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From: Marc Alcobé García on 2 Feb 2010 09:08
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)?
From: Ross A. Finlayson on 2 Feb 2010 22:00
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


From: Jan Burse on 3 Feb 2010 12:42
Marc Alcob� Garc�a schrieb:
> 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)?

Well I think there is a contrast between transfinite induction and
transfinite recursion I guess.

Transfinite induction only postulates that a predicate holds for all
sets, provided that the predicate respects the transfinite induction step.

On the other hand, transfinite recursion postulates the existence
of an indexed family of sets, where the family members are linked
by the transfinite recursion step.

By transfinite recursion we can constructs sets which allow us to show
consistency of some math done only with transfinite induction. This is
at least how I roughly interpret:
http://en.wikipedia.org/wiki/Reverse_mathematics#Arithmetical_Transfinite_Recursion_ATR0

Bye
From: MoeBlee on 3 Feb 2010 12:54
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
From: MoeBlee on 3 Feb 2010 12:56
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

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