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From: Rupert on 20 Jun 2010 09:11
In his thesis "The Search for New Axioms" Peter Koellner writes

"There is no known example of a natural sentence phi of first-order
arithmetic such that (1) phi is known to be independent of ZF and (2)
it is not known whether phi is true."

Is there any example at all of a "natural" sentence of first-order
arithmetic which is known to be independent of ZF?
From: William Elliot on 21 Jun 2010 01:41
On Sun, 20 Jun 2010, Rupert wrote:

> In his thesis "The Search for New Axioms" Peter Koellner writes
>
> "There is no known example of a natural sentence phi of first-order
> arithmetic such that (1) phi is known to be independent of ZF and (2)
> it is not known whether phi is true."
>
> Is there any example at all of a "natural" sentence of first-order
> arithmetic which is known to be independent of ZF?
>
Isn't that Godel's statement of his incompleteness theorem?

BTW, have you heard about Goodstein's theorem?

From: Rupert on 21 Jun 2010 01:54
On Jun 21, 3:41 pm, William Elliot <ma...(a)rdrop.remove.com> wrote:
> On Sun, 20 Jun 2010, Rupert wrote:
> > In his thesis "The Search for New Axioms" Peter Koellner writes
>
> > "There is no known example of a natural sentence phi of first-order
> > arithmetic such that (1) phi is known to be independent of ZF and (2)
> > it is not known whether phi is true."
>
> > Is there any example at all of a "natural" sentence of first-order
> > arithmetic which is known to be independent of ZF?
>
> Isn't that Godel's statement of his incompleteness theorem?
>
> BTW, have you heard about Goodstein's theorem?

The Goedel sentence for ZF is known to be independent of ZF. (More
precisely it can be proved to be independent of ZF on the assumption
that ZF is 1-consistent. It can, of course, also be proved to be true
on the assumption that ZF is 1-consistent.) But I do not know whether
this would be considered a "natural" example.

Peter Koellner also says that the qualifier "natural" is necessary in
the statement of the full fact since "one can use Goedelian methods to
cook up artificial counterexamples." I would be interested in
clarification of this remark.

Yes, I have heard of Goodstein's theorem.
From: Rupert on 21 Jun 2010 02:07
On Jun 21, 3:54 pm, Rupert <rupertmccal...(a)yahoo.com> wrote:
> On Jun 21, 3:41 pm, William Elliot <ma...(a)rdrop.remove.com> wrote:
>
> > On Sun, 20 Jun 2010, Rupert wrote:
> > > In his thesis "The Search for New Axioms" Peter Koellner writes
>
> > > "There is no known example of a natural sentence phi of first-order
> > > arithmetic such that (1) phi is known to be independent of ZF and (2)
> > > it is not known whether phi is true."
>
> > > Is there any example at all of a "natural" sentence of first-order
> > > arithmetic which is known to be independent of ZF?
>
> > Isn't that Godel's statement of his incompleteness theorem?
>
> > BTW, have you heard about Goodstein's theorem?
>
> The Goedel sentence for ZF is known to be independent of ZF. (More
> precisely it can be proved to be independent of ZF on the assumption
> that ZF is 1-consistent. It can, of course, also be proved to be true
> on the assumption that ZF is 1-consistent.) But I do not know whether
> this would be considered a "natural" example.
>
> Peter Koellner also says that the qualifier "natural" is necessary in
> the statement of the full fact since "one can use Goedelian methods to
> cook up artificial counterexamples." I would be interested in
> clarification of this remark.
>
> Yes, I have heard of Goodstein's theorem.

I think I have figured it out. Consider the following sentence.

If it is consistent with ZF that there is a supercompact cardinal,
then a witness that I am refutable in ZF is found before a witness is
found that I am provable in ZF. If it is not consistent with ZF that
there is a supercompact cardinal, then a witness that I am provable in
ZF is found before a witness is found that I am refutable in ZF.

I can construct an arithmetical sentence like this using the diagonal
lemma.

I can prove on the assumption that ZF is consistent that this sentence
is independent of ZF. But I cannot know whether it is true without
knowing whether or not it is consistent with ZF that there is a
supercompact cardinal (and Peter Koellner says that we do not know
whether this is true).

So this shows that Peter Koellner would not consider this an example
of a "natural" sentence. But I suppose it is possible that he would
consider the Goedel sentence "natural". It is not really clear.
From: Herman Jurjus on 21 Jun 2010 06:37
On 6/21/2010 8:07 AM, Rupert wrote:
> On Jun 21, 3:54 pm, Rupert<rupertmccal...(a)yahoo.com> wrote:
>> On Jun 21, 3:41 pm, William Elliot<ma...(a)rdrop.remove.com> wrote:
>>
>>> On Sun, 20 Jun 2010, Rupert wrote:
>>>> In his thesis "The Search for New Axioms" Peter Koellner writes
>>
>>>> "There is no known example of a natural sentence phi of first-order
>>>> arithmetic such that (1) phi is known to be independent of ZF and (2)
>>>> it is not known whether phi is true."
>>
>>>> Is there any example at all of a "natural" sentence of first-order
>>>> arithmetic which is known to be independent of ZF?
>>
>>> Isn't that Godel's statement of his incompleteness theorem?
>>
>>> BTW, have you heard about Goodstein's theorem?
>>
>> The Goedel sentence for ZF is known to be independent of ZF. (More
>> precisely it can be proved to be independent of ZF on the assumption
>> that ZF is 1-consistent. It can, of course, also be proved to be true
>> on the assumption that ZF is 1-consistent.) But I do not know whether
>> this would be considered a "natural" example.
>>
>> Peter Koellner also says that the qualifier "natural" is necessary in
>> the statement of the full fact since "one can use Goedelian methods to
>> cook up artificial counterexamples." I would be interested in
>> clarification of this remark.
>>
>> Yes, I have heard of Goodstein's theorem.
>
> I think I have figured it out. Consider the following sentence.
>
> If it is consistent with ZF that there is a supercompact cardinal,
> then a witness that I am refutable in ZF is found before a witness is
> found that I am provable in ZF. If it is not consistent with ZF that
> there is a supercompact cardinal, then a witness that I am provable in
> ZF is found before a witness is found that I am refutable in ZF.
>
> I can construct an arithmetical sentence like this using the diagonal
> lemma.
>
> I can prove on the assumption that ZF is consistent that this sentence
> is independent of ZF. But I cannot know whether it is true without
> knowing whether or not it is consistent with ZF that there is a
> supercompact cardinal (and Peter Koellner says that we do not know
> whether this is true).
>
> So this shows that Peter Koellner would not consider this an example
> of a "natural" sentence. But I suppose it is possible that he would
> consider the Goedel sentence "natural". It is not really clear.

Is it clear to you what Harvey Friedman is doing, and why?
(Finding atrithmetical sentences that require large cardinal axioms for
their proof, and that are -relevant to the practicing mathematician-.)

Is there any reason for thinking that Koellner is talking about
something else?

--
Cheers,
Herman Jurjus
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