Godel’s Incompleteness work depends heavily on the assumption
that the theory T in question be consistent. But how could we know
if T is consistent, for any given general T? It seems like we don’t
have a precision procedure to determine that. So Godel’s results
are _hypothetical_ results: it doesn’t assert that a particular T
is in fact incomplete, or G(T) is in fact true; it only assert
them hypothetically so. In fact, if T later turns out to be
inconsistent, T would be complete for example.
Two questions seem inevitable:
(1) Why would Godel’s work have a huge impact in modern mathematical
formalism? [Arguably, GIT would be more complicate than "If we know
a counter example of GC, then ~GC is provable in PA". But the fact
that both of them are hypothetical means that T might be no more
factually incomplete than ~GC might be factually provable in PA.]
(2) Is it true then there is *a real danger* that there might exist an
inconsistent theory T in the sense that for all the theorems that
we – as human being – could conceivably prove, their negation
counterparts would have proof lengths that are beyond human capacity
to comprehend?
–
—————————————————-
Time passes, there is no way we can hold it back.
Why then do thoughts linger, long after everything
else is gone?
Ryokan
—————————————————-
On Fri, 16 Dec 2005 06:01:05 GMT, Nam Nguyen <namducngu…@shaw.ca> said:
> Godel’s Incompleteness work depends heavily on the assumption
> that the theory T in question be consistent.
No, it doesn’t.
> But how could we know if T is consistent, for any given general T?
In general, we can’t. That is one of the consequences of Gödel’s work.
> It seems like we don’t have a precision procedure to determine that.
It doesn’t merely seem like it.
> (2) Is it true then there is *a real danger* that there might exist an
> inconsistent theory T in the sense that for all the theorems that
> we – as human being – could conceivably prove, their negation
> counterparts would have proof lengths that are beyond human capacity
> to comprehend?
That is a very strange sense of inconsistency.
Chris Menzel wrote:
> On Fri, 16 Dec 2005 06:01:05 GMT, Nam Nguyen <namducngu…@shaw.ca> said:
>>Godel’s Incompleteness work depends heavily on the assumption
>>that the theory T in question be consistent.
> No, it doesn’t.
If T is not consistent, wouldn’t it be complete?
>>But how could we know if T is consistent, for any given general T?
> In general, we can’t. That is one of the consequences of Gödel’s work.
>>It seems like we don’t have a precision procedure to determine that.
> It doesn’t merely seem like it.
I probably did it in a "wordy" way but I think I basically stated the
same thing: in general we can’t know whether or not T would be
consistent. I just needed to state something to that effect for the
"then" in question (2) below.
>>(2) Is it true then there is *a real danger* that there might exist an
>> inconsistent theory T in the sense that for all the theorems that
>> we – as human being – could conceivably prove, their negation
>> counterparts would have proof lengths that are beyond human capacity
>> to comprehend?
> That is a very strange sense of inconsistency.
I didn’t change the definition of inconsistency though, which is that
for a given formula F, both F and ~F would be provable from T. What I
stated is simply a *suspicion* of something like the following:
Suppose for discussion ZFC is genuienly inconsistent. Obvisouly There
have been only a finite number of theories that we have proven so far,
or that we could prove as long as humanity could last. Take any of such
theorem F, isn’t it possible that the proof of ~F could be too long for
us to *know*? and hence as far as we understand the nature of humanity,
ZFC would exist as an inconsistent theory without us ever knowing that?
[That's just a suspicion I tried to stipulate.]
–
—————————————————-
Time passes, there is no way we can hold it back.
Why then do thoughts linger, long after everything
else is gone?
Ryokan
—————————————————-
- Hide quoted text — Show quoted text -
Nam Nguyen wrote:
> Chris Menzel wrote:
>> On Fri, 16 Dec 2005 06:01:05 GMT, Nam Nguyen <namducngu…@shaw.ca> said:
>>> Godel’s Incompleteness work depends heavily on the assumption
>>> that the theory T in question be consistent.
>> No, it doesn’t.
> If T is not consistent, wouldn’t it be complete?
>>> But how could we know if T is consistent, for any given general T?
>> In general, we can’t. That is one of the consequences of Gödel’s work.
>>> It seems like we don’t have a precision procedure to determine that.
>> It doesn’t merely seem like it.
> I probably did it in a "wordy" way but I think I basically stated the
> same thing: in general we can’t know whether or not T would be
> consistent. I just needed to state something to that effect for the
> "then" in question (2) below.
>>> (2) Is it true then there is *a real danger* that there might exist an
>>> inconsistent theory T in the sense that for all the theorems that
>>> we – as human being – could conceivably prove, their negation
>>> counterparts would have proof lengths that are beyond human capacity
>>> to comprehend?
>> That is a very strange sense of inconsistency.
> I didn’t change the definition of inconsistency though, which is that
> for a given formula F, both F and ~F would be provable from T. What I
> stated is simply a *suspicion* of something like the following:
> Suppose for discussion ZFC is genuienly inconsistent. Obvisouly There
> have been only a finite number of theories that we have proven so far,
Sorry for the typo, I meant "… number of theorems…"
> or that we could prove as long as humanity could last. Take any of such
> theorem F, isn’t it possible that the proof of ~F could be too long for
> us to *know*? and hence as far as we understand the nature of humanity,
> ZFC would exist as an inconsistent theory without us ever knowing that?
> [That's just a suspicion I tried to stipulate.]
–
—————————————————-
Time passes, there is no way we can hold it back.
Why then do thoughts linger, long after everything
else is gone?
Ryokan
—————————————————-
On Fri, 16 Dec 2005, Nam Nguyen wrote:
> (2) Is it true then there is *a real danger* that there might exist an
> inconsistent theory T in the sense that for all the theorems that
> we – as human being – could conceivably prove, their negation
> counterparts would have proof lengths that are beyond human capacity
> to comprehend?
Yes, there are several now in use and many more no longer in use nor
recognized and most likely many more to come. However they are basically
variants upon the general inconsistent theory, the one true religion.
Nam Nguyen wrote:
> (2) Is it true then there is *a real danger* that there might exist an
> inconsistent theory T in the sense that for all the theorems that
> we – as human being – could conceivably prove, their negation
> counterparts would have proof lengths that are beyond human capacity
> to comprehend?
I don’t know about "real danger" but it’s certainly conceiveable that
there exists a proof of a contradiction in e.g. ZFC that is so
complicated and long that no human will ever comprehend it. There’s no
reason to suppose this is the case, though.
I don’t recall if it was Friedman or some one else who conjectured that
naturally occuring inconsistent theories have simple contradictions. It
seems empirical evidence supports this: contradictions found in actually
proposed theories are rather simple in the sense that they don’t require
heavy mathematical machinery beyond but a few experts (I’d be very
interested to see a counterexample – my knowledge of proposed
inconsistencies is probably not exhaustive). Of course, this might be
just a coincidence and all our theories might be contradictory in
horribly complex ways we just haven’t noticed, but the lack of
contradictions in the middle ground between simple and horribly complex
suggests there might be something to this. Or perhaps it’s just because
so few people are seriously looking for a contradiction in ZFC or PA or
what you have.
–
Aatu Koskensilta (aatu.koskensi…@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
– Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Aatu Koskensilta wrote:
> I don’t recall if it was Friedman or some one else who conjectured that
> naturally occuring inconsistent theories have simple contradictions.
I searched the FOM archives and didn’t find Friedman saying anything
like this. In fact, in one post he writes
It’s a nice idea – that if something of a fundamentally set theoretic
nature isn’t very quickly seen to be inconsistent, then it is in fact
consistent.
The trouble is that the number of data points that seem relevant to
this is very small.
The applicability of the concept of evidence to mathematical contexts
is well known to be a highly contentious matter.
So I was probably just imagining the whole thing.
The context of Friedman’s reply is as follows. The following statement
#) There exists an inaccessible kappa, s.t. there is an elementary
embedding j: V_kappa –> V_kappa
is inconsistent with ZFC, as was shown by Kunen. Friedman suggests that
it might be possible to work from the inconsistent premise Con(ZFC+#) to
derive consistency of ZFC+P for some P in a non-trivial manner (i.e.
without reasoning like Con(ZFC+#) is false, and hence trivially
Con(ZFC+#) –> Con(ZFC+anything)):
I could well imagine that we could perform the following experiment
with some real creativity.
The experiment is to stay well clear of Kunen’s inconsistency proof,
and try to use Con(ZFC + #), Con(ZFC + #+) to redo some of [Woodin's
relative consistency proofs of form Con(ZF+#)-->Con(ZFC+P)], with
perhaps easier arguments, and even where we casually use choice (even
high up).
Kunen is said to have discovered the inconsistency of ZFC+# while trying
to strenghten a result of Solovay (having to do with GCH above a
strongly compact cardinal). This was one of the first attempts at doing
anything subtantial with ZFC+#, and Steel remarks this might be evidence
that the experiment suggested by Friedman simply cannot be done; one is
bound to run into the inconsistency.
–
Aatu Koskensilta (aatu.koskensi…@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
– Ludwig Wittgenstein, Tractatus Logico-Philosophicus
On 16 Dec 2005 15:18:31 GMT, Chris Menzel <cmen…@remove-this.tamu.edu>
said:
> On Fri, 16 Dec 2005 08:02:36 GMT, Nam Nguyen <namducngu…@shaw.ca> said:
> …
>> Suppose for discussion ZFC is genuienly inconsistent. Obvisouly There
>> have been only a finite number of theories that we have proven so far,
>> or that we could prove as long as humanity could last. Take any of
>> such theorem F, isn’t it possible that the proof of ~F could be too
>> long for us to *know*? and hence as far as we understand the nature of
>> humanity, ZFC would exist as an inconsistent theory without us ever
>> knowing that?
> Sure.
"Possible" in the sense of "conceivable", that is. It’s consistent with
what we know that there are such contradictions in ZFC. This in itself
of course provides no reason for actually believing there are any.
On Fri, 16 Dec 2005 08:02:36 GMT, Nam Nguyen <namducngu…@shaw.ca> said:
>> On Fri, 16 Dec 2005 06:01:05 GMT, Nam Nguyen <namducngu…@shaw.ca>
>> said:
>>>Godel’s Incompleteness work depends heavily on the assumption
>>>that the theory T in question be consistent.
>> No, it doesn’t.
> If T is not consistent, wouldn’t it be complete?
Of course. In what way does this make the theorem "heavily dependent"
on T’s consistency?
- Hide quoted text — Show quoted text -
>>>(2) Is it true then there is *a real danger* that there might exist an
>>> inconsistent theory T in the sense that for all the theorems that
>>> we – as human being – could conceivably prove, their negation
>>> counterparts would have proof lengths that are beyond human capacity
>>> to comprehend?
>> That is a very strange sense of inconsistency.
> I didn’t change the definition of inconsistency though, which is that
> for a given formula F, both F and ~F would be provable from T. What I
> stated is simply a *suspicion* of something like the following:
> Suppose for discussion ZFC is genuienly inconsistent. Obvisouly There
> have been only a finite number of theories that we have proven so far,
> or that we could prove as long as humanity could last. Take any of
> such theorem F, isn’t it possible that the proof of ~F could be too
> long for us to *know*? and hence as far as we understand the nature of
> humanity, ZFC would exist as an inconsistent theory without us ever
> knowing that?
Sure.
Aatu Koskensilta wrote:
> I don’t know about "real danger" but it’s certainly conceiveable that
> there exists a proof of a contradiction in e.g. ZFC that is so
> complicated and long that no human will ever comprehend it.
Then use a computer.
> There’s no reason to suppose this is the case, though.
> I don’t recall if it was Friedman or some one else who conjectured that
> naturally occuring inconsistent theories have simple contradictions. It
> seems empirical evidence supports this: contradictions found in actually
> proposed theories are rather simple
Once again, the real-life instantiation of formal logic (computer
programming) yields the answer: It can be years before each software
bug is discovered and removed. Internally, something is inconsistent.
Like dominos, other variables then become inconsistent. But if it
doesn’t reach actual output, we never know. (This is similar to one
noticing funny things happening on his PC, and shutting it down before
it crashes.) Likewise, we might not discover the inconsistency in a
system of Logic for a long time.
But to be precise: What is the longest minimal proof of FALSE in a
system of 1 axiom or rule? 2 axioms and rules? 3? By hand or by
computer, we can get a feel for how fast this function grows.
[That is, for each system of a given size, determine the shortest proof
of FALSE. Remember the longest such proof among all of the systems of
that size.]
If someone will list here a specific set of propositional calculus
axioms and rules of inference, I will throw together a program to
determine the shortest proof of FALSE for each subset of size 1, 2, . .
. I’ll probably actually map (# of axioms,# of rules) => # of steps
in shortest proof of false. [if possible?]
Hmmm . . . When can we prove a bound on this number based on a manual
analysis of the axioms and rules? A generalization of the concept of
inconsistency – the length of the shortest proof of FALSE!
C-B
C-B’s 2nd. Incompleteness Theorem: You can’t compute the minimal
length of a proof of FALSE.
Corollary: Lob’s Theorem (Proof is left as an exercise.)
- Hide quoted text — Show quoted text -
> Aatu Koskensilta (aatu.koskensi…@xortec.fi)
> "Wovon man nicht sprechen kann, daruber muss man schweigen"
> – Ludwig Wittgenstein, Tractatus Logico-Philosophicus
"Chris Menzel" <cmen…@remove-this.tamu.edu> wrote in message
news:slrndq5ne7.1r4v.cmenzel@philebus.tamu.edu…
- Hide quoted text — Show quoted text -
> On 16 Dec 2005 15:18:31 GMT, Chris Menzel <cmen…@remove-this.tamu.edu>
> said:
>> On Fri, 16 Dec 2005 08:02:36 GMT, Nam Nguyen <namducngu…@shaw.ca> said:
>> …
>>> Suppose for discussion ZFC is genuienly inconsistent. Obvisouly There
>>> have been only a finite number of theories that we have proven so far,
>>> or that we could prove as long as humanity could last. Take any of
>>> such theorem F, isn’t it possible that the proof of ~F could be too
>>> long for us to *know*? and hence as far as we understand the nature of
>>> humanity, ZFC would exist as an inconsistent theory without us ever
>>> knowing that?
>> Sure.
> "Possible" in the sense of "conceivable", that is. It’s consistent with
> what we know that there are such contradictions in ZFC. This in itself
> of course provides no reason for actually believing there are any.
I interpret your question differently.
Are you asking if you could show the proof of ~P would be so long that it
couldn’t be written down with all the atoms of the Universe, then this may
be so, but mathematics doesn’t concern itself with practicalities like that.
When you have infinity to play with, the Universe seems small.
- Hide quoted text — Show quoted text -
Chris Menzel wrote:
> On 16 Dec 2005 15:18:31 GMT, Chris Menzel <cmen…@remove-this.tamu.edu>
> said:
>>On Fri, 16 Dec 2005 08:02:36 GMT, Nam Nguyen <namducngu…@shaw.ca> said:
>> …
>>>Suppose for discussion ZFC is genuienly inconsistent. Obvisouly There
>>>have been only a finite number of theories that we have proven so far,
>>>or that we could prove as long as humanity could last. Take any of
>>>such theorem F, isn’t it possible that the proof of ~F could be too
>>>long for us to *know*? and hence as far as we understand the nature of
>>>humanity, ZFC would exist as an inconsistent theory without us ever
>>>knowing that?
>>Sure.
> "Possible" in the sense of "conceivable", that is. It’s consistent with
> what we know that there are such contradictions in ZFC. This in itself
> of course provides no reason for actually believing there are any.
I’m not quite sure I get what you intended to say here. What did you
really mean by "any" in "…provides no reason for actually believing
there are _any_"? I don’t think you meant there isn’t any pair F,~F that
are both provable from ZFC, since you stated that "we know there are
such contradictions in ZFC". But did you meant my supposing ZFC be
as such still provides no reason to believe there is not _any_ theory T
that, although satisfying the requirements that it be strong enough
to carry out arithmetic concepts, would be as such – i.e. would be
inconsistent without us ever knowing a particular pair F, ~F that are
both theorems? Would you please clarify? Thanks.
–
—————————————————-
Time passes, there is no way we can hold it back.
Why then do thoughts linger, long after everything
else is gone?
Ryokan
—————————————————-
- Hide quoted text — Show quoted text -
Peter Webb wrote:
> "Chris Menzel" <cmen…@remove-this.tamu.edu> wrote in message
> news:slrndq5ne7.1r4v.cmenzel@philebus.tamu.edu…
>>On 16 Dec 2005 15:18:31 GMT, Chris Menzel <cmen…@remove-this.tamu.edu>
>>said:
>>>On Fri, 16 Dec 2005 08:02:36 GMT, Nam Nguyen <namducngu…@shaw.ca> said:
>>> …
>>>>Suppose for discussion ZFC is genuienly inconsistent. Obvisouly There
>>>>have been only a finite number of theories that we have proven so far,
>>>>or that we could prove as long as humanity could last. Take any of
>>>>such theorem F, isn’t it possible that the proof of ~F could be too
>>>>long for us to *know*? and hence as far as we understand the nature of
>>>>humanity, ZFC would exist as an inconsistent theory without us ever
>>>>knowing that?
>>>Sure.
>>"Possible" in the sense of "conceivable", that is. It’s consistent with
>>what we know that there are such contradictions in ZFC. This in itself
>>of course provides no reason for actually believing there are any.
> I interpret your question differently.
> Are you asking if you could show the proof of ~P would be so long that it
> couldn’t be written down with all the atoms of the Universe, then this may
> be so, but mathematics doesn’t concern itself with practicalities like that.
Imho, mathematicians should concern themselves with such practicalities
because mathematics *does* concern itself with such practicalities! The
evidences for such mathematics’ concern are never 100% explicit but they
are there nonetheless. For example:
1) FOL’s formulae must be of *finite* length.
2) FOL’s proofs must be of *finite* length.
3) Shoenfield’s remark: "Proofs deal with _concrete_ objects [formulae]
in a _constructive_ manner are said to be _finitary_."
4) Kreisel’s alleged remark to the effect "a proof is finitary if we
can ‘visualize’ the proof."
Now unless we were confident we could _spell out_ infinite number of
formulae, we have to brace ourselves for the possibility that there are
proofs whose proof length, or an hypothesis’ length, is beyond humanity
grasp! Just in case we forget, suppose ~GC is genuinely true, God never
promise that its value be small enough that all the sub-atomic particles
in a trillion universes would be adequate to represent 1/trillionth of
the value!
> When you have infinity to play with, the Universe seems small.
Right, and that’s why I think we should be concerned with that possible
practicality.
–
—————————————————-
Time passes, there is no way we can hold it back.
Why then do thoughts linger, long after everything
else is gone?
Ryokan
—————————————————-
On Sun, 18 Dec 2005 23:06:50 GMT, Nam Nguyen <namducngu…@shaw.ca> said:
- Hide quoted text — Show quoted text -
> …
>>>>Suppose for discussion ZFC is genuienly inconsistent. Obvisouly
>>>>There have been only a finite number of theories that we have proven
>>>>so far, or that we could prove as long as humanity could last. Take
>>>>any of such theorem F, isn’t it possible that the proof of ~F could
>>>>be too long for us to *know*? and hence as far as we understand the
>>>>nature of humanity, ZFC would exist as an inconsistent theory
>>>>without us ever knowing that?
>>>Sure.
>> "Possible" in the sense of "conceivable", that is. It’s consistent with
>> what we know that there are such contradictions in ZFC. This in itself
>> of course provides no reason for actually believing there are any.
> I’m not quite sure I get what you intended to say here. What did you
> really mean by "any" in "…provides no reason for actually believing
> there are _any_"? I don’t think you meant there isn’t any pair F,~F
> that are both provable from ZFC, since you stated that "we know there
> are such contradictions in ZFC".
You seem to have a reading comprehension problem.
Or perhaps you write talking point memos for the Republican party.
- Hide quoted text — Show quoted text -
Chris Menzel wrote:
> On Sun, 18 Dec 2005 23:06:50 GMT, Nam Nguyen <namducngu…@shaw.ca> said:
>>…
>>>>>Suppose for discussion ZFC is genuienly inconsistent. Obvisouly
>>>>>There have been only a finite number of theories that we have proven
>>>>>so far, or that we could prove as long as humanity could last. Take
>>>>>any of such theorem F, isn’t it possible that the proof of ~F could
>>>>>be too long for us to *know*? and hence as far as we understand the
>>>>>nature of humanity, ZFC would exist as an inconsistent theory
>>>>>without us ever knowing that?
>>>>Sure.
>>>"Possible" in the sense of "conceivable", that is. It’s consistent with
>>>what we know that there are such contradictions in ZFC. This in itself
>>>of course provides no reason for actually believing there are any.
>>I’m not quite sure I get what you intended to say here. What did you
>>really mean by "any" in "…provides no reason for actually believing
>>there are _any_"? I don’t think you meant there isn’t any pair F,~F
>>that are both provable from ZFC, since you stated that "we know there
>>are such contradictions in ZFC".
> You seem to have a reading comprehension problem.
Why is it that you could ask questions and I couldn’t, in a
conversation?
> Or perhaps you write talking point memos for the Republican party.
–
—————————————————-
Time passes, there is no way we can hold it back.
Why then do thoughts linger, long after everything
else is gone?
Ryokan
—————————————————-
Charlie-Boo wrote:
> Aatu Koskensilta wrote:
>>I don’t know about "real danger" but it’s certainly conceiveable that
>>there exists a proof of a contradiction in e.g. ZFC that is so
>>complicated and long that no human will ever comprehend it.
> Then use a computer.
It’s certainly conveicevable that there exists a proof of a
contradiction in e.g. ZFC that is so complicated and long that no human
will ever discover it even with the aid of all computational resources
available in the universe.
> Once again, the real-life instantiation of formal logic (computer
> programming) yields the answer: It can be years before each software
> bug is discovered and removed.
This is not particularly relevant to the fictionary conjecture.
–
Aatu Koskensilta (aatu.koskensi…@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
– Ludwig Wittgenstein, Tractatus Logico-Philosophicus
On Mon, 19 Dec 2005 02:11:49 GMT, Nam Nguyen <namducngu…@shaw.ca>
wrote:
- Hide quoted text — Show quoted text -
>Chris Menzel wrote:
>> On Sun, 18 Dec 2005 23:06:50 GMT, Nam Nguyen <namducngu…@shaw.ca> said:
>>>…
>>>>>>Suppose for discussion ZFC is genuienly inconsistent. Obvisouly
>>>>>>There have been only a finite number of theories that we have proven
>>>>>>so far, or that we could prove as long as humanity could last. Take
>>>>>>any of such theorem F, isn’t it possible that the proof of ~F could
>>>>>>be too long for us to *know*? and hence as far as we understand the
>>>>>>nature of humanity, ZFC would exist as an inconsistent theory
>>>>>>without us ever knowing that?
>>>>>Sure.
>>>>"Possible" in the sense of "conceivable", that is. It’s consistent with
>>>>what we know that there are such contradictions in ZFC. This in itself
>>>>of course provides no reason for actually believing there are any.
>>>I’m not quite sure I get what you intended to say here. What did you
>>>really mean by "any" in "…provides no reason for actually believing
>>>there are _any_"? I don’t think you meant there isn’t any pair F,~F
>>>that are both provable from ZFC, since you stated that "we know there
>>>are such contradictions in ZFC".
>> You seem to have a reading comprehension problem.
>Why is it that you could ask questions and I couldn’t, in a
>conversation?
The problem is that you don’t seem to be reading what he
writes in reply to your questions – under those circumstances
continuing to reply to new questions seems pointless.
In particular, two things I noticed right away although I
haven’t been following this:
(i) you say that he said "we know there are such contradictions
in ZFC". He didn’t say that. He said "It’s consistent with
what we know that there are such contradictions in ZFC."
There’s a big difference.
(ii) You claim not to understand what he meant by "any"
in ""…provides no reason for actually believing
there are any"? What he meant was _exactly_ what he said:
The fact that it’s consistent with what we know that there
are contradictions in ZFC does not provide any reason for
believing that there _are_ any contradictions in ZFC.
I had the same problem when I was trying to discuss something
with you a little while ago – I’d say exactly what I meant,
in perfectly clear language, and you’d start speculating
about what I meant, not seeming to consider the possibility
that what I meant was exactly what I’d said. It happens a
lot that when _you_ write something here it’s not at all
clear what you mean, because if we take what you say
literally it doesn’t make any sense. But you shouldn’t
necessarily assume that that applies to everyone else –
there _are_ people who do manage to say exactly what they
mean most of the time.
>> Or perhaps you write talking point memos for the Republican party.
************************
David C. Ullrich
- Hide quoted text — Show quoted text -
David C. Ullrich wrote:
> On Mon, 19 Dec 2005 02:11:49 GMT, Nam Nguyen <namducngu…@shaw.ca>
> wrote:
>>Chris Menzel wrote:
>>>On Sun, 18 Dec 2005 23:06:50 GMT, Nam Nguyen <namducngu…@shaw.ca> said:
>>>>…
>>>>>>>Suppose for discussion ZFC is genuienly inconsistent. Obvisouly
>>>>>>>There have been only a finite number of theories that we have proven
>>>>>>>so far, or that we could prove as long as humanity could last. Take
>>>>>>>any of such theorem F, isn’t it possible that the proof of ~F could
>>>>>>>be too long for us to *know*? and hence as far as we understand the
>>>>>>>nature of humanity, ZFC would exist as an inconsistent theory
>>>>>>>without us ever knowing that?
>>>>>>Sure.
>>>>>"Possible" in the sense of "conceivable", that is. It’s consistent with
>>>>>what we know that there are such contradictions in ZFC. This in itself
>>>>>of course provides no reason for actually believing there are any.
>>>>I’m not quite sure I get what you intended to say here. What did you
>>>>really mean by "any" in "…provides no reason for actually believing
>>>>there are _any_"? I don’t think you meant there isn’t any pair F,~F
>>>>that are both provable from ZFC, since you stated that "we know there
>>>>are such contradictions in ZFC".
>>>You seem to have a reading comprehension problem.
>>Why is it that you could ask questions and I couldn’t, in a
>>conversation?
> The problem is that you don’t seem to be reading what he
> writes in reply to your questions – under those circumstances
> continuing to reply to new questions seems pointless.
> In particular, two things I noticed right away although I
> haven’t been following this:
> (i) you say that he said "we know there are such contradictions
> in ZFC". He didn’t say that. He said "It’s consistent with
> what we know that there are such contradictions in ZFC."
> There’s a big difference.
> (ii) You claim not to understand what he meant by "any"
> in ""…provides no reason for actually believing
> there are any"? What he meant was _exactly_ what he said:
> The fact that it’s consistent with what we know that there
> are contradictions in ZFC does not provide any reason for
> believing that there _are_ any contradictions in ZFC.
First of all. Both he and you are wrong on this. What he and you
really meant is:
"The fact that it’s consistent with what we know *about*
conceivable contradictions in ZFC does not provide any reason for
believing that there _actually are_ any contradictions in ZFC."
The key here is the preposition "about", or "of" of the verb "know",
which has 2 modes: transitive and intransitive.
(1) "I know Titan has life"
(2) "I know about [of] Titan having life"
There is a great difference in these 2 modes. If I were a renown
astro-biologist and stated (1), then the press would take that as a fact
and would have this headline "Beware, we are not alone!". If I only
stated (2), they should probably be careful and not publish that
headline: I probably only knew the rumor/suspicion/discussion *about*
Titan harbouring life – but that should not constitute a fact!
So before he or you accuse me of having "reading comprehension problem",
don’t you think that both he and you ought to reflect on what you
exactly meant vs. what you exactly said, here? Don’t you think that
in discussing logic, both the object language-expression and the
meta-language-expression have to be *precise*? [Or at least, when asked
one should be prepared to clarify it as a courtesy, instead of getting
mad!]
Secondly, remember that one of my two key questions is
> (2) Is it true then there is *a real danger* that there might exist an
> inconsistent theory T in the sense that for all the theorems that
> we – as human being – could conceivably prove, their negation
> counterparts would have proof lengths that are beyond human capacity
> to comprehend?
and remember that I clarified the question by *hypothetically
"supposing" ZFC is inconsistent?
So what happens to my key question about "a real danger"? I’m sorry:
I already supposed ZFC be genuinely inconsistent; so that meant his
"conclusion" about there is no reason to believe there are any
contradiction in ZFC is a bit off of my expectation for an answer
about the "danger"! I made a tacit silence and not asked him whether
he actually understood the main questions and the main theme of my op.
Perhaps he should realize that more than accusing me when I simply
asked for a clarification.
> I had the same problem when I was trying to discuss something
> with you a little while ago – I’d say exactly what I meant,
> in perfectly clear language, and you’d start speculating
> about what I meant, not seeming to consider the possibility
> that what I meant was exactly what I’d said. It happens a
> lot that when _you_ write something here it’s not at all
> clear what you mean, because if we take what you say
> literally it doesn’t make any sense. But you shouldn’t
> necessarily assume that that applies to everyone else –
> there _are_ people who do manage to say exactly what they
> mean most of the time.
I don’t agree: the precision in mathematics and logic requires
us to say exactly what we mean *all the time*. If we have to be
informal in order to make the conversation easy then that’s fine.
But that doesn’t mean we could abandon the courtesy of clarifying
things upon request. Humans (experts or otherwise) do have typo’s,
misreading, overlook, mistakes… Naturally.
>>>Or perhaps you write talking point memos for the Republican party.
> ************************
> David C. Ullrich
–
—————————————————-
Time passes, there is no way we can hold it back.
Why then do thoughts linger, long after everything
else is gone?
Ryokan
—————————————————-
Nam Nguyen wrote:
> (2) Is it true then there is *a real danger* that there might exist an
> inconsistent theory T in the sense that for all the theorems that
> we – as human being – could conceivably prove, their negation
> counterparts would have proof lengths that are beyond human capacity
> to comprehend?
Here’s an another answer to this question that might be more on the
lines of what you’re looking for. There is no effective way to bound the
length of proofs of contradiction; if there were we could effectively
test whether a theory is contradictory or not by going trough all proofs
< the bound and this would solve the halting problem. So there most
certainly are inconsistent theories T in which the shortest proofs of
contradiction are beyond human powers to produce or recognize.
–
Aatu Koskensilta (aatu.koskensi…@xortec.fi)
"Wovon man nicht sprechen kann, darüber muss man schweigen"
– Ludwig Wittgenstein, Tractatus Logico-Philosophicus
On Tue, 20 Dec 2005 06:29:28 GMT, Nam Nguyen <namducngu…@shaw.ca>
wrote:
- Hide quoted text — Show quoted text -
>David C. Ullrich wrote:
>> On Mon, 19 Dec 2005 02:11:49 GMT, Nam Nguyen <namducngu…@shaw.ca>
>> wrote:
>>>Chris Menzel wrote:
>>>>On Sun, 18 Dec 2005 23:06:50 GMT, Nam Nguyen <namducngu…@shaw.ca> said:
>>>>>…
>>>>>>>>Suppose for discussion ZFC is genuienly inconsistent. Obvisouly
>>>>>>>>There have been only a finite number of theories that we have proven
>>>>>>>>so far, or that we could prove as long as humanity could last. Take
>>>>>>>>any of such theorem F, isn’t it possible that the proof of ~F could
>>>>>>>>be too long for us to *know*? and hence as far as we understand the
>>>>>>>>nature of humanity, ZFC would exist as an inconsistent theory
>>>>>>>>without us ever knowing that?
>>>>>>>Sure.
>>>>>>"Possible" in the sense of "conceivable", that is. It’s consistent with
>>>>>>what we know that there are such contradictions in ZFC. This in itself
>>>>>>of course provides no reason for actually believing there are any.
>>>>>I’m not quite sure I get what you intended to say here. What did you
>>>>>really mean by "any" in "…provides no reason for actually believing
>>>>>there are _any_"? I don’t think you meant there isn’t any pair F,~F
>>>>>that are both provable from ZFC, since you stated that "we know there
>>>>>are such contradictions in ZFC".
>>>>You seem to have a reading comprehension problem.
>>>Why is it that you could ask questions and I couldn’t, in a
>>>conversation?
>> The problem is that you don’t seem to be reading what he
>> writes in reply to your questions – under those circumstances
>> continuing to reply to new questions seems pointless.
>> In particular, two things I noticed right away although I
>> haven’t been following this:
>> (i) you say that he said "we know there are such contradictions
>> in ZFC". He didn’t say that. He said "It’s consistent with
>> what we know that there are such contradictions in ZFC."
>> There’s a big difference.
>> (ii) You claim not to understand what he meant by "any"
>> in ""…provides no reason for actually believing
>> there are any"? What he meant was _exactly_ what he said:
>> The fact that it’s consistent with what we know that there
>> are contradictions in ZFC does not provide any reason for
>> believing that there _are_ any contradictions in ZFC.
>First of all. Both he and you are wrong on this. What he and you
>really meant is:
>"The fact that it’s consistent with what we know *about*
>conceivable contradictions in ZFC does not provide any reason for
>believing that there _actually are_ any contradictions in ZFC."
That’s certainly not what _I_ meant. I can’t even figure out
what that "sentence" _means_ – the referent of "it" is unclear.
What I meant was exactly what I said.
- Hide quoted text — Show quoted text -
>The key here is the preposition "about", or "of" of the verb "know",
>which has 2 modes: transitive and intransitive.
>(1) "I know Titan has life"
>(2) "I know about [of] Titan having life"
>There is a great difference in these 2 modes. If I were a renown
>astro-biologist and stated (1), then the press would take that as a fact
>and would have this headline "Beware, we are not alone!". If I only
>stated (2), they should probably be careful and not publish that
>headline: I probably only knew the rumor/suspicion/discussion *about*
>Titan harbouring life – but that should not constitute a fact!
>So before he or you accuse me of having "reading comprehension problem",
>don’t you think that both he and you ought to reflect on what you
>exactly meant vs. what you exactly said, here?
Before you tell us not to accuse you of reading comprehension
problems you should reflect on the fact that you explicitly
stated that CM said things that he simply never said.
There’s another doozy below:
- Hide quoted text — Show quoted text -
>Don’t you think that
>in discussing logic, both the object language-expression and the
>meta-language-expression have to be *precise*? [Or at least, when asked
>one should be prepared to clarify it as a courtesy, instead of getting
>mad!]
>Secondly, remember that one of my two key questions is
> > (2) Is it true then there is *a real danger* that there might exist an
> > inconsistent theory T in the sense that for all the theorems that
> > we – as human being – could conceivably prove, their negation
> > counterparts would have proof lengths that are beyond human capacity
> > to comprehend?
>and remember that I clarified the question by *hypothetically
>"supposing" ZFC is inconsistent?
>So what happens to my key question about "a real danger"? I’m sorry:
>I already supposed ZFC be genuinely inconsistent; so that meant his
>"conclusion" about there is no reason to believe there are any
>contradiction in ZFC is a bit off of my expectation for an answer
>about the "danger"! I made a tacit silence and not asked him whether
>he actually understood the main questions and the main theme of my op.
>Perhaps he should realize that more than accusing me when I simply
>asked for a clarification.
>> I had the same problem when I was trying to discuss something
>> with you a little while ago – I’d say exactly what I meant,
>> in perfectly clear language, and you’d start speculating
>> about what I meant, not seeming to consider the possibility
>> that what I meant was exactly what I’d said. It happens a
>> lot that when _you_ write something here it’s not at all
>> clear what you mean, because if we take what you say
>> literally it doesn’t make any sense. But you shouldn’t
>> necessarily assume that that applies to everyone else –
>> there _are_ people who do manage to say exactly what they
>> mean most of the time.
>I don’t agree: the precision in mathematics and logic requires
>us to say exactly what we mean *all the time*.
You think that "the precision in mathematics and logic requires
us to say exactly what we mean *all the time*" somehow
contradicts something I said?
Like he said: You seem to have reading comprehension problems.
>If we have to be
>informal in order to make the conversation easy then that’s fine.
>But that doesn’t mean we could abandon the courtesy of clarifying
>things upon request. Humans (experts or otherwise) do have typo’s,
>misreading, overlook, mistakes… Naturally.
>>>>Or perhaps you write talking point memos for the Republican party.
>> ************************
>> David C. Ullrich
************************
David C. Ullrich
- Hide quoted text — Show quoted text -
Aatu Koskensilta wrote:
> Nam Nguyen wrote:
>> (2) Is it true then there is *a real danger* that there might exist an
>> inconsistent theory T in the sense that for all the theorems that
>> we – as human being – could conceivably prove, their negation
>> counterparts would have proof lengths that are beyond human capacity
>> to comprehend?
> Here’s an another answer to this question that might be more on the
> lines of what you’re looking for. There is no effective way to bound the
> length of proofs of contradiction; if there were we could effectively
> test whether a theory is contradictory or not by going trough all proofs
> < the bound and this would solve the halting problem. So there most
> certainly are inconsistent theories T in which the shortest proofs of
> contradiction are beyond human powers to produce or recognize.
Thanks the information. Let me try to explain the "danger" a little
further. When I mentioned "…all the theorems … human being
…could conceivably prove…" I meant to include the axioms. So
we’d know the formulation of the theory, say, T. It’s just that
any contradiction of any of what we know as theorems would be
just too long. But what about a model M of T? As I understand, an
inconsistent theory has no model, so how would that "rhyme" with
the all the theorems that we could prove? In other words, could there
exist a *partial* model M for the T above, in which all the *known*
theorems could be interpreted as true – but overall M as a ‘structure’
is not considered as a model of T?
My fear of the "danger" is that if such a T and M exist, there’d exist
a high degree of "deception", if not outright "cheating" on the part
of FOL’s framework.
–
—————————————————-
Time passes, there is no way we can hold it back.
Why then do thoughts linger, long after everything
else is gone?
Ryokan
—————————————————-
Nam Nguyen wrote:
> In other words, could there exist a *partial* model M for the [inconsistent] T,
> in which all the *known* theorems could be interpreted as true – but overall M
> as a ‘structure’ is not considered as a model of T?
It’s not clear how to answer this question since the notion of a
"partial model" is not clear. Do you have some specific definition in mind?
–
Aatu Koskensilta (aatu.koskensi…@xortec.fi)
"Wovon man nicht sprechen kann, darüber muss man schweigen"
– Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Aatu Koskensilta wrote:
> Nam Nguyen wrote:
>> In other words, could there exist a *partial* model M for the
>> [inconsistent] T,
>> in which all the *known* theorems could be interpreted as true – but
>> overall M as a ‘structure’ is not considered as a model of T?
> It’s not clear how to answer this question since the notion of a
> "partial model" is not clear. Do you have some specific definition in mind?
I suppose I could try this definition:
A partial "model" M of T is a structure in which there exist formulae
F1, F2 where F1 is interpreted as true, while F2 as false.
Since T is inconsistent, it’s complete; hence both F1 and F2 are
theorems of T. But since we don’t claim M is a [full] model of T,
F2′s being false doesn’t seem to be a contradiction, overall.
–
—————————————————-
Time passes, there is no way we can hold it back.
Why then do thoughts linger, long after everything
else is gone?
Ryokan
—————————————————-
Nam Nguyen wrote:
> Aatu Koskensilta wrote:
>> It’s not clear how to answer this question since the notion of a
>> "partial model" is not clear. Do you have some specific definition in
>> mind?
> I suppose I could try this definition:
> A partial "model" M of T is a structure in which there exist formulae
> F1, F2 where F1 is interpreted as true, while F2 as false.
This makes no obvious sense. What is meant by "interpreted as true" in
the above? Are F1 and F2 theorems of T?
–
Aatu Koskensilta (aatu.koskensi…@xortec.fi)
"Wovon man nicht sprechen kann, darüber muss man schweigen"
– Ludwig Wittgenstein, Tractatus Logico-Philosophicus
- Hide quoted text — Show quoted text -
Aatu Koskensilta wrote:
> Nam Nguyen wrote:
>> Aatu Koskensilta wrote:
>>> It’s not clear how to answer this question since the notion of a
>>> "partial model" is not clear. Do you have some specific definition in
>>> mind?
>> I suppose I could try this definition:
>> A partial "model" M of T is a structure in which there exist formulae
>> F1, F2 where F1 is interpreted as true, while F2 as false.
> This makes no obvious sense. What is meant by "interpreted as true" in
> the above? Are F1 and F2 theorems of T?
Let me present another version:
Let T be a general (1st order) theory, F1, F2 be theorems of T.
A structure M is called a "partial model" of T iff F1 is true in
M but F2 is not.
The following would be an example of a "partial model". Let G’
be a theory by extending the group theory by the following 2 axioms:
A1: ExAy (x=y)
A2: Ex1,x2(~(x1 = x2))
Any singleton (ZF) set could be used as a model M for G’. Let F1 = A1
and F2 = A2, then such an M is a "partial model" of G.
–
—————————————————-
Time passes, there is no way we can hold it back.
Why then do thoughts linger, long after everything
else is gone?
Ryokan
—————————————————-