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Herman Jurjus
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 Posted: Mon Oct 20, 2008 7:46 am Post subject: Independent independence |
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Is there already an example known of a theorem for which the statement
expressing its independence from ZFC is itself independent from ZFC?
--
Cheers,
Herman Jurjus |
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| Posted: Sat Oct 25, 2008 1:21 pm Post subject: Re: Independent independence |
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Herman Jurjus wrote:
[quote]Is there already an example known of a theorem for which the statement
expressing its independence from ZFC is itself independent from ZFC?
[/quote]
"ZFC is consistent" is such an example -- as is anything independent
of ZFC, as it happens. I suspect you>re not asking the question you
really have in mind.
--
Aatu Koskensilta (aatu.koskensilta@uta.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus |
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John Jones
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| Posted: Sun Oct 26, 2008 4:28 pm Post subject: Re: Independent independence |
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Herman Jurjus wrote:
[quote]Is there already an example known of a theorem for which the statement
expressing its independence from ZFC is itself independent from ZFC?
[/quote]
You have mistaken a grammatical convenience for "an example of an
already known theorem".
A "statement" isn>t any sort of entity, theorem, etc. Statements don>t
engage in logical or mathematical deliberations. They are strings that
convey or invoke meaning.
You might as well ask if the ink used to make a statement is independent
from ZFC. |
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Herman Jurjus
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| Posted: Mon Oct 27, 2008 2:49 pm Post subject: Re: Independent independence |
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aatu.koskensilta@xortec.fi wrote:
[quote]Herman Jurjus wrote:
Is there already an example known of a theorem for which the statement
expressing its independence from ZFC is itself independent from ZFC?
"ZFC is consistent" is such an example -- as is anything independent
of ZFC, as it happens. I suspect you>re not asking the question you
really have in mind.
[/quote]
Ah - well, finding exact formulations which make the question
non-trivial is the crucial part of answering the question, i>d say.
Obviously, the question becomes trivial if you think of 'the statement
expressing independence' as something like:
'ZFC |/- A and ZFC |/- -A'.
But the statements proved in the usual independence proofs (of CH, AC
etc.) are all (afaik) of the form:
'if ZF is consistent, then so are ZF + A and ZF + -A'.
Do we know an A for which -that- sentence is provably independent from ZFC?
--
Cheers,
Herman Jurjus |
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Aatu Koskensilta
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| Posted: Tue Oct 28, 2008 3:01 pm Post subject: Re: Independent independence |
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Herman Jurjus <hjurjus@hetnet.nl> writes:
[quote]But the statements proved in the usual independence proofs (of CH, AC
etc.) are all (afaik) of the form:
'if ZF is consistent, then so are ZF + A and ZF + -A'.
Do we know an A for which -that- sentence is provably independent from ZFC?
[/quote]
"ZFC is consistent".
--
Aatu Koskensilta (aatu.koskensilta@uta.fi)
"Wovon man nicht sprechen kann, darĂ¼ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus |
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Herman Jurjus
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| Posted: Wed Oct 29, 2008 12:00 am Post subject: Re: Independent independence |
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Aatu Koskensilta wrote:
[quote]Herman Jurjus <hjurjus@hetnet.nl> writes:
But the statements proved in the usual independence proofs (of CH, AC
etc.) are all (afaik) of the form:
'if ZF is consistent, then so are ZF + A and ZF + -A'.
Do we know an A for which -that- sentence is provably independent from ZFC?
"ZFC is consistent".
[/quote]
Can ZFC -not- prove "if ZF is consistent then so is ZF + Con(ZFC)" ?
Can ZFC -not- prove "if ZF is consistent then so is ZF + -Con(ZFC)" ?
--
Cheers,
Herman Jurjus |
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Daryl McCullough
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| Posted: Wed Oct 29, 2008 2:20 am Post subject: Re: Independent independence |
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Herman Jurjus says...
[quote]
Aatu Koskensilta wrote:
Herman Jurjus <hjurjus@hetnet.nl> writes:
But the statements proved in the usual independence proofs (of CH, AC
etc.) are all (afaik) of the form:
'if ZF is consistent, then so are ZF + A and ZF + -A'.
Do we know an A for which -that- sentence is provably independent from ZFC?
"ZFC is consistent".
Can ZFC -not- prove "if ZF is consistent then so is ZF + Con(ZFC)" ?
[/quote]
I>m pretty sure that it cannot prove that. To prove that ZF + Con(ZFC)
is consistent, you need to know not just that ZF is consistent, but
you need to know that it doesn>t prove any *false* statements of
the form "theory T is inconsistent".
Godel>s theorem shows that ZF does not prove Con(ZFC), but it>s still
possible that ZF proves ~Con(ZFC).
[quote]Can ZFC -not- prove "if ZF is consistent then so is ZF + -Con(ZFC)" ?
[/quote]
Yes, it can prove that. Godel proved "If ZF is consistent, then
so is ZFC", and his incompleteness theorem shows that "if ZFC is
consistent, then ZFC cannot prove Con(ZFC)", from which it follows
that "If ZFC is consistent, then ZFC + ~Con(ZFC) is consistent".
--
Daryl McCullough
Ithaca, NY |
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Herman Jurjus
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| Posted: Wed Oct 29, 2008 3:25 am Post subject: Re: Independent independence |
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Daryl McCullough wrote:
[quote]Herman Jurjus says...
Aatu Koskensilta wrote:
Herman Jurjus <hjurjus@hetnet.nl> writes:
But the statements proved in the usual independence proofs (of CH, AC
etc.) are all (afaik) of the form:
'if ZF is consistent, then so are ZF + A and ZF + -A'.
Do we know an A for which -that- sentence is provably independent from ZFC?
"ZFC is consistent".
Can ZFC -not- prove "if ZF is consistent then so is ZF + Con(ZFC)" ?
I>m pretty sure that it cannot prove that. To prove that ZF + Con(ZFC)
is consistent, you need to know not just that ZF is consistent, but
you need to know that it doesn>t prove any *false* statements of
the form "theory T is inconsistent".
Godel>s theorem shows that ZF does not prove Con(ZFC), but it>s still
possible that ZF proves ~Con(ZFC).
[/quote]
But ZF can prove that Con(ZF) implies Con(ZF + AC).
So if ZF could prove ~Con(ZFC), it could also prove ~Con(ZF).
But then ZFC could also prove ~Con(ZF), and hence ZFC would be
inconsistent, because ZFC can also prove Con(ZF).
BTW, now that i think of it, since ZFC implies Con(ZF), the more
sensible question to ask is whether there is an A such that
'if ZFC is consistent, then so are ZF + A and ZF + -A'
is independent from ZFC.
--
Cheers,
Herman Jurjus |
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Herman Jurjus
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| Posted: Wed Oct 29, 2008 3:33 am Post subject: Re: Independent independence |
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Herman Jurjus wrote:
[quote]Daryl McCullough wrote:
Herman Jurjus says...
Aatu Koskensilta wrote:
Herman Jurjus <hjurjus@hetnet.nl> writes:
But the statements proved in the usual independence proofs (of CH, AC
etc.) are all (afaik) of the form:
'if ZF is consistent, then so are ZF + A and ZF + -A'.
Do we know an A for which -that- sentence is provably independent
from ZFC?
"ZFC is consistent".
Can ZFC -not- prove "if ZF is consistent then so is ZF + Con(ZFC)" ?
I>m pretty sure that it cannot prove that. To prove that ZF + Con(ZFC)
is consistent, you need to know not just that ZF is consistent, but
you need to know that it doesn>t prove any *false* statements of
the form "theory T is inconsistent".
Godel>s theorem shows that ZF does not prove Con(ZFC), but it>s still
possible that ZF proves ~Con(ZFC).
But ZF can prove that Con(ZF) implies Con(ZF + AC).
So if ZF could prove ~Con(ZFC), it could also prove ~Con(ZF).
But then ZFC could also prove ~Con(ZF), and hence ZFC would be
inconsistent, because ZFC can also prove Con(ZF).
BTW, now that i think of it, since ZFC implies Con(ZF)
[/quote]
Sorry; that last one is not true, of course. That shall teach me not to
post so late on the evening. So let me continue this tomorrow.
--
Cheers,
Herman Jurjus |
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Herman Jurjus
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| Posted: Wed Oct 29, 2008 7:25 am Post subject: Re: Independent independence |
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Herman Jurjus wrote:
[quote]Herman Jurjus wrote:
Daryl McCullough wrote:
Herman Jurjus says...
Aatu Koskensilta wrote:
Herman Jurjus <hjurjus@hetnet.nl> writes:
But the statements proved in the usual independence proofs (of CH, AC
etc.) are all (afaik) of the form:
'if ZF is consistent, then so are ZF + A and ZF + -A'.
Do we know an A for which -that- sentence is provably independent
from ZFC?
"ZFC is consistent".
Can ZFC -not- prove "if ZF is consistent then so is ZF + Con(ZFC)" ?
I>m pretty sure that it cannot prove that. To prove that ZF + Con(ZFC)
is consistent, you need to know not just that ZF is consistent, but
you need to know that it doesn>t prove any *false* statements of
the form "theory T is inconsistent".
Godel>s theorem shows that ZF does not prove Con(ZFC), but it>s still
possible that ZF proves ~Con(ZFC).
[/quote]
Right. So indeed Con(ZFC) is a good answer.
So allow me to rephrase the question. (As i said, i consider finding
appropriate formulation(s) as part of answering the original question.)
Second attempt.
Do we know an A for which the two sentences:
'if ZF is consistent, then so is ZF + A'
and
'if ZF is consistent, then so is ZF + ~A'
are both independent from ZFC?
--
Cheers,
Herman Jurjus |
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george
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| Posted: Fri Oct 31, 2008 4:15 am Post subject: Re: Independent independence |
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On Oct 26, 7:28 am, John Jones <jonescard...@aol.com> wrote:
Bitch, PLEASE. GROWN folk are talking. |
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