theoretic one (i. e. a statement
> > about formulas rather than about sets). So, what is its set theoretic
> > content?
>
> > Is it the existence of the delta-approximations? That is, the
> > existence of functions g with domain delta for every delta such that
> > for every alpha < delta, g(alpha) = F(g restricted to alpha)?
>
> ON, the class of ordinals, is syntactic.  So is identity.  Here "ON"
> is for the Ordinals which are like the non-negative, positive
> Integers, of Z, please then defining each limit ordinal.
>
> 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).  In systems maintaining a well-ordering, they go
> through the ordinals.
>
> Here alpha < delta carries variously over limit ordinals in general
> projection.
>
> Here the description is of the well-ordering theorem and the
> transfinite recursion theorem with as well the transfinite induction,
> with ordinals.  Then in the various theories infinite ordinals are
> defined.
>
> Yet seriously I don't feel that well addresses your question.  The
> "set theoretic content" is exactly what it means in a set theoretic
> system, or set theory system.  Set theory is so awesome because really
> getting into set theoretic machinery sees that something like the
> natural integers also has all the combined operations so is that
> structure.
>
> Then, you talk about the total function space on the other side of the
> ordinal, then the structural content is the range which has its
> functions, and they are combined into the total function space.
>
> Still, of course I would not know "delta approximation" that way, yet
> as you define it, it's so that for each alpha < delta by the ordering,
> for transfinite recursion, is restricted to the domain of d.  That's
> part of the transfinite induction schema, there's a d > a for each a,
> or rather it's used in the prover with the collapse over the limit
> ordinals and the Levy collapse and so on.
>
> Regards Marc,
>
> Ross Finlayson

I have tried hard to understand what you are trying to communicate,
but I must confess I do not know what you are talking about.
From: Marc Alcobé García on 3 Feb 2010 15:48
On 3 feb, 18:54, MoeBlee <jazzm...(a)hotmail.com> wrote:
> On Feb 2, 8:08 am, Marc Alcobé García <malc...(a)gmail.com> wrote:
>
> > As usually stated the transfinite recursion theorem on ON seems more a
> > syntactic statement than a truly set theoretic one
>
> There are different transfinite recursion theorems, and there are
> different approaches by different authors regarding these theorems.
>
> So that we can be exact, what particular formulation(s) are you
> referring to in what particular text(s)?
>
> MoeBlee

Sorry, I refer particularly to Kunen's formulation, that I think is
pretty standard: Given a function F:V->V, there exists a G:ON -> V
such that for every alpha G(alpha) = F(G restricted to alpha). Here G
is an explicit formula (otherwise the theorem would be impossible to
express in a first order language, since we can only quantify
existentially over sets and not over classes). This is why I say that
it seems to talk about formulas rather than about sets.
From: MoeBlee on 3 Feb 2010 16:52
On Feb 3, 2:48 pm, Marc Alcobé García <malc...(a)gmail.com> wrote:
> Kunen's formulation

Okay, I'll look at that book tonight.

However, in general, as far as I know, the ordinary transfinite
recursion theorems may be formulated as theorem schemata. In that
sense, yes, a theorem schema mentions formulas and possibly other
syntactical things, but that is in the service of definining a certain
set of formulas each of which is a theorem. Each instance of a theorem
schema is itself a theorem in the language of set theory and thus a
statement about sets, though, yes, the theorem schema itself mentions
formulas and such.

MoeBlee
From: Ross A. Finlayson on 3 Feb 2010 22:41
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.

Regards,

Ross Finlayson
From: Ross A. Finlayson on 3 Feb 2010 22:43
On Feb 3, 12:38 pm, Marc Alcobé García <malc...(a)gmail.com> wrote:
> On 3 feb, 04:00, "Ross A. Finlayson" <ross.finlay...(a)gmail.com> wrote:
>
>
>
> > On Feb 2, 6:08 am, Marc Alcobé García <malc...(a)gmail.com> wrote:
>
> > > As usually stated the transfinite recursion theorem on ON seems more a
> > > syntactic statement than a truly set theoretic one (i. e. a statement
> > > about formulas rather than about sets). So, what is its set theoretic
> > > content?
>
> > > Is it the existence of the delta-approximations? That is, the
> > > existence of functions g with domain delta for every delta such that
> > > for every alpha < delta, g(alpha) = F(g restricted to alpha)?
>
> > ON, the class of ordinals, is syntactic.  So is identity.  Here "ON"
> > is for the Ordinals which are like the non-negative, positive
> > Integers, of Z, please then defining each limit ordinal.
>
> > 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).  In systems maintaining a well-ordering, they go
> > through the ordinals.
>
> > Here alpha < delta carries variously over limit ordinals in general
> > projection.
>
> > Here the description is of the well-ordering theorem and the
> > transfinite recursion theorem with as well the transfinite induction,
> > with ordinals.  Then in the various theories infinite ordinals are
> > defined.
>
> > Yet seriously I don't feel that well addresses your question.  The
> > "set theoretic content" is exactly what it means in a set theoretic
> > system, or set theory system.  Set theory is so awesome because really
> > getting into set theoretic machinery sees that something like the
> > natural integers also has all the combined operations so is that
> > structure.
>
> > Then, you talk about the total function space on the other side of the
> > ordinal, then the structural content is the range which has its
> > functions, and they are combined into the total function space.
>
> > Still, of course I would not know "delta approximation" that way, yet
> > as you define it, it's so that for each alpha < delta by the ordering,
> > for transfinite recursion, is restricted to the domain of d.  That's
> > part of the transfinite induction schema, there's a d > a for each a,
> > or rather it's used in the prover with the collapse over the limit
> > ordinals and the Levy collapse and so on.
>
> > Regards Marc,
>
> > Ross Finlayson
>
> I have tried hard to understand what you are trying to communicate,
> but I must confess I do not know what you are talking about.

Marc,

I wrote back some more in this response to MoeBlee, basically what Jan
says.

Yet, it would be best if you explained it. It's interesting, sure
what are these things?

Regards,

Ross Finlayson
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