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From: Rupert on 21 Jun 2010 17:26
On Jun 21, 8:37 pm, Herman Jurjus <hjm...(a)hetnet.nl> wrote:
> 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?
>

The arithmetical sentences that Harvey Friedman constructs are usually
equivalent in some weak theory to the consistency or 1-consistency of
some extension of ZF by large cardinal axioms. Whether or not we know
such a sentence to be independent of ZF, and whether we know it to be
true, will depend on whether we think we know the extension by large
cardinal axioms in question to be consistent or 1-consistent. However,
the answers to the questions will be the same. So you wouldn't get an
example in this way of a sentence such that we know it to be
independent without knowing whether or not it is true.
From: Herman Jurjus on 21 Jun 2010 17:38
On 6/21/2010 11:26 PM, Rupert wrote:
> On Jun 21, 8:37 pm, Herman Jurjus<hjm...(a)hetnet.nl> wrote:
>> 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?
>>
>
> The arithmetical sentences that Harvey Friedman constructs are usually
> equivalent in some weak theory to the consistency or 1-consistency of
> some extension of ZF by large cardinal axioms. Whether or not we know
> such a sentence to be independent of ZF, and whether we know it to be
> true, will depend on whether we think we know the extension by large
> cardinal axioms in question to be consistent or 1-consistent. However,
> the answers to the questions will be the same. So you wouldn't get an
> example in this way of a sentence such that we know it to be
> independent without knowing whether or not it is true.

Right.

But when Koellner talks about a 'natural' sentence, is there any reason
to suppose that by that he means something else than '(possibly)
relevant to the practicing mathematician'?

--
Cheers,
Herman Jurjus
From: Rupert on 21 Jun 2010 18:20
On Jun 22, 7:38 am, Herman Jurjus <hjm...(a)hetnet.nl> wrote:
> On 6/21/2010 11:26 PM, Rupert wrote:
>
>
>
>
>
> > On Jun 21, 8:37 pm, Herman Jurjus<hjm...(a)hetnet.nl>  wrote:
> >> 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?
>
> > The arithmetical sentences that Harvey Friedman constructs are usually
> > equivalent in some weak theory to the consistency or 1-consistency of
> > some extension of ZF by large cardinal axioms. Whether or not we know
> > such a sentence to be independent of ZF, and whether we know it to be
> > true, will depend on whether we think we know the extension by large
> > cardinal axioms in question to be consistent or 1-consistent. However,
> > the answers to the questions will be the same. So you wouldn't get an
> > example in this way of a sentence such that we know it to be
> > independent without knowing whether or not it is true.
>
> Right.
>
> But when Koellner talks about a 'natural' sentence, is there any reason
> to suppose that by that he means something else than '(possibly)
> relevant to the practicing mathematician'?
>

Yes. That is probably a good interpretation.
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