robinson arithmetic is not incomplete
elsiemelsi
is not incomplete
it is said
http://en.wikipedia.org/wiki/Robinson_arithmetic
robinson arithmetic like PA is incomplete and incompletable in the sense
of Gödel's incompleteness Theorems, and essentially undecidable
http://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_theorem
Gödel's first incompleteness theorem, perhaps the single most celebrated
result in mathematical logic, states that:
For any consistent formal, recursively enumerable theory that proves
basic arithmetical truths, an arithmetical statement that is true, but not
provable in the theory, can be constructed.1 That is, any effectively
generated theory capable of expressing elementary arithmetic cannot be
both consistent and complete.
but godel does not tell us what makes a statement true
until he does we cant tell if robinson arithmetic is incomplete
we need to know what makes a statement true in order to see that it cant
be proven
godel could have said there are gibbly statemets which cant be proven
but untill we know what makes a statement gibbly we have no way of knowing
what his theorem is talking about
so untill he tells us what makes a statement true robinson arthmetic cant
be said to be incomplete
--
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More information at http://www.talkaboutscience.com/faq.html
george
> but godel does not tell us what makes a statement true
>
> until he does we cant tell if robinson arithmetic is incomplete
Of course we can.
> we need to know what makes a
> statement true in order to see that it cant
> be proven
No, we don't.
> godel could have said there are
> gibbly statemets which cant be proven
Right.
> but untill we know what makes a statement gibbly
> we have no way of knowing
> what his theorem is talking about
Wrong.
It DOES NOT MATTER what OTHER attributes the
statement may or may not have. If the theory
proves NEITHER the statement NOR its denial THEN
the theory is incomplete. ONE OR THE OTHER of the
statement and its denial MUST be true, but we DON'T
NEED to know WHICH, or what "truth" "is", in order to know
to know that theory is incomplete.
But since you have never studied this stuff,
you didn't know that.
> so untill he tells us what makes a
> statement true robinson arthmetic cant
> be said to be incomplete
This whole endeavor DOES NOT EVEN CARE about
what's TRUE: it CARES about what's PROVABLE.
And we all HAVE told you that BEING AN AXIOM makes
a statement provable, AS does being INFERRABLE from
the axioms ACCORDING to the INFERENCE RULES of
the LOGIC.
"Truth" in the arithmetical sense that the endeavor is dealing
with IS something that you could be or possibly even have
been told about, if not by Godel then by YOUR ELEMENTARY
SCHOOL math teachers. The theories have a "standard"
model, where 0 is zero, s(0) or (0+1) is 1, s(s(0)) or 1+1 is
two, ad inf. All those numbers (and many different numeration
systems for them, in addition to the two mentioned HERE)
were ALREADY OUT THERE BEFORE Godel started
proving things, and EVERYbody was legitimately expected
to know what made 43*69=3426 true (or false), WITHOUT
any help from Godel or 1st-order logic. So the point is,
WE DO ALREADY KNOW what "makes" these statements
true or false. The ones involving "Every" natural number or
"Some" natural number might require an infinitary process for
confirmation or refutation, but the point is, we KNOW it is
THAT process and NOT some other, so we DO know what
"makes" those statements true, even when we can't
practically perform the investigation.
elsiemelsi
Right.
> but untill we know what makes a statement gibbly
> we have no way of knowing
> what his theorem is talking about
Wrong.
It DOES NOT MATTER what OTHER attributes the
statement may or may not have. If the theory
proves NEITHER the statement NOR its denial THEN
the theory is incomplete. ONE OR THE OTHER of the
statement and its denial MUST be true, but we DON'T
NEED to know WHICH, or what "truth" "is", in order to know
to know that theory is incomplete.
i say
rubbish unless i know what a gibbly i cant not know what his theorem is
talking about
george say
we DON'TNEED to know WHICH, or what "truth" "is", in order to know to
know that theory is incomplete.
rubbish until i know what makes a statement true i cant identify a true
statement thus his theorem is meaningless as with out knowing what a true
statement is i cant say there are true statements which are not provable
george says
This whole endeavor DOES NOT EVEN CARE about
what's TRUE: it CARES about what's PROVABLE.
And we all HAVE told you that BEING AN AXIOM makes
a statement provable, AS does being INFERRABLE from
the axioms ACCORDING to the INFERENCE RULES of
the LOGIC.
i say
rubbish until i know what makes a statement true i cant identify a true
statement thus his theorem is meaningless as with out knowing what a true
statement is i cant say there are true statements which are not provable
goerge says
This whole endeavor DOES NOT EVEN CARE about
what's TRUE:
i say rubbish it depends on there being identifibale true statements if i
cant tell what a true statement is the theorem collapses into
meaninglessness as i cant never find a true statement which i will know is
not provable
if there are no such thing as true statement the theorenm is meaningless
and
if there are true statement but i dont know what the hell they are then
again the theorem is meaningless as i cant make any identifications about
anything ie unprovable statments
Peter_Smith
> The australian philosopher colin leslie dean points out robinson arithmetic
> is not incomplete
Then the australian philosopher colin leslie dean is a buffoon.
Perhaps colin leslie dean would like to prove either (Ax)(0 + x = x)
or its negation in Robinson Arithmetic as defined e.g. in Boolos/
Burgess/Jeffrey edn 4, or in my book.
I'm sure that George and I will chip in for a $1000 prize if he
does ... and Aatu will double it! ;-)
elsiemelsi
The theories have a "standard"
model, where 0 is zero, s(0) or (0+1) is 1, s(s(0)) or 1+1 is
two, ad inf. All those numbers (and many different numeration
systems for them, in addition to the two mentioned HERE)
were ALREADY OUT THERE BEFORE Godel started
proving things, and EVERYbody was legitimately expected
to know what made 43*69=3426 true (or false), WITHOUT
any help from Godel or 1st-order logic. So the point is,
WE DO ALREADY KNOW what "makes" these statements
true or false.
i say
rubbish
you say
43*69=3426 true
it is only true via a proof
but godel is said to have made a distinction between true and proven
you are just giving the old discredired idea of truth
ie the hilbert idea that true statements are proven from axioms
you need to tell us why 43*69=3426 is true proof
quote
http://en.wikipedia.org/wiki/Truth#Truth_in_mathematics
In addition, from at least the time of Hilbert's program at the turn of
the twentieth century to the proof of Gödel's theorem and the development
of the Church-Turing thesis in the early part of that century, true
statements in mathematics were generally assumed to be those statements
which are provable in a formal axiomatic system.
The works of Kurt Gödel, Alan Turing, and others shook this assumption,
with the development of statements that are true but cannot be proven
within the system
Rupert
> The australian philosopher colin leslie dean points out robinson arithmetic
> is not incomplete
>
> it is said
>
> http://en.wikipedia.org/wiki/Robinson_arithmetic
>
> robinson arithmetic like PA is incomplete and incompletable in the sense
> of Gödel's incompleteness Theorems, and essentially undecidable
>
> http://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_theorem
>
> Gödel's first incompleteness theorem, perhaps the single most celebrated
> result in mathematical logic, states that:
>
> For any consistent formal, recursively enumerable theory that proves
> basic arithmetical truths, an arithmetical statement that is true, but not
> provable in the theory, can be constructed.1 That is, any effectively
> generated theory capable of expressing elementary arithmetic cannot be
> both consistent and complete.
>
> but godel does not tell us what makes a statement true
>
> until he does we cant tell if robinson arithmetic is incomplete
>
There are two claims you might make:
(1) Robinson Arithmetic is complete
(2) Robinson Arithmetic has not been proved to be incomplete.
You started by saying (1), but you end up only arguing for (2). Do you
want to go with (1)? Do you think you can give some sort of
completeness proof for Robinson Arithmetic? Or do you just want to
stick with the weaker claim that it hasn't been proved to be
incomplete?
> we need to know what makes a statement true in order to see that it cant
> be proven
>
Two points:
(1) We do have a definition of truth, due to Tarski
(2) In any case, Gödel's argument can be done without using the notion
of truth, there is a purely syntactic version of the argument which
only needs the assumption that Robinson Arithmetic is consistent
> godel could have said there are gibbly statemets which cant be proven
>
The theorem which says "There exists a statement which is true but
unprovable in Robinson Arithmeic" is the semantic version of the
theorem. There is also a syntactic version which just says "If
Robinson Arithmetic is consistent, then there is a sentence which is
undecided by Robinson Arithmetic".
Cheers.
elsiemelsi
Perhaps colin leslie dean would like to prove either (Ax)(0 + x = x)
or its negation in Robinson Arithmetic as defined e.g. in Boolos/
Burgess/Jeffrey edn 4, or in my boo
i say rubbish
godel made a distinction between true and proof
true is independent of a proof
so tell us why Ax)(0 + x = x) is true independent of a proof
In addition, from at least the time of Hilbert's program at the turn of
the twentieth century to the proof of Gödel's theorem and the development
of the Church-Turing thesis in the early part of that century, true
statements in mathematics were generally assumed to be those statements
which are provable in a formal axiomatic system.
The works of Kurt Gödel, Alan Turing, and others shook this assumption,
with the development of statements that are true but cannot be proven
within the system
Rupert
> peter smith says
>
> Perhaps colin leslie dean would like to prove either (Ax)(0 + x = x)
> or its negation in Robinson Arithmetic as defined e.g. in Boolos/
> Burgess/Jeffrey edn 4, or in my boo
>
> i say rubbish
> godel made a distinction between true and proof
> true is independent of a proof
> so tell us why Ax)(0 + x = x) is true independent of a proof
>
Why does he have to tell you that?
You said Robinson Arithmetic is not incomplete. So you must think that
either this sentence or its negation can be proved in Robinson
Arithmetic. He's challenging you to actually do it.
> In addition, from at least the time of Hilbert's program at the turn of
> the twentieth century to the proof of Gödel's theorem and the development
> of the Church-Turing thesis in the early part of that century, true
> statements in mathematics were generally assumed to be those statements
> which are provable in a formal axiomatic system.
>
> The works of Kurt Gödel, Alan Turing, and others shook this assumption,
> with the development of statements that are true but cannot be proven
> within the system
>
> --
Aatu Koskensilta
> I'm sure that George and I will chip in for a $1000 prize if he
> does ... and Aatu will double it! ;-)
Gladly.
--
Aatu Koskensilta (aatu.kos...@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
fr...@live.fr
Who's that Colin Leslie anyway...? Why is he so frustrated by the
incompleteness theorem? weird....
Why do his followers keep claiming that Godel's theorem is false?
They keep claiming that there is a vicious circle in the demonstration
but there's none. There's no faulty circularity at all.
It just happens that we take a proposition that asserts that a well-
formed formula is unprobable. And then it turns out this well-formed
formula is indeed the one by which the proposition itself was
expressed.(cf note 15, page 598, Godel's paper inVan Heijenoort source
book).
Nam D. Nguyen
I'd say the theories have an *assumed* "standard" model, which
would make a difference as far as the validity of GIT, as a meta
theorem, is concerned.
> All those numbers (and many different numeration
> systems for them, in addition to the two mentioned HERE)
> were ALREADY OUT THERE BEFORE Godel started
> proving things, and EVERYbody was legitimately expected
> to know what made 43*69=3426 true (or false), WITHOUT
> any help from Godel or 1st-order logic.
> So the point is,
> WE DO ALREADY KNOW what "makes" these statements
> true or false. The ones involving "Every" natural number or
> "Some" natural number might require an infinitary process for
> confirmation or refutation, but the point is, we KNOW it is
> THAT process and NOT some other, so we DO know what
> "makes" those statements true, even when we can't
> practically perform the investigation.
Suppose ~GC is not provable in Q, what process would you think
would make us "know" GC be true?
Aatu Koskensilta
> Who's that Colin Leslie anyway...? Why is he so frustrated by the
> incompleteness theorem? weird....
Colin Leslie Dean is an esteemed "australia philosopher" and poet.
> Why do his followers keep claiming that Godel's theorem is false?
His "followers" seem to consist of himself only. He is on a noble
quest, valiantly attempting to show the meaninglessness of everything.
Aatu Koskensilta
> Suppose ~GC is not provable in Q, what process would you think
> would make us "know" GC be true?
Ordinary mathematical research. What's the point of expressing
"suppose Goldbach's conjecture is true" as "suppose ~GC is not
provable in Robinson arithmetic"?
elsiemelsi
You said Robinson Arithmetic is not incomplete. So you must think that
either this sentence or its negation can be proved in Robinson
Arithmetic. He's challenging you to actually do it.
i say
i could not care less
robinson arithmetic has not been proved to be incomplete untill we are
not told what make a statement true
the theorem depend on two things consistency and the presence of true
statements
but if you cant tell us what makes a statement true-independent of proof-
then the theorem is meaningles
regardless if (Ax)(0 + x = x)
or its negation in Robinson Arithmetic can be prooved or not
and even if it was proved
what makes it true anyway
Nam D. Nguyen
> On 2008-04-12, in sci.logic, Nam D. Nguyen wrote:
>> Suppose ~GC is not provable in Q, what process would you think
>> would make us "know" GC be true?
>
> Ordinary mathematical research.
Care to explain that in details?
> What's the point of expressing
> "suppose Goldbach's conjecture is true" as "suppose ~GC is not
> provable in Robinson arithmetic"?
>
My question doesn't say anything about "suppose Goldbach's conjecture
is true". And my question is a technical question. Care to directly
answer that?
elsiemelsi
says
> Suppose ~GC is not provable in Q, what process would you think
> would make us "know" GC be true?
Ordinary mathematical research. What's the point of expressing
"suppose Goldbach's conjecture is true" as "suppose ~GC is not
provable in Robinson arithmetic"?
i say
not good enough tell s us what would makes us know GC is true even if it
is unprovable
tell us what would make GC true
Nam D. Nguyen
For what it's worth, "suppose Goldbach's conjecture is true" still
requires an *unsaid assumption* that Q be consistent. In my
"suppose ~GC is not provable in Robinson arithmetic", the consistency
of Q would be a corollary, hence my assumption would be more compacted,
among other benefits I've not stated.
Rupert
wrote:
> On 2008-04-12, in sci.logic, Peter_Smith wrote:
>
> > I'm sure that George and I will chip in for a $1000 prize if he
> > does ... and Aatu will double it! ;-)
>
> Gladly.
>
> --
>
> "Wovon man nicht sprechen kann, daruber muss man schweigen"
> - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
If someone could somehow convince me that that sentence or its
negation was provable in Robinson Arithmetic, I'd gladly pay 5000
Australian dollars for that, at the very least.
Rupert
> rupert said
>
> You said Robinson Arithmetic is not incomplete. So you must think that
> either this sentence or its negation can be proved in Robinson
> Arithmetic. He's challenging you to actually do it.
>
> i say
>
> i could not care less
> robinson arithmetic has not been proved to be incomplete untill we are
> not told what make a statement true
>
Well, this is what I was saying before. You can say it's not
incomplete, as indeed you did, or else you can say it hasn't been
proved to be incomplete, which is a weaker claim. You should make it
clear which claim you are making.
And, with regard to your argument, as I was saying we do have a
definition of truth, but we can give an argument that Robinson
Arithmetic is incomplete without using the notion of truth at all.
> the theorem depend on two things consistency and the presence of true
> statements
> but if you cant tell us what makes a statement true-independent of proof-
> then the theorem is meaningles
> regardless if (Ax)(0 + x = x)
> or its negation in Robinson Arithmetic can be prooved or not
>
If neither the sentence nor its negation can be proved in Robinson
Arithmetic, then Robinson Arithmetic is certainly incomplete and so
you must retract your statement that it's not. As I'm saying, you must
be clear about what you are claiming.
> and even if it was proved
> what makes it true anyway
>
> --
Rupert
> Aatu
> says
>
> > Suppose ~GC is not provable in Q, what process would you think
> > would make us "know" GC be true?
>
> Ordinary mathematical research. What's the point of expressing
> "suppose Goldbach's conjecture is true" as "suppose ~GC is not
> provable in Robinson arithmetic"?
>
> i say
> not good enough tell s us what would makes us know GC is true even if it
> is unprovable
>
> tell us what would make GC true
>
> --
> More information athttp://www.talkaboutscience.com/faq.html
Aatu's point was: we can prove in a very weak theory that ~GC is
unprovable in Robinson Arithmetic if and only if GC. And he's saying,
to find out whether GC is true, we just attack the problem in the way
mathematicians ordinarily do. We might succeed in showing that it can
be proved in ZFC, for example.
Rupert
> Aatu Koskensilta wrote:
> > On 2008-04-12, in sci.logic, Nam D. Nguyen wrote:
> >> Suppose ~GC is not provable in Q, what process would you think
> >> would make us "know" GC be true?
>
> > Ordinary mathematical research.
>
> Care to explain that in details?
>
We might show that GC is provable in ZFC, for example. That's what
we're trying to do at the moment.
> > What's the point of expressing
> > "suppose Goldbach's conjecture is true" as "suppose ~GC is not
> > provable in Robinson arithmetic"?
>
> My question doesn't say anything about "suppose Goldbach's conjecture
> is true".
"Q does not prove ~GC" is equivalent to GC in PRA.
Nam D. Nguyen
> On Apr 13, 12:57 am, "Nam D. Nguyen" <namducngu...@shaw.ca> wrote:
>> Aatu Koskensilta wrote:
>>> On 2008-04-12, in sci.logic, Nam D. Nguyen wrote:
>>>> Suppose ~GC is not provable in Q, what process would you think
>>>> would make us "know" GC be true?
>>> Ordinary mathematical research.
>> Care to explain that in details?
>>
>
> We might show that GC is provable in ZFC, for example. That's what
> we're trying to do at the moment.
Except that if ZFC turns out to be inconsistent, ~GC would also be provable
in this case!
Something about a formal system - as an axiom set - that has seemed to escape
our attention: it's supposed to reflect the entire underlying set of concepts;
hence the assumption of its consistency is a must however *assumed* it might
be. Consequently, any meta assertion that *must depend on the consistency*
must clearly state so to be a valid meta theorem!
There's actually nothing "sacred" about using Q as the encoding theory,
or about assuming N as the standard arithmetic model of Q, in GIT.
One could equally *assumes* ZFC be consistent, and perform certain
"set-ization", instead of "numerization", and still arrive at GIT,
as a hypothetical meta theorem. "Hypothetical" because of the assumption
ZFC be consistent in this case. It's only a matter of how explicit or implicit
we'd make this assumption for the encoding theory!
>
>>> What's the point of expressing
>>> "suppose Goldbach's conjecture is true" as "suppose ~GC is not
>>> provable in Robinson arithmetic"?
>> My question doesn't say anything about "suppose Goldbach's conjecture
>> is true".
>
> "Q does not prove ~GC" is equivalent to GC in PRA.
Will PRA be consistent?
Rupert
> Rupert wrote:
> > On Apr 13, 12:57 am, "Nam D. Nguyen" <namducngu...@shaw.ca> wrote:
> >> Aatu Koskensilta wrote:
> >>> On 2008-04-12, in sci.logic, Nam D. Nguyen wrote:
> >>>> Suppose ~GC is not provable in Q, what process would you think
> >>>> would make us "know" GC be true?
> >>> Ordinary mathematical research.
> >> Care to explain that in details?
>
> > We might show that GC is provable in ZFC, for example. That's what
> > we're trying to do at the moment.
>
> Except that if ZFC turns out to be inconsistent, ~GC would also be provable
> in this case!
>
We might prove GC in ZFC, then find an inconsistency proof for ZFC,
and so re-assess our belief in GC. Certainly that could happen. In
that sense no mathematical knowledge we obtain is completely certain.
So what?
> Something about a formal system - as an axiom set - that has seemed to escape
> our attention: it's supposed to reflect the entire underlying set of concepts;
> hence the assumption of its consistency is a must however *assumed* it might
> be. Consequently, any meta assertion that *must depend on the consistency*
> must clearly state so to be a valid meta theorem!
>
> There's actually nothing "sacred" about using Q as the encoding theory,
> or about assuming N as the standard arithmetic model of Q, in GIT.
> One could equally *assumes* ZFC be consistent, and perform certain
> "set-ization", instead of "numerization", and still arrive at GIT,
> as a hypothetical meta theorem. "Hypothetical" because of the assumption
> ZFC be consistent in this case. It's only a matter of how explicit or implicit
> we'd make this assumption for the encoding theory!
>
>
>
> >>> What's the point of expressing
> >>> "suppose Goldbach's conjecture is true" as "suppose ~GC is not
> >>> provable in Robinson arithmetic"?
> >> My question doesn't say anything about "suppose Goldbach's conjecture
> >> is true".
>
> > "Q does not prove ~GC" is equivalent to GC in PRA.
>
> Will PRA be consistent?
>
Seems quite likely...
Nam D. Nguyen
> On Apr 13, 2:41 am, "Nam D. Nguyen" <namducngu...@shaw.ca> wrote:
>> Rupert wrote:
>>> On Apr 13, 12:57 am, "Nam D. Nguyen" <namducngu...@shaw.ca> wrote:
>>>> Aatu Koskensilta wrote:
>>>>> On 2008-04-12, in sci.logic, Nam D. Nguyen wrote:
>>>>>> Suppose ~GC is not provable in Q, what process would you think
>>>>>> would make us "know" GC be true?
>>>>> Ordinary mathematical research.
>>>> Care to explain that in details?
>>> We might show that GC is provable in ZFC, for example. That's what
>>> we're trying to do at the moment.
>> Except that if ZFC turns out to be inconsistent, ~GC would also be provable
>> in this case!
>>
>
> We might prove GC in ZFC, then find an inconsistency proof for ZFC,
> and so re-assess our belief in GC. Certainly that could happen. In
> that sense no mathematical knowledge we obtain is completely certain.
I'd not say "no mathematical knowledge...", only "some mathematical knowledge...".
But right, my main point here is the foundation of *current* mathematics
is based on the "all-knowing" human mind even in dealing with infinity,
which imho is quite wrong. Isn't it true that the essence Lindenbaum's
Compactness would sort of encourage us to be a little conservative,
as far as our knowledge is concerned?
>
> So what?
So that we could *improve* our reasoning, to cope with the ever increasing
demand that mathematics is "the language" of natural sciences!
>
>> Something about a formal system - as an axiom set - that has seemed to escape
>> our attention: it's supposed to reflect the entire underlying set of concepts;
>> hence the assumption of its consistency is a must however *assumed* it might
>> be. Consequently, any meta assertion that *must depend on the consistency*
>> must clearly state so to be a valid meta theorem!
>>
>> There's actually nothing "sacred" about using Q as the encoding theory,
>> or about assuming N as the standard arithmetic model of Q, in GIT.
>> One could equally *assumes* ZFC be consistent, and perform certain
>> "set-ization", instead of "numerization", and still arrive at GIT,
>> as a hypothetical meta theorem. "Hypothetical" because of the assumption
>> ZFC be consistent in this case. It's only a matter of how explicit or implicit
>> we'd make this assumption for the encoding theory!
>>
>>
>>
>>>>> What's the point of expressing
>>>>> "suppose Goldbach's conjecture is true" as "suppose ~GC is not
>>>>> provable in Robinson arithmetic"?
>>>> My question doesn't say anything about "suppose Goldbach's conjecture
>>>> is true".
>>> "Q does not prove ~GC" is equivalent to GC in PRA.
>> Will PRA be consistent?
>>
>
> Seems quite likely...
If mathematical reasoning is always based on "seems ... likely" there
wouldn't be the words "formal logic". And in such case, we might as well
become Zen masters, to know the "unutterable" truths, which would be the
only truths!
george
> i say rubbish it depends on there
> being identifibale true statements
OF COURSE there are identifiable true statements.
But the point is that there is also a statement with
the property that neither it nor its denial IS identified as a true
statement.
> cant tell what a true statement is the theorem collapses into
> meaninglessness
No, it doesn't, since the theorem basically PROVES you CAN'T
always tell which statements are "true". The theorem CONSTRUCTS
a statement that COULD be EITHER of true or false, as far as
the axioms are concerned. There are models BOTH WAYS.
elsiemelsi
If neither the sentence nor its negation can be proved in Robinson
Arithmetic, then Robinson Arithmetic is certainly incomplete and so
you must retract your statement that it's not. As I'm saying, you must
be clear about what you are claiming.
i say
rubbish
it is not proven to be complete untill you tell us what makes a statement
true
you talk about tarski
so tel us what makes a statement true
elsiemelsi
We might show that GC is provable in ZFC, for example. That's what
we're trying to do at the moment.
it not about proving it
it about telling us why it is true
since godel truth is independent of proof
elsiemelsi
We might prove GC in ZFC, then find an inconsistency proof for ZFC,
i have shown him ZFC is inconsistent ie skolem paradox
quote
Using the Löwenheim-Skolem Theorem, we can get a model of set theory
which only contains a countable number of objects. However, it must
contain the aforementioned uncountable sets, which appears to be a
contradiction.
dont say apears -as you or any one else cant disprove skolem
elsiemelsi
The theorem CONSTRUCTS
a statement that COULD be EITHER of true or false, as far as
the axioms are concerned. There are models BOTH WAYS.
tells us what makes a statement true then
you talk of tarski
so tell us what makes a statement true
if you cant
RA has not been proved to be incomplete
elsiemelsi
The theorem CONSTRUCTS
a statement that COULD be EITHER of true or false, as far as
the axioms are concerned. There are models BOTH WAYS.
i say
you talk about tarski
so
tell us what makes
1+1=2 true independent of proof
Aatu Koskensilta
> tell us what makes
> 1+1=2 true independent of proof
A putrid marshmallow I once found in my backyard.
--
Aatu Koskensilta (aatu.kos...@xortec.fi)
elsiemelsi
> tell us what makes
> 1+1=2 true independent of proof
A putrid marshmallow I once found in my backyard.
as i say
you cant tell us what makes a statement true
ie 1+1=2
this RA is not incomplete as you cant show it to be untill you cant tells
us
what makes a statement true
ie 1+1=2
elsiemelsi
well this is what tarskis theory of truth amounts to
quote
http://en.wikipedia.org/wiki/Truth#Semantic_theory_of_truth
The semantic theory of truth has as its general case for a given
language:
'P' is true if and only if P
where 'P' is a reference to the sentence (the sentence's name), and P is
just the sentence itself.
Logician and philosopher Alfred Tarski developed the theory for formal
languages (such as formal logic). Here he restricted it in this way: no
language could contain its own truth predicate, that is, the expression is
true could only apply to sentences in some other language. The latter he
called an object language, the language being talked about. (It may, in
turn, have a truth predicate that can be applied to sentences in still
another language.) The reason for his restriction was that languages that
contain their own truth predicate will contain paradoxical sentences like
the Liar:
note
Bertrand Russell is credited with noticing the existence of such paradoxes
even in the best symbolic formalizations of mathematics in his day,
so please rupert
show us how tarski
tells us why
1+1=2 is true
it would seem from
'P' is true if and only if P
"1+1=2 is true" only if 1+1=2 is true
but then according to tarski
Here he restricted it in this way: no language could contain its own truth
predicate, that is, the expression is true could only apply to sentences in
some other language. The latter he called an object language, the language
being talked about
so mathematics cant contain its own truth predicate
so we have two languages to determine truth
the object ie mathematics and the other language
so what is this other language which deterimins the truth
and as i have asked you time and time again
tell us via tarski
why 1+1=2 is true
Rupert
> rupert says
>
> If neither the sentence nor its negation can be proved in Robinson
> Arithmetic, then Robinson Arithmetic is certainly incomplete and so
> you must retract your statement that it's not. As I'm saying, you must
> be clear about what you are claiming.
>
> i say
> rubbish
>
Yes. You say "rubbish" a lot. Unfortunately, what you quote me saying
above is absolutely trivial and beyond rational dispute. And you
appear not to have understood it, which suggests a lack of basic
reading comprehension skills.
> it is not proven to be complete
Perhaps you meant to say "incomplete" here?
> untill you tell us what makes a statement
> true
There is a perfectly good proof of its incompleteness in Primitive
Recursive Arithmetic. No semantic notions are required. If you want to
understand the proof, maybe you should start by learning what a first-
order language is.
> you talk about tarski
> so tel us what makes a statement true
>
Tarski's work isn't really relevant here. Let's begin at the
beginning. First, do something about your inability to understand
plain English. When you've resolved that problem, then why don't you
get yourself a copy of Moshé Machover's "Set theory, logic, and their
limitations" and have a look at that. I can help you if you have any
questions.
Rupert
> rupert says
>
> We might show that GC is provable in ZFC, for example. That's what
> we're trying to do at the moment.
>
> it not about proving it
> it about telling us why it is true
> since godel truth is independent of proof
>
> --
> More information athttp://www.talkaboutscience.com/faq.html
Proof is the method by which we convince ourselves that mathematical
statements are true. Gödel's work doesn't change that.
Rupert
> rupert says
>
> We might prove GC in ZFC, then find an inconsistency proof for ZFC,
>
> i have shown him ZFC is inconsistent
No.
Rupert
> rupert say
>
> The theorem CONSTRUCTS
> a statement that COULD be EITHER of true or false, as far as
> the axioms are concerned. There are models BOTH WAYS.
>
I think George said that, didn't he?
> tells us what makes a statement true then
> you talk of tarski
> so tell us what makes a statement true
> if you cant
> RA has not been proved to be incomplete
>
As I say, there is an incompleteness proof which doesn't use semantic
notions. You want to understand it, then you'd better get down to
learning some logic.
Rupert
> Rupert wrote:
> > On Apr 13, 2:41 am, "Nam D. Nguyen" <namducngu...@shaw.ca> wrote:
> >> Rupert wrote:
> >>> On Apr 13, 12:57 am, "Nam D. Nguyen" <namducngu...@shaw.ca> wrote:
> >>>> Aatu Koskensilta wrote:
> >>>>> On 2008-04-12, in sci.logic, Nam D. Nguyen wrote:
> >>>>>> Suppose ~GC is not provable in Q, what process would you think
> >>>>>> would make us "know" GC be true?
> >>>>> Ordinary mathematical research.
> >>>> Care to explain that in details?
> >>> We might show that GC is provable in ZFC, for example. That's what
> >>> we're trying to do at the moment.
> >> Except that if ZFC turns out to be inconsistent, ~GC would also be provable
> >> in this case!
>
> > We might prove GC in ZFC, then find an inconsistency proof for ZFC,
> > and so re-assess our belief in GC. Certainly that could happen. In
> > that sense no mathematical knowledge we obtain is completely certain.
>
> I'd not say "no mathematical knowledge...", only "some mathematical knowledge...".
I would say: there is no serious doubt that ZFC is consistent, say,
but nevertheless complete certainty does not exist. You maintain that
it does exist, can you tell me which examples you have in mind?
> But right, my main point here is the foundation of *current* mathematics
> is based on the "all-knowing" human mind even in dealing with infinity,
> which imho is quite wrong. Isn't it true that the essence Lindenbaum's
> Compactness would sort of encourage us to be a little conservative,
> as far as our knowledge is concerned?
>
What are we talking about here? The compactness theorem for first-
order logic?
>
>
> > So what?
>
> So that we could *improve* our reasoning, to cope with the ever increasing
> demand that mathematics is "the language" of natural sciences!
>
To me, the elementary point that complete certainty does not exist is
no evidence that our reasoning stands in need of improvement. We will
never change the fact that complete certainty does not exist.
>
>
>
>
> >> Something about a formal system - as an axiom set - that has seemed to escape
> >> our attention: it's supposed to reflect the entire underlying set of concepts;
> >> hence the assumption of its consistency is a must however *assumed* it might
> >> be. Consequently, any meta assertion that *must depend on the consistency*
> >> must clearly state so to be a valid meta theorem!
>
> >> There's actually nothing "sacred" about using Q as the encoding theory,
> >> or about assuming N as the standard arithmetic model of Q, in GIT.
> >> One could equally *assumes* ZFC be consistent, and perform certain
> >> "set-ization", instead of "numerization", and still arrive at GIT,
> >> as a hypothetical meta theorem. "Hypothetical" because of the assumption
> >> ZFC be consistent in this case. It's only a matter of how explicit or implicit
> >> we'd make this assumption for the encoding theory!
>
> >>>>> What's the point of expressing
> >>>>> "suppose Goldbach's conjecture is true" as "suppose ~GC is not
> >>>>> provable in Robinson arithmetic"?
> >>>> My question doesn't say anything about "suppose Goldbach's conjecture
> >>>> is true".
> >>> "Q does not prove ~GC" is equivalent to GC in PRA.
> >> Will PRA be consistent?
>
> > Seems quite likely...
>
> If mathematical reasoning is always based on "seems ... likely" there
> wouldn't be the words "formal logic".
Well, I'm afraid it is, in the sense that the certainty it provides is
virtual certainty, not complete certainty. That's just life. It's time
to face up to it like an adult. The phrase "formal logic" has nothing
to do with it.
> And in such case, we might as well
> become Zen masters, to know the "unutterable" truths, which would be the
> only truths!
>
Zen has nothing to do with it either. Complete certainty does not
exist. That is just an obvious fact of life.
elsiemelsi
Proof is the method by which we convince ourselves that mathematical
statements are true. Gödel's work doesn't change that.
i say rubbish
godel is said to have destroyed the hilbet idea of true being proven from
axioms
thus true is independent of proof
quote
In addition, from at least the time of Hilbert's program at the turn of
the twentieth century to the proof of Gödel's theorem and the development
of the Church-Turing thesis in the early part of that century, true
statements in mathematics were generally assumed to be those statements
which are provable in a formal axiomatic system.
The works of Kurt Gödel, Alan Turing, and others shook this assumption,
with the development of statements that are true but cannot be proven
within the system
and if you insist true is what is proven
then godel theorem is in contradiction
for it would read
there are true statements-which are true becuase they are proven - which
cant be proven
blatant contradiction
for this would
For any consistent formal, recursively enumerable theory that proves basic
arithmetical truths, an arithmetical statement that is true, but not
provable in the theory, can be constructed.1 That is, any effectively
generated theory capable of expressing elementary arithmetic cannot be
both consistent and complete.
would read under your notion of truth
For any consistent formal, recursively enumerable theory that proves basic
arithmetical truths -which are proven-, an arithmetical statement that is
true ie proven , but not provable in the theory, can be constructed.1 That
is, any effectively generated theory capable of expressing elementary
arithmetic cannot be
both consistent and complete.
Rupert
> rupert tells us that tarski gave a theory of truth
>
> well this is what tarskis theory of truth amounts to
>
> quote
>
> http://en.wikipedia.org/wiki/Truth#Semantic_theory_of_truth
>
> The semantic theory of truth has as its general case for a given
> language:
>
> 'P' is true if and only if P
>
This is the disquotationalist theory of truth, I'm talking about
Tarski's basic semantic definition for first-order languages.
But, as I say, it's irrelevant. There is an incompleteness proof for
Robinson Arithmetic which does not use any semantic notions. You don't
have the faintest glimmering of how it goes. You want to understand
it; well, you've got a long way to go. The first step would be to
improve your ability to read and understand plain English.
> where 'P' is a reference to the sentence (the sentence's name), and P is
> just the sentence itself.
>
> Logician and philosopher Alfred Tarski developed the theory for formal
> languages (such as formal logic). Here he restricted it in this way: no
> language could contain its own truth predicate, that is, the expression is
> true could only apply to sentences in some other language. The latter he
> called an object language, the language being talked about. (It may, in
> turn, have a truth predicate that can be applied to sentences in still
> another language.) The reason for his restriction was that languages that
> contain their own truth predicate will contain paradoxical sentences like
> the Liar:
>
> note
>
> Bertrand Russell is credited with noticing the existence of such paradoxes
> even in the best symbolic formalizations of mathematics in his day,
>
> so please rupert
> show us how tarski
> tells us why
> 1+1=2 is true
>
Why "1+1=2" is true? Well, remember how I once compared talking with
you to trying to teach advanced mathematics to my five-year-old
cousin? I once had a conversation with my sister when she was about
seven, saying "Can you show me why 2+2=4?" So she got out two beads,
counted them "One, two", then held up two fingers and said "Two". Then
she got out another two beads... you get the idea.
You're making some kind of thesis in the epistemology of mathematics
here? Well, the question is not "Why is 1+1=2 true?", the question is
"How is it possible to entertain the slightest doubt that 1+1=2
without also entertaining doubts about your own sanity?"
You want me to show you how Tarski's basic semantic definition applies
to the case of the sentence "1+1=2"? Well, sure, we can go through
that, although it hasn't got very much to do with understanding why
Robinson Arithmetic is incomplete.
But where do we start? Do you know what a first-order language is?
> it would seem from
>
> 'P' is true if and only if P
>
> "1+1=2 is true" only if 1+1=2 is true
>
> but then according to tarski
>
> Here he restricted it in this way: no language could contain its own truth
> predicate, that is, the expression is true could only apply to sentences in
> some other language. The latter he called an object language, the language
> being talked about
>
> so mathematics cant contain its own truth predicate
>
> so we have two languages to determine truth
> the object ie mathematics and the other language
> so what is this other language which deterimins the truth
>
> and as i have asked you time and time again
> tell us via tarski
> why 1+1=2 is true
>
Is a proof in ZFC good enough for you?
Rupert
> Aatu wrote
>
> > tell us what makes
> > 1+1=2 true independent of proof
>
> A putrid marshmallow I once found in my backyard.
>
> as i say
> you cant tell us what makes a statement true
> ie 1+1=2
> this RA is not incomplete as you cant show it to be untill you cant tells
> us
>
> what makes a statement true
> ie 1+1=2
>
> --
> More information athttp://www.talkaboutscience.com/faq.html
Poke! :)
Rupert
> rupert says
>
> Proof is the method by which we convince ourselves that mathematical
> statements are true. Gödel's work doesn't change that.
>
> i say rubbish
>
Well. What a big surprise. :)
> godel is said to have destroyed the hilbet idea of true being proven from
> axioms
> thus true is independent of proof
>
He shows that truth cannot be identified with provability.
Nevertheless, what I said stands.
> quote
>
> In addition, from at least the time of Hilbert's program at the turn of
> the twentieth century to the proof of Gödel's theorem and the development
> of the Church-Turing thesis in the early part of that century, true
> statements in mathematics were generally assumed to be those statements
> which are provable in a formal axiomatic system.
>
> The works of Kurt Gödel, Alan Turing, and others shook this assumption,
> with the development of statements that are true but cannot be proven
> within the system
>
> and if you insist true is what is proven
Which I don't...
[blah blah blah...]
herbzet
Aatu Koskensilta wrote:
> elsiemelsi wrote:
> > tell us what makes
> > 1+1=2 true independent of proof
>
> A putrid marshmallow I once found in my backyard.
Strangely, this remark was also made by Chao-chou (b.778, d.897 AD)
in answer to the question "What is the Buddha-nature?".
--
hz
elsiemelsi
You're making some kind of thesis in the epistemology of mathematics
here? Well, the question is not "Why is 1+1=2 true?", the question is
"How is it possible to entertain the slightest doubt that 1+1=2
without also entertaining doubts about your own sanity?"
haha so truth in maths is based on psychological necessity
so mathematician have too get a clearance from a psychiatrist before
practicing maths hahaha
herbzet
elsiemelsi wrote:
> rupert says
>
> You're making some kind of thesis in the epistemology of mathematics
> here? Well, the question is not "Why is 1+1=2 true?", the question is
> "How is it possible to entertain the slightest doubt that 1+1=2
> without also entertaining doubts about your own sanity?"
>
> haha so truth in maths is based on psychological necessity
>
> so mathematician have too get a clearance from a psychiatrist before
> practicing maths hahaha
Not a bad idea in the present case.
--
hz
elsiemelsi
He shows that truth cannot be identified with provability.
Nevertheless, what I said stands
i say
haha proof and provablity are different things
gee talk about semantic nonsence
fact is you cant tell us what makes a statement true independant of proof
thus RA is not incomplete as you have not told us what make a statement
true
oh you say provability
but provability
is the same as prove only it is the noun
dictionary say provability is
http://www.thefreedictionary.com/provability
Noun 1. provability - capability of being demonstrated or logically proved
Nam D. Nguyen
> On Apr 13, 3:27 am, "Nam D. Nguyen" <namducngu...@shaw.ca> wrote:
>> Rupert wrote:
>>> On Apr 13, 2:41 am, "Nam D. Nguyen" <namducngu...@shaw.ca> wrote:
>>>> Rupert wrote:
>>>>> On Apr 13, 12:57 am, "Nam D. Nguyen" <namducngu...@shaw.ca> wrote:
>>>>>> Aatu Koskensilta wrote:
>>>>>>> On 2008-04-12, in sci.logic, Nam D. Nguyen wrote:
>>>>>>>> Suppose ~GC is not provable in Q, what process would you think
>>>>>>>> would make us "know" GC be true?
>>>>>>> Ordinary mathematical research.
>>>>>> Care to explain that in details?
>>>>> We might show that GC is provable in ZFC, for example. That's what
>>>>> we're trying to do at the moment.
>>>> Except that if ZFC turns out to be inconsistent, ~GC would also be provable
>>>> in this case!
>>> We might prove GC in ZFC, then find an inconsistency proof for ZFC,
>>> and so re-assess our belief in GC. Certainly that could happen. In
>>> that sense no mathematical knowledge we obtain is completely certain.
>> I'd not say "no mathematical knowledge...", only "some mathematical knowledge...".
>
> I would say: there is no serious doubt that ZFC is consistent, say,
> but nevertheless complete certainty does not exist. You maintain that
> it does exist, can you tell me which examples you have in mind?
Classical misconception! I only maintain that we don't have knowledge one
way or the other! If you maintain ZFC be consistent you would give a proof.
If you believe ZFC be inconsistent you would give a proof. You could also
state "I don't know how to prove one way or the other". It's that simple!
>> But right, my main point here is the foundation of *current* mathematics
>> is based on the "all-knowing" human mind even in dealing with infinity,
>> which imho is quite wrong. Isn't it true that the essence Lindenbaum's
>> Compactness would sort of encourage us to be a little conservative,
>> as far as our knowledge is concerned?
>>
>
> What are we talking about here? The compactness theorem for first-
> order logic?
The theorem that says "Every consistent set of formulas can be extended
into a maximally consistent one".
>>> So what?
>> So that we could *improve* our reasoning, to cope with the ever increasing
>> demand that mathematics is "the language" of natural sciences!
>
> To me, the elementary point that complete certainty does not exist is
> no evidence that our reasoning stands in need of improvement.
You might be aware of the fact around the year 2000, there was a prize
worth 1 million US dollar for proving or disproving GC. Naturally, many
of them tried proving "GC is true" - to no avail. What they (and perhaps
quite a few of us here) don't seem to realize is that if, genuinely,
~GC is not provable in Q, there's no way we could prove GC in any
as-strong-as-arithmetic theory. A reasoning framework that has such
a nightmare which could cause waste of time on intellectual effort
would indeed be in need for improving. Wouldn't you think so?
> We will never change the fact that complete certainty does not exist.
Another misconception. I've *never* asked for a change that a complete certainty
would exist! You could better the reasoning by acknowledging there is
a limitation of what you know, instead of pretending you'd know everything!
>>
>>>> Something about a formal system - as an axiom set - that has seemed to escape
>>>> our attention: it's supposed to reflect the entire underlying set of concepts;
>>>> hence the assumption of its consistency is a must however *assumed* it might
>>>> be. Consequently, any meta assertion that *must depend on the consistency*
>>>> must clearly state so to be a valid meta theorem!
>>>> There's actually nothing "sacred" about using Q as the encoding theory,
>>>> or about assuming N as the standard arithmetic model of Q, in GIT.
>>>> One could equally *assumes* ZFC be consistent, and perform certain
>>>> "set-ization", instead of "numerization", and still arrive at GIT,
>>>> as a hypothetical meta theorem. "Hypothetical" because of the assumption
>>>> ZFC be consistent in this case. It's only a matter of how explicit or implicit
>>>> we'd make this assumption for the encoding theory!
>>>>>>> What's the point of expressing
>>>>>>> "suppose Goldbach's conjecture is true" as "suppose ~GC is not
>>>>>>> provable in Robinson arithmetic"?
>>>>>> My question doesn't say anything about "suppose Goldbach's conjecture
>>>>>> is true".
>>>>> "Q does not prove ~GC" is equivalent to GC in PRA.
>>>> Will PRA be consistent?
>>> Seems quite likely...
>> If mathematical reasoning is always based on "seems ... likely" there
>> wouldn't be the words "formal logic".
>
> Well, I'm afraid it is, in the sense that the certainty it provides is
> virtual certainty, not complete certainty.
If we're "honest" to ourself, we could say "there's not complete certainty"
and stop at that: there's nothing wrong with that. Saying certainty is
"virtual certainty" is quite "funny"!
> That's just life. It's time to face up to it like an adult.
To face it up like an adult is to accept what we don't and can't know.
It's indeed childish to take refuge in an illusion instead of accepting
the truth, however unpleasant it might be!
>
>> And in such case, we might as well
>> become Zen masters, to know the "unutterable" truths, which would be the
>> only truths!
>>
>
> Zen has nothing to do with it either.
Which is the essence of Zen, when it has something to do with it by
having nothing to do with it!
> Complete certainty does not exist.
Neither does the certainty of knowledge of the natural numbers that
you'd believe in!
> That is just an obvious fact of life.
And of reasoning.
elsiemelsi
Nam D. Nguyen say
If you believe ZFC be inconsistent you would give a proof.
i say
skolem gave that proof ie skolems paradox
Using the Löwenheim-Skolem Theorem, we can get a model of set theory
which only contains a countable number of objects. However, it must
contain the aforementioned uncountable sets, which appears to be a
contradiction.
note it say set theory is in contradiction-which means inconsistent
dont say it only appears to be a contradiction
as no one has been able to disprove skolem
Rupert
> Nam D. Nguyen say
>
> If you believe ZFC be inconsistent you would give a proof.
>
> i say
> skolem gave that proof ie skolems paradox
>
WRONG
WRONG WRONG WRONG WRONG
So very, very wrong.
:)
Rupert
Why? I know I can't.
> If you believe ZFC be inconsistent you would give a proof. You could also
> state "I don't know how to prove one way or the other". It's that simple!
>
Yes, I can state "I don't know how to prove one way or the other."
But it's not that simple.
I can prove that ZFC is consistent in ZFC+"there exists an inaccesible
cardinal."
And I can prove that Robinson Arithmetic is consistent in PRA.
So what am I supposed to conclude, that I don't know the consistency
of any theory?
I do concede this in the sense that I maintain that absolute certainty
does not exist.
But you were maintain that there are some areas in which absolute
certainty does exist, and I was asking you to give examples...
> >> But right, my main point here is the foundation of *current* mathematics
> >> is based on the "all-knowing" human mind even in dealing with infinity,
> >> which imho is quite wrong. Isn't it true that the essence Lindenbaum's
> >> Compactness would sort of encourage us to be a little conservative,
> >> as far as our knowledge is concerned?
>
> > What are we talking about here? The compactness theorem for first-
> > order logic?
>
> The theorem that says "Every consistent set of formulas can be extended
> into a maximally consistent one".
>
All right. What about it?
> >>> So what?
> >> So that we could *improve* our reasoning, to cope with the ever increasing
> >> demand that mathematics is "the language" of natural sciences!
>
> > To me, the elementary point that complete certainty does not exist is
> > no evidence that our reasoning stands in need of improvement.
>
> You might be aware of the fact around the year 2000, there was a prize
> worth 1 million US dollar for proving or disproving GC. Naturally, many
> of them tried proving "GC is true" - to no avail. What they (and perhaps
> quite a few of us here) don't seem to realize is that if, genuinely,
> ~GC is not provable in Q, there's no way we could prove GC in any
> as-strong-as-arithmetic theory.
Why?
> A reasoning framework that has such
> a nightmare which could cause waste of time on intellectual effort
> would indeed be in need for improving. Wouldn't you think so?
>
> > We will never change the fact that complete certainty does not exist.
>
> Another misconception. I've *never* asked for a change that a complete certainty
> would exist! You could better the reasoning by acknowledging there is
> a limitation of what you know, instead of pretending you'd know everything!
>
Okay, so I acknowledge that complete certainty doesn't exist. What
next?
All right, fine, let's do that. So, what next?
> Saying certainty is
> "virtual certainty" is quite "funny"!
>
Very funny.
> > That's just life. It's time to face up to it like an adult.
>
> To face it up like an adult is to accept what we don't and can't know.
> It's indeed childish to take refuge in an illusion instead of accepting
> the truth, however unpleasant it might be!
>
What illusion? You've lost me, I'm afraid.
>
>
> >> And in such case, we might as well
> >> become Zen masters, to know the "unutterable" truths, which would be the
> >> only truths!
>
> > Zen has nothing to do with it either.
>
> Which is the essence of Zen, when it has something to do with it by
> having nothing to do with it!
>
> > Complete certainty does not exist.
>
> Neither does the certainty of knowledge of the natural numbers that
> you'd believe in!
>
> > That is just an obvious fact of life.
>
> And of reasoning.
Your point is lost on me, I'm afraid...
Rupert
> rupert says
>
> He shows that truth cannot be identified with provability.
> Nevertheless, what I said stands
>
> i say
>
> haha proof and provablity are different things
> gee talk about semantic nonsence
>
> fact is you cant tell us what makes a statement true independant of proof
> thus RA is not incomplete as you have not told us what make a statement
> true
Um, this is *still* meaningless babble no matter how many times you
say it.
Of course RA is incomplete. It can be proved incomplete in PRA, and
the proof makes no use of semantic notions. You have absolutely no
idea of what the proof is. We can try to help you understand it, but
that's a pretty ambitious task when you can't even understand plain
English.
> oh you say provability
> but provability
> is the same as prove only it is the noun
>
> dictionary say provability ishttp://www.thefreedictionary.com/provability
> Noun 1. provability - capability of being demonstrated or logically proved
>
> --
elsiemelsi
I can prove that ZFC is consistent in ZFC+"there exists an inaccesible
cardinal."
you say skolem is not pointing out the inconsistency of ZFC
then just give us a disproof of skolem
ie of this
Using the Löwenheim-Skolem Theorem, we can get a model of set theory
which only contains a countable number of objects. However, it must
contain the aforementioned uncountable sets, which appears to be a
contradiction.
note it says set theory is in contradiction -ie inconsistent
so just disprove skolem
Rupert
> rupert says
>
> I can prove that ZFC is consistent in ZFC+"there exists an inaccesible
> cardinal."
>
> you say skolem is not pointing out the inconsistency of ZFC
>
> then just give us a disproof of skolem
> ie of this
>
I did before.
I explained how someone might think that Skolem's paradox yielded a
proof in Z_2 that ZF is inconsistent, and I showed where the argument
broke down.
You didn't listen.
> Using the Löwenheim-Skolem Theorem, we can get a model of set theory
> which only contains a countable number of objects. However, it must
> contain the aforementioned uncountable sets, which appears to be a
> contradiction.
>
> note it says set theory is in contradiction -ie inconsistent
> so just disprove skolem
>
> --
elsiemelsi
Of course RA is incomplete.
it is not incomplete until you tell us what makes a statement ie 1+1=2
true
rupert says
It can be proved incomplete in PRA, and
the proof makes no use of semantic notions.
a proof with out the notion of truth is not an incompletness proof as
formulated by godel
if you give a prof that does not include the notion of truth it is not a
godelian incompletness proof
it is something else
and calling RA incomplete is a blantant misrepresentation of the godelian
theorem and thus false
there might be statements that cant be proven in RA but that does not make
it incomplete in a godelian sense unless the notion of truth is involved
Rupert
> rupert say
>
> Of course RA is incomplete.
>
> it is not incomplete until you tell us what makes a statement ie 1+1=2
> true
>
Er... why's that? :)
> rupert says
>
> It can be proved incomplete in PRA, and
> the proof makes no use of semantic notions.
>
> a proof with out the notion of truth is not an incompletness proof as
> formulated by godel
Why's that? :)
> if you give a prof that does not include the notion of truth it is not a
> godelian incompletness proof
Why?
Why?
Why?
:)
> it is something else
> and calling RA incomplete is a blantant misrepresentation of the godelian
> theorem and thus false
Who says it has anything to do with Gödel's theorem?
> there might be statements that cant be proven in RA but that does not make
> it incomplete in a godelian sense unless the notion of truth is involved
>
Do you know what "incomplete" means?
I was thinking this might be something we'd have to talk about sooner
or later.
elsiemelsi
I can prove that ZFC is consistent in ZFC+"there exists an inaccesible
cardinal.
i say
but ZFC is meant to be incomplete and according to godel the consistency
then of ZFC cant be shown
2 incompleteness theorem
quote
For any formal recursively enumerable (i.e. effectively generated) theory
T including basic arithmetical truths and also certain truths about formal
provability, T includes a statement of its own consistency if and only if T
is inconsistent.
if you can prove ZFC consistent then by godel theorem ZFC must be
inconsistent -as skolem says
ZFC can only be
Rupert
> rupert says
>
> I can prove that ZFC is consistent in ZFC+"there exists an inaccesible
> cardinal.
>
> i say
>
> but ZFC is meant to be incomplete and according to godel the consistency
> then of ZFC cant be shown
> 2 incompleteness theorem
>
> quote
>
> For any formal recursively enumerable (i.e. effectively generated) theory
> T including basic arithmetical truths and also certain truths about formal
> provability, T includes a statement of its own consistency if and only if T
> is inconsistent.
>
> if you can prove ZFC consistent then by godel theorem ZFC must be
> inconsistent -as skolem says
> ZFC can only be
>
> --
> More information athttp://www.talkaboutscience.com/faq.html
Gödel's second incompleteness theorem says that, if ZFC is consistent,
then ZFC cannot prove the consistency of ZFC. (This can be proved in
Bounded Arithmetic).
However, ZFC+"there exists an inaccessible cardinal" can prove the
consistency of ZFC. This is not a problem. There is no contradiction
with Gödel's second incompleteness theorem.
elsiemelsi
I explained how someone might think that Skolem's paradox yielded a
proof in Z_2 that ZF is inconsistent, and I showed where the argument
broke down.
i say
you did not give a disproof you just gave skolems attempt at disproving
it
and i showed via peter suber his attempt is not accepted
and as suber states
Most mathematicians [dont ]agree on how to resolve it
elsiemelsi
Do you know what "incomplete" means?
i say
quote
Gödel's first incompleteness theorem, perhaps the single most celebrated
result in mathematical logic, states that:
For any consistent formal, computably enumerable theory that proves basic
arithmetical truths, an arithmetical statement that is true, but not
provable in the theory, can be constructed.1 That is, any effectively
generated theory capable of expressing elementary arithmetic cannot be
both consistent and complete.
in a godeliann sence
incompletness is
For any consistent formal, computably enumerable theory that proves basic
arithmetical truths, an arithmetical statement that is true, but not
provable in the theory
note the notion of true is involved in incompleteness
elsiemelsi
However, ZFC+"there exists an inaccessible cardinal" can prove the
consistency of ZFC. This is not a problem. There is no contradiction
with Gödel's second incompleteness theorem.
rubbish
you cant prove ZFC to be consistent untill you prove
ZFC+ is consistent
and if you use another system to prove ZFC+ consistent you just create the
carrol paradox
Rupert
> rupert says
>
> I explained how someone might think that Skolem's paradox yielded a
> proof in Z_2 that ZF is inconsistent, and I showed where the argument
> broke down.
>
> i say
> you did not give a disproof you just gave skolems attempt at disproving
> it
A disproof of *what*?
You're trying to show ZF is inconsistent.
You haven't done it.
You think you've done it, fine, give us a machine-checkable proof.
I showed you a way where people *might* think the Löwenheim-Skolem
theorem leads to a proof in Z_2 that ZF is inconsistent, and I showed
you why it doesn't work.
You think you've got a proof that ZF is inconsistent, you have to make
some effort to explain your proof to me.
> and i showed via peter suber his attempt is not accepted
>
Did you really?
Show us the writing from Peter Suber which refutes my argument again.
> and as suber states
>
> Most mathematicians [dont ]agree on how to resolve it
>
> --
Rupert
> rupert says
>
> Do you know what "incomplete" means?
>
> i say
>
> quote
> Gödel's first incompleteness theorem, perhaps the single most celebrated
> result in mathematical logic, states that:
>
> For any consistent formal, computably enumerable theory that proves basic
> arithmetical truths, an arithmetical statement that is true, but not
> provable in the theory, can be constructed.1 That is, any effectively
> generated theory capable of expressing elementary arithmetic cannot be
> both consistent and complete.
>
> in a godeliann sence
> incompletness is
>
> For any consistent formal, computably enumerable theory that proves basic
> arithmetical truths, an arithmetical statement that is true, but not
> provable in the theory
>
> note the notion of true is involved in incompleteness
>
> --
> More information athttp://www.talkaboutscience.com/faq.html
Okay, so you don't know what "incomplete" means.
A theory T is said to be incomplete if there exists a sentence S such
that neither S nor the negation of S is in the theory T.
Okay?
Truth has nothing to do with it.
Rupert
> rupert says
>
> However, ZFC+"there exists an inaccessible cardinal" can prove the
> consistency of ZFC. This is not a problem. There is no contradiction
> with Gödel's second incompleteness theorem.
>
> rubbish
> you cant prove ZFC to be consistent untill you prove
> ZFC+ is consistent
>
I can prove in ZFC+"there exists an inaccessible cardinal" that ZFC is
consistent. There is no epistemological value in this proof, because
anyone who entertained serious doubts that ZFC is consistent would
also entertain doubts that ZFC+"there exists an inaccessible cardinal"
is consistent.
I cannot completely remove doubts that ZFC is consistent.
Complete certainty does not exist. That is just a fact of life, as I
discussed elsewhere on this thread.
So what?
> and if you use another system to prove ZFC+ consistent you just create the
> carrol paradox
>
> --
elsiemelsi
i see the problem clearly
you believe all this shit out of psychological necessity
ypou have a whole life invested in this shit
to have it destroyed would destroy you mind
why i say this is because
we debeted for days on skolem
you bring out skolem attempt at resolution
but i showed by quotes that attempt is not acepted
but
you still belive skolem paradox has been resolved and does not make ZFC
inconsistent-all in the face of accepted authorities which show you are
wrong
1
i pointed out that godel made a distinction betwen true and proven
ypou say true is what can be proven
and godel was talking about provability
i pointed out provablity is just the known of prove
so godel was talking about proof even with your attempt at semantic
deciet
yet
you still belive true is what can be proven -in the face of all the
evidence that that view is descredited
these things
show you dont believe via demonstration by via a psychological need to
have these things the way you want- for with out them that way i think you
would collapse and end up on medication
when you are dealing with neuroses logical argument is pointless
elsiemelsi
I can prove in ZFC+"there exists an inaccessible cardinal" that ZFC is
consistent.
i say
you have proved nothing untill you prove ZFC+ is consistent -but that just
leads you into the carrol paradox
so stop saying you have proved ZFC consistent -because you have not
Rupert
> rupert
> i see the problem clearly
> you believe all this shit out of psychological necessity
> ypou have a whole life invested in this shit
> to have it destroyed would destroy you mind
>
*What* do I believe out of psychological necessity?
That 1+1=2?
What do *you* believe?
Do you believe your computer is in front of you?
Your point is not clear to me, I'm afraid.
> why i say this is because
>
> we debeted for days on skolem
No, "debate" is not quite the right word. :)
> you bring out skolem attempt at resolution
> but i showed by quotes that attempt is not acepted
Well, that's not right.
Show us the quotes again.
> but
> you still belive skolem paradox has been resolved and does not make ZFC
> inconsistent-all in the face of accepted authorities which show you are
> wrong
>
Actually, "accepted authorities" go along with me on this one.
As you can confirm by reading Wikipedia.
> 1
> i pointed out that godel made a distinction betwen true and proven
> ypou say true is what can be proven
No...
> and godel was talking about provability
> i pointed out provablity is just the known of prove
> so godel was talking about proof even with your attempt at semantic
> deciet
> yet
> you still belive true is what can be proven
Nooooo....
> -in the face of all the
> evidence that that view is descredited
>
> these things
> show you dont believe via demonstration by via a psychological need to
> have these things the way you want- for with out them that way i think you
> would collapse and end up on medication
>
> when you are dealing with neuroses logical argument is pointless
>
There certainly are some circumstances under which logical argument is
pointless.
Rupert
> rupert says
>
> I can prove in ZFC+"there exists an inaccessible cardinal" that ZFC is
> consistent.
>
> i say
> you have proved nothing untill you prove ZFC+ is consistent -but that just
> leads you into the carrol paradox
> so stop saying you have proved ZFC consistent -because you have not
>
> --
> More information athttp://www.talkaboutscience.com/faq.html
I said what I said, and it was quite correct.
elsiemelsi
you have not told us what makes a statement true
thus RA is not incomplet as it has not been shown what makes a true
statement
dont say there is an incompleteness proof that does not use true
as i have said that is not a incompleteness proof in the godelian sence
which requires two things
consistency
and true
ie read the bloody thing
quote
Gödel's first incompleteness theorem, perhaps the single most celebrated
result in mathematical logic, states that:
For any consistent formal, computably enumerable theory that proves
basic arithmetical truths, an arithmetical statement that is true, but not
provable in the theory, can be constructed.1 That is, any effectively
generated theory capable of expressing elementary arithmetic cannot be
both consistent and complete.
--
elsiemelsi
*What* do I believe out of psychological necessity?
i say
i have just shown you your proof of ZFC being consistent is no proof as
you end in the carroll paradox
and all you say is
quote
I said what I said, and it was quite correct.
in the face of demonstration you like the mathematical version of the
relgious zealot blindly hold to yourt dicredited views
and you ask
*What* do I believe out of psychological necessity?
--
Rupert
> after all this
> you have not told us what makes a statement true
> thus RA is not incomplet as it has not been shown what makes a true
> statement
As I have repeatedly explained to you, this is meaningless babble. We
do not need to define the notion of truth to prove that RA is
incomplete.
I could try to help you understand Tarski's work on the definition of
truth, but it would be a big job. We would have to start by teaching
you how to understand simple statements in plain English.
> dont say there is an incompleteness proof that does not use true
>
Why not?
> as i have said that is not a incompleteness proof in the godelian sence
> which requires two things
> consistency
> and true
>
So what it boils down to is you want to use "incomplete" in a
nonstandard sense. Yes?
Rupert
> rupert says
>
> *What* do I believe out of psychological necessity?
>
> i say
> i have just shown you your proof of ZFC being consistent is no proof
It is a proof. The Carroll paradox has nothing to do with it. I
explicitly said that it lacked epistemological significance.
> as
> you end in the carroll paradox
> and all you say is
>
> quote
> I said what I said, and it was quite correct.
>
Indeed.
> in the face of demonstration you like the mathematical version of the
> relgious zealot blindly hold to yourt dicredited views
Which views of mine have you discredited? None.
> and you ask
>
> *What* do I believe out of psychological necessity?
>
Yes, I'd like to know this too.
elsiemelsi
So what it boils down to is you want to use "incomplete" in a
nonstandard sense. Yes?
i say
i am useing it in its original godelian sense
quote
Gödel's first incompleteness theorem, perhaps the single most celebrated
result in mathematical logic, states that:
For any consistent formal, computably enumerable theory that proves
basic arithmetical truths, an arithmetical statement that is true, but not
provable in the theory, can be constructed.1 That is, any effectively
generated theory capable of expressing elementary arithmetic cannot be
both consistent and complete.
you can play around with that to get your RA proof with out true
but you are misrepresenting godels work and falsifying results when you
use incompletness without reference to true
you say tarski is not relevent
then you say he is
quote
i said
> you talk about tarski
> so tel us what makes a statement true
then you say
Tarski's work isn't really relevant here.
now you say it is relevent
I could try to help you understand Tarski's work on the definition of
truth, but it would be a big job. We would have to start by teaching
you how to understand simple statements in plain English.
you are a dogs dinner of inconsistencies rupert
you make it up as you go along
Rupert
> rupert says
>
> So what it boils down to is you want to use "incomplete" in a
> nonstandard sense. Yes?
>
> i say
> i am useing it in its original godelian sense
>
> quote
>
> Gödel's first incompleteness theorem, perhaps the single most celebrated
> result in mathematical logic, states that:
>
> For any consistent formal, computably enumerable theory that proves
> basic arithmetical truths, an arithmetical statement that is true, but not
> provable in the theory, can be constructed.1 That is, any effectively
> generated theory capable of expressing elementary arithmetic cannot be
> both consistent and complete.
>
You're quoting a semantic version of the incompleteness theorem which
says that given a sound recursively axiomatizable theory in the first-
order language of arithmetic, there exists a true sentence not
provable in the theory. You can't use this to get an alternative
definition of "incompleteness". You can say that a sound theory is
incomplete if and only if there exists a true sentence not provable in
the theory. That's a result *about* incomplete theories which makes
use of the notion of truth. That doesn't mean the notion of truth is
relevant to the definition of incompleteness.
> you can play around with that to get your RA proof with out true
> but you are misrepresenting godels work and falsifying results when you
> use incompletness without reference to true
>
Wrong. *No* definition of incompleteness involves semantic notions.
This is what George was trying to tell you before. You see, this is
the basic problem; you really haven't got the first clue what you're
talking about.
> you say tarski is not relevent
> then you say he is
>
> quote
> i said
>
> > you talk about tarski
> > so tel us what makes a statement true
>
> then you say
> Tarski's work isn't really relevant here.
>
> now you say it is relevent
>
No, I do not. In the quote you provide from me below I do not say
this, as anyone with ordinary English reading comprehension skills can
understand.
> I could try to help you understand Tarski's work on the definition of
> truth, but it would be a big job. We would have to start by teaching
> you how to understand simple statements in plain English.
>
> you are a dogs dinner of inconsistencies rupert
>
Go screw yourself, you pathetic loon.
> you make it up as you go along
>
Only trying to help. If I'm impervious to reason, you can always find
someone else to talk to. :)
Peter_Smith
arithmetic is not incomplete"
He is either (A) wildly misusing standard words or (B) is claiming
that there is no sentence S of the language of RA such that we have
both RA doesn't prove S and RA doesn't prove S.
You reject (A).
In which case you've been provided with a challenge to defend (B). In
particular, show either that RA proves (Ax)(0 + x = x) or that RA
proves not-(Ax)(0 + x = x).
There's £1000 on the table from me that says it can't be done. It's as
simple as that. Put up, provide a proof one way or the other of that
sentence or its negation, or shut up.
elsiemelsi
Wrong. *No* definition of incompleteness involves semantic notions.
This is what George was trying to tell you before. You see, this is
the basic problem; you really haven't got the first clue what you're
talking about.
i say
godel is clear
there are true statements which cant be proven
so
RA is not incomplete untill you tell us what makes a statement true
quote
Gödel's first incompleteness theorem, perhaps the single most celebrated
result in mathematical logic, states that:
For any consistent formal, computably enumerable theory that proves
basic arithmetical truths, an arithmetical statement that is true, but
not
provable in the theory, can be constructed.1 That is, any effectively
generated theory capable of expressing elementary arithmetic cannot be
both consistent and complete.
--
elsiemelsi
He is either (A) wildly misusing standard words or (B) is claiming
that there is no sentence S of the language of RA such that we have
both RA doesn't prove S and RA doesn't prove S.
i say
godels incompleteness theorem is mot jut about unproven statements
it is about true statements which cant be proven
if you just do a proof only showing statements cant be propven
then
it is not a godelian incompleteness proof
read the bloody quote
it is clear
For any consistent formal, computably enumerable theory that provesbasic
arithmetical TRUTHS, an arithmetical STATEMENT THAT IS TRUE but not
provable in the theory, can be constructed
--
Rupert
> rupert says
>
> Wrong. *No* definition of incompleteness involves semantic notions.
> This is what George was trying to tell you before. You see, this is
> the basic problem; you really haven't got the first clue what you're
> talking about.
>
> i say
> godel is clear
> there are true statements which cant be proven
Yes. Given a sound recursively axiomatizable theory T in the first-
order language of arithmetic there exists a true sentence which is not
provable in T.
> so
> RA is not incomplete untill you tell us what makes a statement true
>
Garbage.
Rupert
> peter smith says
>
> He is either (A) wildly misusing standard words or (B) is claiming
> that there is no sentence S of the language of RA such that we have
> both RA doesn't prove S and RA doesn't prove S.
>
> i say
> godels incompleteness theorem is mot jut about unproven statements
>
> it is about true statements which cant be proven
>
But the concept of incompleteness has nothing to do with truth. A
theory T is said to be incomplete if and only if there exists a
sentence S such that neither S nor ~S is provable in T. That means
that if you say Robinson Arithmetic is not incomplete then you must be
claiming that either (Ax)(0+x=x) or ~(Ax)(0+x=x) is provable in
Robinson Arithmetic. If this is so then you ought to be able to show
us the proof.
> if you just do a proof only showing statements cant be propven
> then
> it is not a godelian incompleteness proof
> read the bloody quote
> it is clear
>
Learn the meanings of the bloody terms you are using. No, that quote
does not support your contention. What that quote says is correct; you
are talking babble. What do you think a "Gödelian incompleteness proof
is"? You give us the definition. Tell us what you would accept as a
"Gödelian incompleteness proof" of Robinson Arithmetic.
> For any consistent formal, computably enumerable theory that provesbasic
> arithmetical TRUTHS, an arithmetical STATEMENT THAT IS TRUE but not
> provable in the theory, can be constructed
>
> --
Rupert
>
> - Show quoted text -
Here, let me help you out here.
So, you're talking about the semantic version of Gödel's
incompleteness theorem, which says:
Given a sound recursively axiomatizable theory T in the first-order
language of arithmetic, there exists a true sentence not provable in
T.
To put it another way, given any theory T at all, either
(1) T is not a sound recursively axiomatizable theory in the first-
order language of arithmetic, or
(2) there exists a true sentence in the first-order language of
arithmetic not provable in T.
So your challenge could be: Well, Robinson Arithmetic is a theory. So
prove either (1) or (2) in the case where T is Robinson Arithmetic. No
problem. I'll prove (2). Consider the sentence (Ax)(0+x=x). This is a
true sentence in the first-order language of arithmetic not provable
in Robinson Arithmetic.
And then your objection might be: I haven't found the definition of
truth as yet in my random skimming of Wikipedia articles, so I've
decided to come along and be abusive and pretend I've found a flaw in
a major theorem which has been accepted for over 70 years. All right.
So you want the definition of truth.
Well, Moshé Machover's book "Set theory, logic, and their limitations"
is quite good, and Chapters 7-10 would be relevant here. (Note: I'm
saying the definition of truth *is* relevant to your *newly formulated
objection*. It's not relevant to the question of whether Robinson
Arithmetic is incomplete. So my former comments still stand).
So you might want to consider getting hold of a copy of that book and
trying to study it, and I could help you if you have any questions.
I'm not too optimistic about the prospects of your achieving great
learning outcomes, because I can observe that you're often unable to
understand simple sentences in plain English with no technical content
whatever. So there's a long way to go. But we could have a try.
I am happy to quote you the definition of truth from Chapters 8 and 10
of Machover's book if you want. Let me know.
elsiemelsi
I am happy to quote you the definition of truth from Chapters 8 and 10
of Machover's book if you want. Let me know.
sure qive us the quote
and show how it relates to 1+1=2 being true
and while you are at it
tell us is that the definition of truth godel was useing
or any of those other hacks who spoke about true statements in the other
incompleteness proof
if Moshé Machover's theory of truth is not what godel was useing or the
other hacks
then his theory of truth is not relevant
as it was not being used by godel etal hacks anyt way
the issue is not traskis theory or kripkies or Moshé Machover's the issue
is what theory was godel et al the hacks useing
i note moshes book is published (May 31, 1996)
so it can only relate any way to incompleteness theorems proven after 1996
- if those theorems where useing moshes definition
so theorems pre to 1996 are of no relevance to moshes defintion as it was
not around for the hacks to use
SO WHAT WAS THE DEFINITION GODEL AND THE PRE 1996 HACKS USEING
Rupert
> rupert say
>
> I am happy to quote you the definition of truth from Chapters 8 and 10
> of Machover's book if you want. Let me know.
>
> sure qive us the quote
>
Let me fix this up for you.
"Thank you very much, Rupert. It really is extraordinarily kind of you
to take so much time and trouble to attempt the herculean task of
educating me, when I cannot be bothered doing the slightest bit of
research myself beyond skimming Wikipedia articles. I appreciate it
very much."
Also, you really need to buy the book. We're going to have to look at
practically all of Chapters 7 and 8. I can't post all of them here.
That violates copyright.
> and show how it relates to 1+1=2 being true
>
Okay, let's fix this up too.
"Also, I apologize for making further requests of you when I have
already put you to so much trouble, but I wonder if you could also
take the time and trouble to show how the definition of truth applies
in the case of the sentence '1+1=2'."
Sure, no problem.
> and while you are at it
>
> tell us is that the definition of truth godel was useing
Yes, it is.
> or any of those other hacks who spoke about true statements in the other
> incompleteness proof
I don't know whom you are talking about here.
> if Moshé Machover's theory of truth is not what godel was useing or the
> other hacks
> then his theory of truth is not relevant
Tarski's definition of truth, which Moshé Machover expounds in his
very nice textbook. Yes, this is relevant to understanding the
statement of the semantic version of the incompleteness theorem which
you keep quoting. And I, for some reason, am going to attempt the
futile task ot trying to educate you to the point where you can
actually understand it. Funny old world, isn't it?
> as it was not being used by godel etal hacks anyt way
>
> the issue is not traskis theory or kripkies or Moshé Machover's the issue
> is what theory was godel et al the hacks useing
This one. When they stated their theorems in semantic form. (Which
they usually didn't, by the way).
Gödel's 1931 paper predates Tarki's definition of truth, and for that
reason Gödel took care to state his results in a form which didn't
require any semantic notions, because he knew that a precise
definition of the requisite semantic notions hadn't yet been given.
Then Tarski came along and cleared it all up, and now we have the
semantic versions of the incompleteness theorems as well.
>
> i note moshes book is published (May 31, 1996)
> so it can only relate any way to incompleteness theorems proven after 1996
You really have a strong predilection for talking pitiful drivel,
don't you?
> - if those theorems where useing moshes definition
Sigh. It's not his definition, it's Tarsrki's, I don't know the date
but I'm sure it's in the first half of the twentieth century. Now yes,
Tarski's work does come after Gödel's 1931 paper (I believe). Which is
why in that paper Gödel took care to state his results using syntactic
rather than semantic notions. As one can easily confirm by reading the
paper. Also, there is a famous footnote 48a where Gödel makes some
remarks which suggest that he had independently come up with Tarski's
method of defining truth.
> so theorems pre to 1996 are of no relevance to moshes defintion as it was
> not around for the hacks to use
> SO WHAT WAS THE DEFINITION GODEL AND THE PRE 1996 HACKS USEING
>
You want to talk about Gödel's 1931 paper? In that case, we have to
stop talking about the semantic version of the incompleteness theorem
and we have to get back to trying to get you to acknowledge my point
that the definition of truth is in no way relevant to the notion of
"incompleteness" that Gödel was using. You have to actually look at
the paper, not a Wikipedia soundbite. Do you have any conception at
all of what is involved in doing serious scholarship?
So make up your mind. Are we going to talk about the non-semantic
version of the incompleteness theorem given in Gödel's 1931 paper, or
are we going to talk about the semantic version and Tarski's work on
how to define truth?
elsiemelsi
if Moshé Machover's theory of truth is not what godel was useing or the
> other hacks
> then his theory of truth is not relevant
rupert says
Tarski's definition of truth, which Moshé Machover expounds in his very
nice textbook.
but you have told has when i asked tarskis work is not relevent
again rupert inconsistencies
we cant tell what you are talking about at all
quote
i said
> you talk about tarski
> so tel us what makes a statement true
then you say
Tarski's work isn't really relevant here.
--
elsiemelsi
Machover's book
so is this theory of truth mosches
or
tarskis
if it is tarskis
you have said when i asked show how tarski tell us why 1+1=2 is true
that tarski is not relevent
you say godel was useing tarskis theory of truth
so now you show where in his proof he tells us that
if he does not tell us that
you cant asssume he did
and for that matter if tarski is the theory of truth for mathematics show
how wiles used it in his proof of fermats theorem
Rupert
> i said
> if Moshé Machover's theory of truth is not what godel was useing or the
>
> > other hacks
> > then his theory of truth is not relevant
>
> rupert says
>
> Tarski's definition of truth, which Moshé Machover expounds in his very
> nice textbook.
>
> but you have told has when i asked tarskis work is not relevent
>
> again rupert inconsistencies
> we cant tell what you are talking about at all
>
Actually, the problem is we can't tell what *you* are talking about.
First you say you want to talk about the semantic version of Gödel's
incompleteness theorem, then you say you want to talk about what Gödel
did in 1931. These are two different things, and the question of which
of these we want to talk about has a bearing on whether Tarski's
theory of truth is relevant. This is what I have been trying to convey
to you. I am sorry you are having so much trouble understanding, but I
am afraid it is not due to any deficiency on my part.
> quote
> i said
>
> > you talk about tarski
> > so tel us what makes a statement true
>
> then you say
> Tarski's work isn't really relevant here.
>
It's not if you want to talk about the non-semantic version of the
incompleteness theorem, which is what Gödel did in 1931, or the
question of whether Robinson Arithmetic is incomplete. (You keep
making the absurd statement "Robinson Arithmetic is not incomplete").
elsiemelsi
godel published his theorem in 1931-32
but tarski did not publish his work on truth untill 1933
so godel could not have been using it
quote
http://plato.stanford.edu/entries/tarski-truth/
In 1933 the Polish logician Alfred Tarski published a paper in which he
discussed the criteria that a definition of ‘true sentence’ should
meet, and gave examples of several such definitions for particular formal
languages.
but
n 1956 he and his colleague Robert Vaught published a revision of one of
the 1933 truth definitions, to serve as a truth definition for
model-theoretic languages
you are full of shit
so tell us according to godel what makes a statement true
Rupert
> i asked you what makes a statement true and you refered me to Moshé
> Machover's book
>
> so is this theory of truth mosches
> or
> tarskis
>
Okay, listening very carefully here?
Tarski's.
I said that a number of times. I said it right from the word go. I
don't know what would have led you to believe that this material was
originated by Machover. He's just writing a textbook for
undergraduates about stuff which has been quite standard for a long
time. Clear on that now?
> if it is tarskis
> you have said when i asked show how tarski tell us why 1+1=2 is true
> that tarski is not relevent
>
Because at the time we were discussing your stupid statement that
Robinson Arithmetic is not incomplete. Tarski's work is not relevant
to that issue.
> you say godel was useing tarskis theory of truth
When Gödel wrote his 1931 paper, the notion of truth had not yet been
given a precise mathematical definition. In the introduction, Gödel
discusses a semantic version of the incompleteness theorem, where he
makes use of an intuitive notion of "truth". However, he was aware
that this concept did not have a precise mathematical definition at
the time of writing. So, in the main part of the paper, he gave a non-
semantic version of the incompleteness theorem, which did not use the
notion of truth. So if you want to talk about this version of the
theorem, the notion of truth is not in any way relevant, as we have
all been telling you from day one. This is why you must read Gödel's
actual paper if you want to do serious scholarship, not rely on
Wikipedia soundbites. There is a famous footnote 48a to Gödel's paper
which seems to suggest that he had already independently come up with
some of Tarski's ideas about how to define truth. We're not sure. But
in any case, the results stated in the main part of the paper do not
use semantic notions. Later on, when various people including Gödel
started discussing semantic versions of the theorem, everyone agreed
that Tarski's definition of truth was spot-on. That's the definition
we always use whenever we talk about the semantics for a first-order
theory in classical logic.
Glad to be helping you out with understanding the basics of the
subject.
> so now you show where in his proof he tells us that
> if he does not tell us that
> you cant asssume he did
>
See above.
> and for that matter if tarski is the theory of truth for mathematics show
> how wiles used it in his proof of fermats theorem
>
I haven't read Wiles' proof of Fermat's last theorem, and it would
take me many months if not years of study to be able to understand it.
I have no reason to believe that he had any need to use Tarski's
definition of truth. It is not usually used outside mathematical
logic. He may or may not have done. I doubt it. The notion of truth is
not used ubiquitously. It is used when we are considering a
mathematical theory which is now itself an object of mathematical
investigation, and we want to discuss the semantics for this theory.
elsiemelsi
Rupert
> you say godel is using tarskis theory of truth
> godel published his theorem in 1931-32
>
> but tarski did not publish his work on truth untill 1933
>
> so godel could not have been using it
>
> In 1933 the Polish logician Alfred Tarski published a paper in which he
> discussed the criteria that a definition of 'true sentence' should
> meet, and gave examples of several such definitions for particular formal
> languages.
>
Yes, thank you.
Here, quite a few posts ago,
http://groups.google.com/group/sci.logic/msg/0d462c7ab9d76aaa?dmode=source
I wrote
"Gödel's 1931 paper predates Tarki's definition of truth, and for that
reason Gödel took care to state his results in a form which didn't
require any semantic notions, because he knew that a precise
definition of the requisite semantic notions hadn't yet been given.
Then Tarski came along and cleared it all up, and now we have the
semantic versions of the incompleteness theorems as well."
And here, just now,
http://groups.google.com/group/sci.logic/msg/2ae5a644ae73e2bb?dmode=source
I wrote
"When Gödel wrote his 1931 paper, the notion of truth had not yet
been
given a precise mathematical definition. In the introduction, Gödel
discusses a semantic version of the incompleteness theorem, where he
makes use of an intuitive notion of "truth". However, he was aware
that this concept did not have a precise mathematical definition at
the time of writing. So, in the main part of the paper, he gave a
non-
semantic version of the incompleteness theorem, which did not use the
notion of truth. So if you want to talk about this version of the
theorem, the notion of truth is not in any way relevant, as we have
all been telling you from day one. This is why you must read Gödel's
actual paper if you want to do serious scholarship, not rely on
Wikipedia soundbites. There is a famous footnote 48a to Gödel's paper
which seems to suggest that he had already independently come up with
some of Tarski's ideas about how to define truth. We're not sure. But
in any case, the results stated in the main part of the paper do not
use semantic notions. Later on, when various people including Gödel
started discussing semantic versions of the theorem, everyone agreed
that Tarski's definition of truth was spot-on. That's the definition
we always use whenever we talk about the semantics for a first-order
theory in classical logic."
And now you re-state part of what I wrote as if it is news and refutes
me.
Hmmmm.
It looks like you are not listening very carefully.
What a surprise.
> but
>
> n 1956 he and his colleague Robert Vaught published a revision of one of
> the 1933 truth definitions, to serve as a truth definition for
> model-theoretic languages
>
That's interesting, I'll have to look into that further. I said that I
didn't know the date when Tarski's truth definition for first-order
languages first appears, but I conjectured that it was probably in the
first half of the twentieth century. No doubt we'll clear this all up
before too long.
Here's what I attempted to make clear a number of times. We'll say it
all just one more time. Try to listen carefully.
(1) When Gödel wrote his paper in 1931, a precise definition of truth
had not yet been given. There was a widely shared intuitive notion of
truth, but at the time it lacked a precise definition. For that
reason, in the introduction to his paper Gödel sketched a proof for a
semantic version of his incompleteness theorem, but in the main part
of the paper gave a version of the theorem using purely syntactic
notions, with a precise and detailed proof.
(2) For that reason, if you want to talk about what Gödel did in 1931,
you cannot talk about the semantic version of the incompleteness
theorem. Furthermore, the notion of "incompleteness" is not a semantic
one at all, contrary to your inane babbling. A theory T is said to be
incomplete if there exists a sentence S in the language of the theory
T such that neither S nor ~S is provable in the theory T. This is a
purely syntactic notion. You were told this in your very first reply.
When we are addressing the question of whether Robinson Arithmetic is
incomplete, semantic notions are irrelevant.
(3) If you like, we can talk about the semantic version of the
incompleteness theorem and its application to Robinson Arithmetic. We
can say
Given any sound recursively axiomatizable theory T in the first-order
language of arithmetic, there exists a true sentence in the first-
order language of arithmetic which is not provable in the theory T.
That's the semantic version of the incompleteness theorem. We can re-
state it like this.
Given any theory T, one of the following is true:
(A) T is not a sound recursively axiomatizable theory in the first-
order language of arithmetic.
(B) There exists a true sentence in the first-order language of
arithmetic which is not provable in the theory T.
You might challenge me to prove one of these statements in the case
where T is Robinson Arithmetic. Fine. I'll go with (B). Consider the
sentence (Ax)(0+x=x). This is a true sentence in the first-order
language of arithmetic. It is not provable in Robinson Arithmetic. (We
can prove, for example, in Primitive Recursive Arithmetic, that this
sentence is not provable in Robinson Arithmetic).
But then you might say "Well, hold on a minute. In the course of my
random skimming of Wikipedia soundbites, I haven't yet grasped the
definition of truth which has been universally accepted by logicians
for 50 years at the very least. So that means I'm entitled to come
here and start being abusive to people who are trying to educate me,
and claim that I've refuted a major theorem which has been universally
accepted by the mathematical community for the past 70 years, and be
very pleased with what I've achieved."
So then we might patiently say "Well, we can have a go at teaching you
Tarski's theory of truth if you like. Just for the hell of it."
But note: If we do all this, then we are talking about the *semantic*
version of the incompleteness theorem, which is not what Gödel did in
1931. As I've said very, very many times now, but you evidently
haven't been listening. But what the hell, we may as well say it just
one more time.
> you are full of shit
>
Yes, thank you for evaluating my efforts to educate you in such a
thoughtful and fair-minded way.
> so tell us according to godel what makes a statement true
>
All right. So just one more time:
In the main part of his paper, Gödel does not use the notion of truth.
Because it did not have a precise definition at the time.
Okay?
So you need to tell me where you want to go from here.
elsiemelsi
In the main part of his paper, Gödel does not use the notion of truth.
Because it did not have a precise definition at the time.
the quote is clear
quote
Gödel's first incompleteness theorem, perhaps the single most celebrated
result in mathematical logic, states that:
For any consistent formal, recursively enumerable theory that proves
basic arithmetical truths, an arithmetical statement that is true, but not
provable in the theory, can be constructed.1 That is, any effectively
generated theory capable of expressing elementary arithmetic cannot be
both consistent and complete.
--
elsiemelsi
For that
reason, in the introduction to his paper Gödel sketched a proof for a
semantic version of his incompleteness theorem, but in the main part
of the paper gave a version of the theorem using purely syntactic
notions, with a precise and detailed proof.
i say
there is a semantic version and a syntactic
so when you say
For that reason, if you want to talk about what Gödel did in 1931,you
cannot talk about the semantic version of the incompleteness
theorem
you are wrong as you have said there are two version
and noted that godel had no idea of what truth was in his semantic yet
talks about true statements
i rest my case you cant tell us what godel meant by true statements as you
admitt he did not know
thus his theorem is meaningless
and further you say taskis idea of truth is what is used by mathematics
it is up to you to show just how that theory of truth conditions what
mathematicians do - not what the talk about -but how tarskis theory is
brought into any thing a mathematician does
ie wiles
or any one else who happens to prove something
for if you cant then any old theory of truth can be mouthed of as what
mathematics uses
EXAMPLE
i can quite easily tell you based on a coherence theory of truth just
what godel could have meant by true statement
ie a true statement is one that chohers with all the statements in the
system ie is not in contradiction
so easy
elsiemelsi
the syntactical version of the incompeteness theorem is not about true
statements thus truth
i say rubbish
Gödel represented statements by numbers
it is all about the godel statement being true
assuming the theory is consistent (as done in the theorem's hypothesis)
there is no such number, and the Gödel statement is true, but the theory
cannot prove it
it is still about truth
and untill godel tells us what makes his statement true
-which you say he has no theory of truth- then his theorem is meaningless
quote
http://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_theorems#First_incompleteness_theorem
In order to prove the first incompleteness theorem, Gödel represented
statements by numbers. Then the theory at hand, which is already about
numbers, also pertains to statements, including its own. Questions about
the provability of statements are represented as questions about the
properties of numbers, which would be decidable by the theory if it were
complete. In these terms, the Gödel sentence is a claim that there does
not exist a natural number with a certain property. A number with that
property would encode a proof of inconsistency of the theory. If there
were such a number then the theory would be inconsistent, contrary to
hypothesis. So, assuming the theory is consistent (as done in the
theorem's hypothesis) there is no such number, and the Gödel statement is
true, but the theory cannot prove it. An important conceptual point is that
we must assume that the theory is consistent in order to state that this
statement is true.
elsiemelsi
a semantic and a syntactic
and that godel had no idea what makes a statement true
quote
"When Gödel wrote his 1931 paper, the notion of truth had not yet
been
given a precise mathematical definition. In the introduction, Gödel
discusses a semantic version of the incompleteness theorem, where he makes
use of an intuitive notion of "truth". However, he was aware that this
concept did not have a precise mathematical definition at
the time of writing. So, in the main part of the paper, he gave a
non-semantic version of the incompleteness theorem, which did not use the
notion of truth.
note he say in the syntactic there is no notion of truth
so we have now reached the endgame
i have shown in what rupert calles the semantic godel uses the notion of
truth
quote
Gödel's first incompleteness theorem, perhaps the single most celebrated
result in mathematical logic, states that:
For any consistent formal, recursively enumerable theory that proves
basic arithmetical truths, an arithmetical statement that is true, but not
provable in the theory, can be constructed.1 That is, any effectively
generated theory capable of expressing elementary arithmetic cannot be
both consistent and complete.
now seeing rupert admitts godel had no idea of truth
then this semantic version is meaningless as he cant tell us what makes a
statement true
now in what rupert calls the syntactic i have shown the notion of truth is
used
ie
Gödel represented statements by numbers
it is all about the godel statement being true
assuming the theory is consistent (as done in the theorem's
hypothesis)there is no such number, and the Gödel statement is true, but
the theory cannot prove it
note it is still about truth
and rupert has admited godel has no idea what truth is
thus untill godel tells us what makes his statement true
then this syntactic version is meaningless
quote
http://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_theorems#First_incompleteness_theorem
In order to prove the first incompleteness theorem, Gödel represented
statements by numbers. Then the theory at hand, which is already about
numbers, also pertains to statements, including its own. Questions about
the provability of statements are represented as questions about the
properties of numbers, which would be decidable by the theory if it were
complete. In these terms, the Gödel sentence is a claim that there does
not exist a natural number with a certain property. A number with that
property would encode a proof of inconsistency of the theory. If there
were such a number then the theory would be inconsistent, contrary to
hypothesis. So, assuming the theory is consistent (as done in the
theorem's hypothesis) there is no such number, and the Gödel statement is
true, but the theory cannot prove it. An important conceptual point is that
we must assume that the theory is consistent in order to state that this
statement is true.
thus we have the situation that both godels versions are meaningles as he
cant tell us what makes a statement true-as according to rupert godel had
no idea of what truth is
so
both godels theorems is meaningless
and RA is not incomplete as it cant be shown whjat makes statements
true
Rupert
> rupert says
>
> In the main part of his paper, Gödel does not use the notion of truth.
> Because it did not have a precise definition at the time.
>
> the quote is clear
> quote
>
> Gödel's first incompleteness theorem, perhaps the single most celebrated
> result in mathematical logic, states that:
>
> For any consistent formal, recursively enumerable theory that proves
> basic arithmetical truths, an arithmetical statement that is true, but not
> provable in the theory, can be constructed.1 That is, any effectively
> generated theory capable of expressing elementary arithmetic cannot be
> both consistent and complete.
>
> --
> More information athttp://www.talkaboutscience.com/faq.html
Sigh. Yes, this is the *semantic* version of the incompleteness
theorem. But that is *not* the version which Gödel presented in the
main part of his 1931 paper, as I have explained countless times, and
as you can easily check by looking at the translation of the paper
which is available on-line for free. Do you want me to give you the
link *again*? I was under the impression that you'd made some effort
to read the paper.
This is why you have to engage in serious scholarship instead of
relying on Wikipedia soundbites if you want to gain any insight into
what you are talking about.
elsiemelsi
then this semantic version is meaningless as he cant tell us what makes a
statement true
now in what rupert calls the syntactic i have shown the notion of truth
is
used
ie
Gödel represented statements by numbers
it is all about the godel statement being true
assuming the theory is consistent (as done in the theorem's
hypothesis)there is no such number, and the Gödel statement is true, but
the theory cannot prove it
note it is still about truth
and rupert has admited godel has no idea what truth is
thus untill godel tells us what makes his statement true
then this syntactic version is meaningless
--
Rupert
> rupert says
>
> For that
> reason, in the introduction to his paper Gödel sketched a proof for a
> semantic version of his incompleteness theorem, but in the main part
> of the paper gave a version of the theorem using purely syntactic
> notions, with a precise and detailed proof.
>
> i say
> there is a semantic version and a syntactic
> so when you say
>
> For that reason, if you want to talk about what Gödel did in 1931,you
> cannot talk about the semantic version of the incompleteness
> theorem
>
> you are wrong as you have said there are two version
No, I'm not wrong. Sorry about that. :)
There are two versions *now*. In 1931, Gödel presented the *syntactic*
version in the main part of his paper and gave it a precise proof. He
also presented a sketch proof of the *semantic* version in the
introduction, but that was not precise at the time, because the
relevant semantic notions did not have precise definitions at the
time.
And of course I've said all this countless times.
Who knows, we may eventually clear away a cobweb or two eventually.
> and noted that godel had no idea of what truth was in his semantic yet
> talks about true statements
>
Informally, yes, in the introduction. But he acknowledged that truth
was not a precise mathematical concept at the time, and for that
reason gave the syntactic version in the main part of his paper. As
you could check easily enough if you actually read the paper...
> i rest my case you cant tell us what godel meant by true statements as you
> admitt he did not know
> thus his theorem is meaningless
>
Sorry, wrong again. :)
You go and look up the translation of Gödel's paper which is available
on-line for free. Have a look at the statement of Proposition VI. It
makes no mention of truth and a version of it can be proved in
Primitive Recursive Arithmetic. Any trouble you have in understand it,
let me know and I'm happy to help. I'm such a nice guy.
> and further you say taskis idea of truth is what is used by mathematics
>
Tarski's notion of truth for sentences in a first-order language (with
respect to a given interpretation of the language) is now the standard
one, yes.
> it is up to you to show just how that theory of truth conditions what
> mathematicians do
Actually, it's not up to me to do anything. All I'm doing is wasting
my time trying to reason with a complete moron, because it provides me
with mild entertainment. I certainly have no obligations whatsoever
towards you.
> - not what the talk about -but how tarskis theory is
> brought into any thing a mathematician does
> ie wiles
Wiles' work has nothing to do with Tarski's work.
> or any one else who happens to prove something
You don't have to refer to Tarski's work on truth every time you do
mathematics. Similarly, you don't have to ponder the question of what
truth is every time you do work in history, or philosophy, or physics.
> for if you cant then any old theory of truth can be mouthed of as what
> mathematics uses
>
As I mentioned last time, you have a very strong predilection for
talking inane rubbish.
> EXAMPLE
> i can quite easily tell you based on a coherence theory of truth just
> what godel could have meant by true statement
> ie a true statement is one that chohers with all the statements in the
> system ie is not in contradiction
> so easy
>
Yeah, you could say that. You would have the problem that Proposition
VI as given in the main part of the paper makes absolutely no mention
of truth at all, so people who had actually read the paper would say
"What are you talking about? Go and read the paper." Or you could try
to make some inane babblings about how the coherence theory of truth
is assumed in the proof of the semantic version of the incompleteness
theorem (which was not precisely formulated until after Gödel's 1931
paper; say it often enough and you may get it one day). Now, the
problem there would be that the coherence theory of truth is a
philosophical theory about the notion of truth in informal languages,
it's not a precise mathematical definition of a notion of truth for
formal languages. So anyone with the slightest knowledge of the
subject would immediately know you were talking complete nonsense. But
then, that happens to you quite a lot, so it would be nothing new.
Rupert
> rupert say
> the syntactical version of the incompeteness theorem is not about true
> statements thus truth
>
I'm not sure whether this is a fragment of a sentence I actually
wrote. If it is, then clearly you should give the full sentence, at
the very least.
> i say rubbish
>
> Gödel represented statements by numbers
> it is all about the godel statement being true
>
No. Go and read the paper. Examine the statement of Proposition VI.
Let me know if you need any help.
Gödel did make the point that, for example, the Gödel sentence for P
is equivalent (in quite a weak theory) to a sentence which may be said
to assert that P is consistent. Hence (informally) he made the point
that, if we are convinced that P is consistent, then we ought to also
be convinced that the Gödel sentence is true. However, this was not a
part of the argument which was done with mathematical precision at the
time of the writing of the paper (though it can be so done now). Gödel
was clear about what he could and could not state and prove with
mathematical precision at the time.
> assuming the theory is consistent (as done in the theorem's hypothesis)
> there is no such number, and the Gödel statement is true, but the theory
> cannot prove it
>
Yes, he said this. This is an informal part of the argument and is not
meant to be a precise mathematical statement. The actual formal
statement of Proposition VI makes no mention of truth at all. Go and
have a look.
> it is still about truth
> and untill godel tells us what makes his statement true
> -which you say he has no theory of truth- then his theorem is meaningless
The theorem of Proposition VI. There is absolutely nothing wrong with
Proposition VI, and no theory of truth is needed to make it precise.
(Although we do now have a theory of truth, by the way).
> quote
>
> http://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_theorems#Fir...
>
> In order to prove the first incompleteness theorem, Gödel represented
> statements by numbers. Then the theory at hand, which is already about
> numbers, also pertains to statements, including its own. Questions about
> the provability of statements are represented as questions about the
> properties of numbers, which would be decidable by the theory if it were
> complete. In these terms, the Gödel sentence is a claim that there does
> not exist a natural number with a certain property. A number with that
> property would encode a proof of inconsistency of the theory.
This last sentence is wrong, by the way. This is why there are
problems with relying on soundbites from Wikipedia.
The article on Wikipedia about Gödel's incompleteness theorem is not
very good; it has a lot of rubbish in it and falsely states that
Hofstadter gave a new proof of the theorem in his popular book about
the subject.
> If there
> were such a number then the theory would be inconsistent, contrary to
> hypothesis. So, assuming the theory is consistent (as done in the
> theorem's hypothesis) there is no such number, and the Gödel statement is
> true, but the theory cannot prove it.
That's right; if the theory is consistent then the Gödel sentence is
true, and Gödel did make this point informally, but he was aware that
he could not formulate this as a precise mathematical statement at the
time. We can now. He did not do it back then. He made the assertion
that if the theory is consistent then the Gödel sentence is true
informally; it did not play a role in the mathematically precise part
of his argument. The mathematically precise part of his argument makes
use of purely syntactic notions; no notion of truth is needed.
Say it often enough and who knows, one day we might drill it into his
thick skull.
> An important conceptual point is that
> we must assume that the theory is consistent in order to state that this
> statement is true.
>
Yes, that is an important point, the hypothesis that the theory is
consistent is required. The Gödel sentence for P is true if and only
if P is consistent. We can state this precisely and prove it
rigorously now, because we now have a precise definition of truth. We
did not have one at the time when Gödel wrote his paper. Hence in the
mathematically precise part of his argument he made use of purely
syntactic notions.
Say it often and clearly enough and who knows, we might succeed in
clearing away just a tiny part of the dense network of cobwebs.
Rupert
> rupert says godel gave two version of the incompleteness proof
> a semantic and a syntactic
>
> and that godel had no idea what makes a statement true
>
I wrote what you quote me as writing below. I think this is a better
way of saying it. There was a widely shared intuitive notion of truth
at the time. Gödel had some idea about what truth is. He did not know
how to give a mathematically precise definition of it. Or he may have
done, as footnote 48a suggests, but he chose not to do it in his 1931
paper, formulating the mathematically precise part of his argument
using purely syntactic notions instead.
> quote
>
> "When Gödel wrote his 1931 paper, the notion of truth had not yet
> been
> given a precise mathematical definition. In the introduction, Gödel
> discusses a semantic version of the incompleteness theorem, where he makes
> use of an intuitive notion of "truth". However, he was aware that this
> concept did not have a precise mathematical definition at
> the time of writing. So, in the main part of the paper, he gave a
> non-semantic version of the incompleteness theorem, which did not use the
> notion of truth.
>
> note he say in the syntactic there is no notion of truth
>
> so we have now reached the endgame
>
> i have shown in what rupert calles the semantic godel uses the notion of
> truth
> quote
> Gödel's first incompleteness theorem, perhaps the single most celebrated
> result in mathematical logic, states that:
>
> For any consistent formal, recursively enumerable theory that proves
> basic arithmetical truths, an arithmetical statement that is true, but not
> provable in the theory, can be constructed.1 That is, any effectively
> generated theory capable of expressing elementary arithmetic cannot be
> both consistent and complete.
>
This is a soundbite from Wikipedia which presents the semantic version
of Gödel's incompleteness theorem which is *not* the theorem Gödel
stated in his 1931 paper. To find the statement of Proposition VI in
Gödel's 1931 paper, one should look at the translation of the paper
which is available on-line for free. You need the link again, do you?
It's very easy to find on Google. *That* is where you must look if you
have a fetish for focussing on the question of what Gödel achieved in
1931, as opposed to what has been done since. If that is what you want
to talk about, then you must look at what Gödel actually wrote. Not
soundbites from popular expositions in Wikipedia.
> now seeing rupert admitts godel had no idea of truth
See above. I wrote was I wrote. What I wrote is correct.
> then this semantic version is meaningless as he cant tell us what makes a
> statement true
>
The semantic version is not meaningless because we do *now* have a
precise definition of truth. Tarski provided it, in 1933 as you kindly
told us. The semantic version was not formulated as a precise
mathematical statement in 1931. Gödel explicitly acknowledges this in
the paper. In the part of the argument which is intended to have
mathematical precision, he presents a version of the theorem which
uses purely syntactic notions. This appears as Proposition VI in the
paper. One can read the statement of Proposition VI in the translation
of the paper which is available on-line for free. Google is your
friend. You will find that it makes no mention of any semantic
notions. It uses the notion of omega-consistency, which is a syntactic
notion. It also uses the concept "primitive recursively
axiomatizable". This is a concept which Gödel introduced and gave a
precise definition of in the early part of his paper. Again, it is a
purely syntactic notion.
Some people might have the crazy idea that if you want to comment on
what Gödel did in 1931 you must try to read the paper...
> now in what rupert calls the syntactic i have shown the notion of truth is
> used
>
No, you have not.
Google for the translation of Gödel's paper which is available on-line
for free. Find the statement of Proposition VI, a purely syntactic
theorem. Quote it for us. Show us where the notion of truth is used in
the statement of Proposition VI. It isn't.
> ie
> Gödel represented statements by numbers
> it is all about the godel statement being true
>
We've been over all this. Gödel's theorem is Proposition VI. No
mention is made of the "truth" of the Gödel sentence. Gödel stated
informally that the Gödel sentence for P was true if and only if P was
consistent. He did not view this as a precise mathematical statement;
it was a point he made informally. He could, however state precisely
and prove that a certain sentence which intuitively can be thought of
as asserting the consistency of P could be proved in P to be
equivalent to the Gödel sentence for P. This is an important point for
the proof of the second incompleteness theorem. (That one appears as
Proposition XI, I believe. Or maybe it's Proposition X. We can check).
In the actual formally stated *theorems*, no mention of truth is made.
The notion of truth is not needed. The formally stated theorems of
Gödel's 1931 paper are syntactic results.
> assuming the theory is consistent (as done in the theorem's
> hypothesis)there is no such number, and the Gödel statement is true, but
> the theory cannot prove it
>
This is a soundbite from a popular exposition from Wikipedia. It is
correct, by the way, and we can make it all precise and prove it
rigorously now that Tarski has done his work on truth. It is not a
statement of what Gödel claimed to be able to prove rigorously in
1931. To find out what he claimed to prove rigorously in 1931, you
must look at the actual paper, not soundbites from popular expositions
in Wikipedia. Let me know if you need any help with reading the paper.
> note it is still about truth
>
> and rupert has admited godel has no idea what truth is
> thus untill godel tells us what makes his statement true
> then this syntactic version is meaningless
>
You haven't read the syntactic version yet so you are not competent to
say anything about it in this point. The syntactic version appears as
Proposition VI in the paper. A translation of the paper is available
on-line for free. Google is your friend.
[snip blah]
> and RA is not incomplete
Absolute garbage.
It can be proved in Bounded Arithmetic that if Robinson Arithmetic is
consistent it is incomplete. One does it by showing in Bounded
Arithmetic that if Robinson Arithmetic is consistent then it does not
prove (Ax)(0+x=x) or ~(Ax)(0+x=x). To achieve this, one constructs
interpretations of the two theories T_1=RA+(Ax)(0+x=x) and T_2=RA+~(Ax)
(0+x=x) in RA, (and one proves in BA that they are interpretations).
Constructing an interpretation of T_2 in RA is a very easy
undergraduate exercise, the only insight required is how to construct
a non-standard model of RA in which ~(Ax)(0+x=x). Constructing an
interpretation of T_1 in RA is a little more difficult; I think it was
Solovay who first showed how to do things like this, it is all
discussed in Edward Nelson's "Predicative Arithmetic". Perhaps there
is an easier way of doing it, I don't know.
And it can be proved in Primitive Recursive Arithmetic that Robinson
Arithmetic is consistent. This is done in Shoenfield's "Mathematical
Logic". One reformulates RA as an open theory and then applies
Gentzen's Hauptsatz.
None of this reasoning in any way uses the notion of truth and you
have done absolutely nothing to cast the slightest doubt on it.
Robinson Arithmetic is incomplete. Continually babbling that it isn't
because Gödel didn't know how to define truth is just the height of
inanity.
> as it cant be shown whjat makes statements
> true
>
> --
elsiemelsi
> i say rubbish
> Gödel represented statements by numbers
> it is all about the godel statement being true
No. Go and read the paper. Examine the statement of Proposition VI.
Let me know if you need any help.
it is commonly accepted that godel is talking about true statements in his
theorem
the syntactic version
quote
http://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_theorems#First_incompleteness_theorem
In order to prove the first incompleteness theorem, Gödel represented
statements by numbers ....
So, assuming the theory is consistent (as done in the theorem's
hypothesis) there is no such number, and the Gödel statement is true
now rupert says
Gödel did make the point that, for example, the Gödel sentence for P is
equivalent (in quite a weak theory) to a sentence which may be said
to assert that P is consistent. Hence (informally) he made the point that,
if we are convinced that P is consistent, then we ought to also be
convinced that the Gödel sentence is true. However, this was not a part
of the argument which was done with mathematical precision at the time of
the writing of the pape
so rupert still admit godel talks about truth-even though accoroding to
rupert this was not a part of the argument which was done with
mathematical precision at the time of the writing of the paper
stiff shit if he could not give precision -fact is he used a notion of
truth-which you have said he had no idea of what truth is
rupert says
You go and look up the translation of Gödel's paper which is available
on-line for free. Have a look at the statement of Proposition VI. It makes
no mention of truth
proposition v1 says
http://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_theorems#First_incompleteness_theorem
The general result about the existence of undecidable propositions reads
as follows:
"Theorem VI. For every ω-consistent recursive class κ of FORMULAS
there are recursive CLASS SIGNS r, such that neither v Gen r nor Neg(v Gen
r) belongs to Flg(κ) (where v is the FREE VARIABLE of r).2 (van Heijenoort
translation and typsetting 1967:607. "Flg" is from "Folgerungsmenge = set
of consequences" and "Gen" is from "Generalisation = generalization" (cf
Meltzer and Braithwaite 1962, 1992 edition:33-34)
to which the commonaly accepted interpretation interpretation of that is
quote
Gödel's first incompleteness theorem, perhaps the single most celebrated
result in mathematical logic, states that:
For any consistent formal, recursively enumerable theory that proves
basic arithmetical truths, an arithmetical statement that is true, but not
provable in the theory, can be constructed.1 That is, any effectively
generated theory capable of expressing elementary arithmetic cannot be
both consistent and complete.
that godel is talking about true statments as distinct from proven
statements accounts for the claim that godel destroyed the hibert idea
that true is proven from axioms
quote
http://en.wikipedia.org/wiki/Truth#Truth_in_mathematics
In addition, from at least the time of Hilbert's program at the turn of
the twentieth century to the proof of Gödel's theorem and the development
of the Church-Turing thesis in the early part of that century, true
statements in mathematics were generally assumed to be those statements
which are provable in a formal axiomatic system.
The works of Kurt Gödel, Alan Turing, and others shook this assumption,
with the development of statements that are true but cannot be proven
within the system.
elsiemelsi
Yes, that's right. And we actually succeeded in proving the epsilon
conjecture, so the proof is fine.
i say
the proof is not fine
why
because untill maths is proven consistent
then it could be inconsistent
in which case anything and every thing could be proved about any ad hoc
conjecture put forward to make a proof work
the proof of the ad hoc conjecture rest on maths being consistent
and that is what is wrong with useing ad hoc for any ad hoc could be
proven if maths is inconsistent
Rupert
> i said
>
> > i say rubbish
> > Gödel represented statements by numbers
> > it is all about the godel statement being true
>
> No. Go and read the paper. Examine the statement of Proposition VI.
> Let me know if you need any help.
>
> it is commonly accepted that godel is talking about true statements in his
> theorem
>
It is not commonly accepted that Gödel makes any mention of the notion
of truth in the statement of Proposition VI by people who have
actually read the paper. And there's a very good reason for that.
> the syntactic version
> quotehttp://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_theorems#Fir...
>
> In order to prove the first incompleteness theorem, Gödel represented
> statements by numbers ....
> So, assuming the theory is consistent (as done in the theorem's
> hypothesis) there is no such number, and the Gödel statement is true
>
You have not given us the syntactic version of Gödel's incompleteness
theorem here. I've never suggested this to you before, but some people
might have the crazy idea that the place to look for a statement of
that is in Gödel's paper, not Wikipedia.
> now rupert says
> Gödel did make the point that, for example, the Gödel sentence for P is
> equivalent (in quite a weak theory) to a sentence which may be said
> to assert that P is consistent. Hence (informally) he made the point that,
> if we are convinced that P is consistent, then we ought to also be
> convinced that the Gödel sentence is true. However, this was not a part
> of the argument which was done with mathematical precision at the time of
> the writing of the pape
>
> so rupert still admit godel talks about truth-even though accoroding to
> rupert this was not a part of the argument which was done with
> mathematical precision at the time of the writing of the paper
>
Yes, this is correct. I grant that Gödel talks about truth in an
informal way in his paper. I contend that in the part of the argument
which is intended to be mathematically precise, he makes no mention of
truth. Obviously evaluating this claim of mine would involve actually
reading the paper. Did I mention it might be a good idea to read the
paper?
> stiff shit if he could not give precision
He did. The mathematically precise part of his argument is just that;
perfectly precise.
> -fact is he used a notion of
> truth-which you have said he had no idea of what truth is
>
Well, see, there's a problem here.
In the mathematically precise part of the argument, he doesn't use a
notion of truth. I've never pointed this out to you before, of course.
Also, I've never suggested this utterly crazy idea, but has it
occurred to you that if you want to know what Gödel did in 1931 you
might want to consider reading the paper?
You see, there are limitations to what you can achieve if you rely on
soundbites from Wikipedia. I've never made this point before, of
course.
> rupert says
>
> You go and look up the translation of Gödel's paper which is available
> on-line for free. Have a look at the statement of Proposition VI. It makes
> no mention of truth
>
> proposition v1 says
>
> http://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_theorems#Fir...
>
> The general result about the existence of undecidable propositions reads
> as follows:
>
> "Theorem VI. For every ω-consistent recursive class κ of FORMULAS
> there are recursive CLASS SIGNS r, such that neither v Gen r nor Neg(v Gen
> r) belongs to Flg(κ) (where v is the FREE VARIABLE of r).2 (van Heijenoort
> translation and typsetting 1967:607. "Flg" is from "Folgerungsmenge = set
> of consequences" and "Gen" is from "Generalisation = generalization" (cf
> Meltzer and Braithwaite 1962, 1992 edition:33-34)
>
Yes. Good. Very good.
> to which the commonaly accepted interpretation interpretation of that is
>
> quote
>
> Gödel's first incompleteness theorem, perhaps the single most celebrated
> result in mathematical logic, states that:
>
> For any consistent formal, recursively enumerable theory that proves
> basic arithmetical truths, an arithmetical statement that is true, but not
> provable in the theory, can be constructed.1 That is, any effectively
> generated theory capable of expressing elementary arithmetic cannot be
> both consistent and complete.
>
Er, no, actually that happens to be wrong.
You've just stated a completely different theorem to Proposition VI
which you gave above. Proposition VI is the syntactic version. You've
just given the semantic version. Of course I've never once made any
attempt to point out to you that these two are different. I'm sorry
I've been being so unhelpful.
No, seriously, the two are different.
Would you like me to have a go at explaining the difference?
> that godel is talking about true statments as distinct from proven
> statements accounts for the claim that godel destroyed the hibert idea
> that true is proven from axioms
>
The work done in 1931 certainly does problematize any attempt to
identify truth with provability. That was more of a philosophical
point back in 1931. Later, in 1933, Tarski came up with a
mathematically precise definition of truth and was able to state and
prove as a precise mathematical theorem that truth is not the same as
provability. This was now a mathematical theorem, as opposed to a
philosophical point. Tarski's argument for this conclusion relied
heavily on Gödel's work.
> quotehttp://en.wikipedia.org/wiki/Truth#Truth_in_mathematics
> In addition, from at least the time of Hilbert's program at the turn of
> the twentieth century to the proof of Gödel's theorem and the development
> of the Church-Turing thesis in the early part of that century, true
> statements in mathematics were generally assumed to be those statements
> which are provable in a formal axiomatic system.
>
> The works of Kurt Gödel, Alan Turing, and others shook this assumption,
> with the development of statements that are true but cannot be proven
> within the system.
>
> --