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From: Aatu Koskensilta on 24 May 2010 15:18
Frode Bj�rdal <fbenlightenment4all(a)gmail.com> writes:

> To quote myself:
>
>>> If we follow the scarequoted terminology of the Wikipedia entry on
>>> large countable ordinals, this ordinal would be an "unprovable"
>>> ordinal..

My reply was based on the mistaken idea that the Wikipedia entry spoke
of unprovable ordinals in the sense of proof theory. My apologies. I
wonder, though, whether this rather unfortunate Wikipedian usage is
based on something observed in the wild, in the set theoretic
literature?

--
Aatu Koskensilta (aatu.koskensilta(a)uta.fi)

"Wovon man nicht sprechan kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Frode Bjørdal on 24 May 2010 15:28
On 24 Mai, 21:14, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:
> Frode Bjørdal <fbenlightenment4...(a)gmail.com> writes:
> > To quote myself:
>
> >>> If we follow
> >>> the scarequoted terminology of the Wikipedia entry on large countable
> >>> ordinals, this ordinal would be an "unprovable" ordinal..
>
> My reply was based on the mistaken idea that the Wikipedia entry spoke
> of unprovable ordinals in the sense of proof theory. My apologies. I
> wonder, though, whether this rather unfortunate usage is found somewhere
> in the literature?

The usage may be unfortunate, and I am not aware of occurrences in the
literature. You seem to have had
a notion of 'unprovable ordinals in the sense of prrof theory' in
mind. Could you expand, as I do not understand.

>
> Aatu Koskensilta (aatu.koskensi...(a)uta.fi)
>
> "Wovon man nicht sprechan kann, darüber muss man schweigen"
>  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus

From: Frode Bjørdal on 24 May 2010 15:32
On 24 Mai, 21:18, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:
> Frode Bjørdal <fbenlightenment4...(a)gmail.com> writes:
> > To quote myself:
>
> >>> If we follow the scarequoted terminology of the Wikipedia entry on
> >>> large countable ordinals, this ordinal would be an "unprovable"
> >>> ordinal..
>
> My reply was based on the mistaken idea that the Wikipedia entry spoke
> of unprovable ordinals in the sense of proof theory. My apologies. I
> wonder, though, whether this rather unfortunate Wikipedian usage is
> based on something observed in the wild, in the set theoretic
> literature?

So - in my previous reply, i.e. to the first of your nearly identical
postings, I had the whole literature, both set theoretical and proof
theoretical in mind. I do NOT pretend to know it all, but one picks up
on certain things,,,,


> --
> Aatu Koskensilta (aatu.koskensi...(a)uta.fi)
>
> "Wovon man nicht sprechan kann, darüber muss man schweigen"
>  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus

From: Aatu Koskensilta on 24 May 2010 15:34
Frode Bj�rdal <fbenlightenment4all(a)gmail.com> writes:

> You seem to have had a notion of 'unprovable ordinals in the sense of
> prrof theory' in mind. Could you expand, as I do not understand.

In proof theory a recursive ordinal alpha is said to be a provable
ordinal of a theory T if T proves that transfinite induction holds along
some recursive ordering on the naturals with order-type alpha.

--
Aatu Koskensilta (aatu.koskensilta(a)uta.fi)

"Wovon man nicht sprechan kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Frode Bjørdal on 24 May 2010 15:46
On 24 Mai, 21:34, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:
> Frode Bjørdal <fbenlightenment4...(a)gmail.com> writes:
> > You seem to have had a notion of 'unprovable ordinals in the sense of
> > prrof theory' in mind. Could you expand, as I do not understand.
>
> In proof theory a recursive ordinal alpha is said to be a provable
> ordinal of a theory T if T proves that transfinite induction holds along
> some recursive ordering on the naturals with order-type alpha.

This makes sense. I failed to make the connection between prrof
theoretic ordinal and this notion.
Also, the Wikipedia entry suggests some sort of indexical usage of the
term "unprovable". But there will in addition be recursive ordinals
which are not provable for ZF.




> --
> Aatu Koskensilta (aatu.koskensi...(a)uta.fi)
>
> "Wovon man nicht sprechan kann, darüber muss man schweigen"
>  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus

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