practical application of Godel's Incompleteness Theorem
slartibartfast
other than the Godel statenent for a theory T,-
that is provable in T+G(T) but not provable in T?
would the statement:
S: "G(T) is true but unprovable in T AND 1+1=2"
qualify as such a statment?
(If there does not exist such an S, then since the only thing that T+G
(T) would prove in addition to what T proves would be just G(T)
itself !)
what practical application has the godel result?
Contrariwise, if there DOES exist such an S,
Can you give me an example of such a S,T and G?
david petry
> what practical application has the godel result?
None.
Nam Nguyen
> Is there any statement S, -
>
> other than the Godel statenent for a theory T,-
>
> that is provable in T+G(T) but not provable in T?
Yes, trivially so. Here is one example of such S:
S = G(T) \/ G(T+G(T))
>
> would the statement:
> S: "G(T) is true but unprovable in T AND 1+1=2"
> qualify as such a statment?
>
> (If there does not exist such an S, then since the only thing that T+G
> (T) would prove in addition to what T proves would be just G(T)
> itself !)
>
> what practical application has the godel result?
As the other poster said: "None". The way it looks, there's a good
chance it's "Never".
>
> Contrariwise, if there DOES exist such an S,
>
> Can you give me an example of such a S,T and G?
As the above mentioned S.
--
"To discover the proper approach to mathematical logic,
we must therefore examine the methods of the mathematician."
(Shoenfield, "Mathematical Logic")
Jan Burse
> what practical application has the godel result?
A lot. Starting with coding/arithmetic stuff,
continuing with recursion/complexity stuff, and
further some proof stuff.
And last but not least, giving brain food to
people from all kinds of adjacent areas. Even
those that are rather not attracted.
Bye
Peter_Smith
> Is there any statement S, -
>
> other than the Godel statenent for a theory T,-
>
> that is provable in T+G(T) but not provable in T?
>
> would the statement:
On standard assumptions, T doesn't prove Con(T), but T + G(T) proves
Con(T).
> what practical application has the godel result?
What counts as a "practical application"??
MoeBlee
The result lets us know that we'd be wasting our time trying to find
an algorithm to determine whether any given formula of arithmetic is
true (by 'true', I mean true in the standard model). Then that
suggests that instead we look for algorithms for only subsets of the
set of formulas. In particular, a subsequent result shows up for even
just Diophantine equations.
Also, the methods used in the incompleteness proof led to the very
invention of the branch of mathematics that is recursion theory and
theory of computability. I'm not prepared to document particular
cases, but I can imagine that concepts from this field have
contributed to actual computing practices, designs, and inventions.
Moreover, as I understand (and I don't mind being corrected if my
understanding is not correct) one of the outstanding mathematical
problems with one of the biggest prizes is P/NP and that the prize is
motivated by businessnes that wish to know how best to allocate their
research for developing algorithms pertaining to their products and
services; and that the P/NP problem arose in the context of this
subject of computability that was engendered by Godel's methods and
those of other logicians, from the 1930s to today.
MoeBlee
Nam Nguyen
> On Feb 3, 8:09 pm, david petry <david_lawrence_pe...@yahoo.com> wrote:
>> On Feb 3, 3:54 pm, slartibartfast <tomokane2...@yahoo.ie> wrote:
>>
>>> what practical application has the godel result?
>> None.
>
> The result lets us know that we'd be wasting our time trying to find
> an algorithm to determine whether any given formula of arithmetic is
> true (by 'true', I mean true in the standard model). Then that
> suggests that instead we look for algorithms for only subsets of the
> set of formulas. In particular, a subsequent result shows up for even
> just Diophantine equations.
Whatever. The fact remains Godel's work assumes we knew what we meant by
"the standard [arithmetic] model", which we actually don't. (E.g. Is GC
true in that model?)
>
> Also, the methods used in the incompleteness proof led to the very
> invention of the branch of mathematics that is recursion theory and
> theory of computability.
You seem to mix up and answer and the question. The result and the methods
to get the results are *not* the same. If One use many good methods to build
a road that leads to no where. How useful and practical is the built road?
Jesse F. Hughes
> Whatever. The fact remains Godel's work assumes we knew what we meant by
> "the standard [arithmetic] model", which we actually don't. (E.g. Is GC
> true in that model?)
Right. For exactly the same reason, I don't know what is meant by
"Barack Obama" (e.g., is the statement "Obama has an ingrown toenail"
true?)
--
"There are people [...] who think it's socially acceptable to level
accusations of mental illness in insulting exchanges to make
points[...] [They] are rather sick [them]selves, and in reality, are
sociopathic." --- James Harris, evidently a self-described sociopath
MoeBlee
> MoeBlee wrote:
> > On Feb 3, 8:09 pm, david petry <david_lawrence_pe...@yahoo.com> wrote:
> >> On Feb 3, 3:54 pm, slartibartfast <tomokane2...@yahoo.ie> wrote:
>
> >>> what practical application has the godel result?
> >> None.
>
> > The result lets us know that we'd be wasting our time trying to find
> > an algorithm to determine whether any given formula of arithmetic is
> > true (by 'true', I mean true in the standard model). Then that
> > suggests that instead we look for algorithms for only subsets of the
> > set of formulas. In particular, a subsequent result shows up for even
> > just Diophantine equations.
>
> Whatever. The fact remains Godel's work assumes we knew what we meant by
> "the standard [arithmetic] model", which we actually don't. (E.g. Is GC
> true in that model?)
If I am not mistaken, Godel's original proofs require defining 'the
standard model of PA'. However, once we have defined 'the standard
model of PA', then we can apply Godel-Rosser and Church-Turing thesis
to see that there is no algorithm for determing whether a given
formula is a member of the theory of the standard model of PA.
And, in, e.g., formal Z set theory, we do have a formal definition of
'the standard model of PA'. That we don't know whether GC is true in
that model reflects the very remark I started with: There is no
algorithm for determining whether a given formula is true in the
standard model of PA (i.e., whether a given formula is a member of the
theory of the standard model of PA).
> > Also, the methods used in the incompleteness proof led to the very
> > invention of the branch of mathematics that is recursion theory and
> > theory of computability.
>
> You seem to mix up and answer and the question. The result and the methods
> to get the results are *not* the same.
Of course I know that the result and the methods used to prove the
result are different things, and I take it for granted that that is
clear to any knowledgable person reading my remarks. Clearly, yes, I
am going beyond the bounds of the original question as to the theorem
itself as I am going on to remark about a related matter: the methods
and context created by the proof of the theorem.
MoeBlee
Aatu Koskensilta
> If I am not mistaken, Godel's original proofs require defining 'the
> standard model of PA'.
You are mistaken. No such definition is either required or present in
Gödel's original proof. The proof of course makes use of the notion of
a natural number, being in this respect pretty much like any proof in
mathematics.
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechen kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
MoeBlee
> MoeBlee <jazzm...@hotmail.com> writes:
> > If I am not mistaken, Godel's original proofs require defining 'the
> > standard model of PA'.
>
> You are mistaken.
No, typo of omission. I meant:
Godel's original proofs do NOT require defining 'the standard model of
PA'.
I was rebutting Nam Nguyen, not agreeing with him.
Reading the paper (as translation provided in the 'Frege To Godel'
book should make that clear.
We are in complete accord modulo my typo of omission.
MoeBlee
Nam Nguyen
> Nam Nguyen <namduc...@shaw.ca> writes:
>
>> Whatever. The fact remains Godel's work assumes we knew what we meant by
>> "the standard [arithmetic] model", which we actually don't. (E.g. Is GC
>> true in that model?)
>
> Right. For exactly the same reason, I don't know what is meant by
> "Barack Obama" (e.g., is the statement "Obama has an ingrown toenail"
> true?)
>
Would "Obama has an ingrown toenail"'s being true or false have
anything to do with truth or falsehood of "The US economy is very
bad"? Unless one works for a tabloid, the answer would be a "No".
Your analogy is incorrect in that GC and G(PA) [or CON(PA)] are bound
tightly together in truth/falsehood, in computability, in the rules
of inference, in definition of models, ... (to name a few), while
the statements about Obama's toenail and US economy aren't bound by
any reasoning framework, and one is free to assign any truth value
to them all, at will! [It might be of your interest to note that
in his "Gödel Theorem" TF spent some effort defining "Goldbach-like"
statements, in conjunction the Incompleteness!].
To see the relationship between the truth *value* of, say, [the encoded]
G(T) and that of GC one could notice that both would depend on the
existence of *any infinite subset of the primes*. It's of course obvious
why this is the case for GC; but it's a simple observation it's also
the case for G(T): although finite, the number of non-logical symbols of
T could be arbitrarily large (because T is general enough), so an infinite
number of primes is needed.
Now let cGC = "There exist infinitely many counter examples of GC" be
a 1st order formula in L(Q). The long and short of it is:
- If GC is true the so would ~cGC; but if cGC is true, so would ~GC.
- If GC is true, it'd yield an infinite subset of primes (say S1)
for the encoding.
- But if cGC is true it'd yield a *different* infinite subset of primes
(say S2).
So the existence of S1 would exclude S2, and vice versa. And so the
question is: which of S1 and S2 would lead one to the "useful" conclusion
that G(T) is true?
*****
In brief, I'd suggest to you (and Aatu, Moeblee, et al.) to think twice,
before thinking that we know enough about the natural numbers to prove
G(T) is true, or that GC's truth value has no relationship to Godel's
results.
Recall that TF mentioned in "Gödel Theorem" that "Every statement of the
form 'The Diophantine equation D(x1,..,xn)=0 has no solution in non-negative
integers' is a Goldbach-like statement". And of course GC is a Goldbach-like
statement
Nam Nguyen
> MoeBlee <jazz...@hotmail.com> writes:
>
>> If I am not mistaken, Godel's original proofs require defining 'the
>> standard model of PA'.
>
> You are mistaken. No such definition is either required or present in
> Gödel's original proof. The proof of course makes use of the notion of
> a natural number, being in this respect pretty much like any proof in
> mathematics.
First of all, had Godel not "made use" the notion of the naturals, would
he have been able to prove GIT at all? (And isn't it true that we'd typically
regards the naturals as the standard model of arithmetics, or of PA?)
Secondly from the theory {AxP(x)} one can have a *syntactical* proof that
ExP(x). So your notion that "any" proof would make use the notion of the
naturals is wrong.
Nam Nguyen
> Jesse F. Hughes wrote:
>> Nam Nguyen <namduc...@shaw.ca> writes:
>>
>>> Whatever. The fact remains Godel's work assumes we knew what we meant by
>>> "the standard [arithmetic] model", which we actually don't. (E.g. Is GC
>>> true in that model?)
>>
>> Right. For exactly the same reason, I don't know what is meant by
>> "Barack Obama" (e.g., is the statement "Obama has an ingrown toenail"
>> true?)
>>
>
> Would "Obama has an ingrown toenail"'s being true or false have
> anything to do with truth or falsehood of "The US economy is very
> bad"? Unless one works for a tabloid, the answer would be a "No".
>
> Your analogy is incorrect in that GC and G(PA) [or CON(PA)] are bound
> tightly together in truth/falsehood, in computability, in the rules
> of inference, in definition of models, ... (to name a few), while
> the statements about Obama's toenail and US economy aren't bound by
> any reasoning framework, and one is free to assign any truth value
> to them all, at will! [It might be of your interest to note that
> in his "Gödel Theorem" TF spent some effort defining "Goldbach-like"
> statements, in conjunction the Incompleteness!].
>
> To see the relationship between the truth *value* of, say, [the encoded]
> G(T) and that of GC one could notice that both would depend on the
> existence of *any infinite subset of the primes*.
A typo: it should have been "*an infinite subset of the primes*".
> It's of course obvious
> why this is the case for GC; but it's a simple observation it's also
> the case for G(T): although finite, the number of non-logical symbols of
> T could be arbitrarily large (because T is general enough), so an infinite
> number of primes is needed.
>
> Now let cGC = "There exist infinitely many counter examples of GC" be
> a 1st order formula in L(Q). The long and short of it is:
>
> - If GC is true the so would ~cGC; but if cGC is true, so would ~GC.
>
> - If GC is true, it'd yield an infinite subset of primes (say S1)
> for the encoding.
>
> - But if cGC is true it'd yield a *different* infinite subset of primes
> (say S2).
>
> So the existence of S1 would exclude S2, and vice versa. And so the
> question is: which of S1 and S2 would lead one to the "useful" conclusion
> that G(T) is true?
>
> *****
>
> In brief, I'd suggest to you (and Aatu, Moeblee, et al.) to think twice,
> before thinking that we know enough about the natural numbers to prove
> G(T) is true, or that GC's truth value has no relationship to Godel's
> results.
>
> Recall that TF mentioned in "Gödel Theorem" that "Every statement of the
> form 'The Diophantine equation D(x1,..,xn)=0 has no solution in non-
> negative integers' is a Goldbach-like statement". And of course GC is a
> Goldbach-like statement.
I'd like to borrow this opportunity to add a few more comments on something
related to Godel's work and results. And that something is the "meaning" of
CON(T), which iirc has also be debated in the past.
Post Godel FOL has been so
Nam Nguyen
Sorry that I accidentally pushed the "send" button prematurely. I'll continue
this post when I could.
MoeBlee
> Aatu Koskensilta wrote:
> > MoeBlee <jazzm...@hotmail.com> writes:
>
> >> If I am not mistaken, Godel's original proofs require defining 'the
> >> standard model of PA'.
For the record, that was a typo of omission. I meant to say:
Godel's original proofs do NOT require definining 'the standard model
of PA'.
> > You are mistaken. No such definition is either required or present in
> > Gödel's original proof. The proof of course makes use of the notion of
> > a natural number, being in this respect pretty much like any proof in
> > mathematics.
>
> First of all, had Godel not "made use" the notion of the naturals, would
> he have been able to prove GIT at all?
I don't see how he could have. So?
> (And isn't it true that we'd typically
> regards the naturals as the standard model of arithmetics, or of PA?)
A model of PA is requires not just a set, but rather, a set, a
particular member of the set, a unary operation on the set, and two
binary operations on the set. The standard model is a mapping from the
non-logical symbols of language of first order PA plus to the
universal quantifier, so that the universal quantifier maps to the set
of natural numbers, the symbol '0' to the natural number 0, the symbol
'S' to the successor operation on natural numbers, and the symbols '+'
and '*' to the operations of addition and multiplication on natural
numbers, respectively.
> Secondly from the theory {AxP(x)} one can have a *syntactical* proof that
> ExP(x). So your notion that "any" proof would make use the notion of the
> naturals is wrong.
The part after 'So' follows how?
MoeBlee
MoeBlee
>
> >> Whatever. The fact remains Godel's work assumes we knew what we meant by
> >> "the standard [arithmetic] model", which we actually don't. (E.g. Is GC
> >> true in that model?)
>
> > Right. For exactly the same reason, I don't know what is meant by
> > "Barack Obama" (e.g., is the statement "Obama has an ingrown toenail"
> > true?)
>
> Would "Obama has an ingrown toenail"'s being true or false have
> anything to do with truth or falsehood of "The US economy is very
> bad"? Unless one works for a tabloid, the answer would be a "No".
>
> Your analogy is incorrect in that GC and G(PA) [or CON(PA)] are bound
> tightly together in truth/falsehood, in computability, in the rules
> of inference, in definition of models, ... (to name a few), while
> the statements about Obama's toenail and US economy aren't bound by
> any reasoning framework, and one is free to assign any truth value
> to them all, at will! [It might be of your interest to note that
> in his "Gödel Theorem" TF spent some effort defining "Goldbach-like"
> statements, in conjunction the Incompleteness!].
Jesse's just saying that we don't have to know every attribute of an
object just to identity the object. We can identify the standard model
of PA while not knowing whether it has the attribute of GC being true
in it.
> To see the relationship between the truth *value* of, say, [the encoded]
> G(T) and that of GC one could notice that both would depend on the
> existence of *any infinite subset of the primes*. It's of course obvious
> why this is the case for GC; but it's a simple observation it's also
> the case for G(T): although finite, the number of non-logical symbols of
> T could be arbitrarily large (because T is general enough), so an infinite
> number of primes is needed.
>
> Now let cGC = "There exist infinitely many counter examples of GC" be
> a 1st order formula in L(Q). The long and short of it is:
>
> - If GC is true the so would ~cGC; but if cGC is true, so would ~GC.
>
> - If GC is true, it'd yield an infinite subset of primes (say S1)
> for the encoding.
>
> - But if cGC is true it'd yield a *different* infinite subset of primes
> (say S2).
>
> So the existence of S1 would exclude S2, and vice versa. And so the
> question is: which of S1 and S2 would lead one to the "useful" conclusion
> that G(T) is true?
>
> *****
>
> In brief, I'd suggest to you (and Aatu, Moeblee, et al.) to think twice,
> before thinking that we know enough about the natural numbers to prove
> G(T) is true, or that GC's truth value has no relationship to Godel's
> results.
I might have missed it, but I don't know what you mean by 'G(T)'. And
I didn't say that the truth value of GC has no "relationship" with
Godel's results, whatever you mean by "relationship".
> Recall that TF mentioned in "Gödel Theorem" that "Every statement of the
> form 'The Diophantine equation D(x1,..,xn)=0 has no solution in non-negative
> integers' is a Goldbach-like statement". And of course GC is a Goldbach-like
> statement
MoeBlee
Nam Nguyen
> On Feb 7, 4:36 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
>> Aatu Koskensilta wrote:
>>> MoeBlee <jazzm...@hotmail.com> writes:
>>>> If I am not mistaken, Godel's original proofs require defining 'the
>>>> standard model of PA'.
>
> For the record, that was a typo of omission. I meant to say:
>
> Godel's original proofs do NOT require definining 'the standard model
> of PA'.
>
>>> You are mistaken. No such definition is either required or present in
>>> Gödel's original proof. The proof of course makes use of the notion of
>>> a natural number, being in this respect pretty much like any proof in
>>> mathematics.
>> First of all, had Godel not "made use" the notion of the naturals, would
>> he have been able to prove GIT at all?
>
> I don't see how he could have. So?
So the fact he couldn't have plus a correct to my question below in the
parenthesis would mean you and Aatu were wrong in believing that Gödel's
original proofs don't require the standard model of PA, which is the
natural numbers.
>
>> (And isn't it true that we'd typically
>> regards the naturals as the standard model of arithmetics, or of PA?)
Isn't-it-true here means the answer is either a Yes or No. Whatever
you said below is not an answer to the question.
>
> A model of PA is requires not just a set, but rather, a set, a
> particular member of the set, a unary operation on the set, and two
> binary operations on the set. The standard model is a mapping from the
> non-logical symbols of language of first order PA plus to the
> universal quantifier, so that the universal quantifier maps to the set
> of natural numbers, the symbol '0' to the natural number 0, the symbol
> 'S' to the successor operation on natural numbers, and the symbols '+'
> and '*' to the operations of addition and multiplication on natural
> numbers, respectively.
So, answer the question and you'd understand why you didn't need to ask
"So?" above.
>
>> Secondly from the theory {AxP(x)} one can have a *syntactical* proof that
>> ExP(x). So your notion that "any" proof would make use the notion of the
>> naturals is wrong.
>
> The part after 'So' follows how?
So my *extremely simple* example here counters his statement that
*any proof* would make use the notions of the naturals (which doesn't
seem to take one a lot to understand from what he said and from my
example!)
Nam Nguyen
> On Feb 7, 11:11 am, Nam Nguyen <namducngu...@shaw.ca> wrote:
>> Jesse F. Hughes wrote:
>>> Nam Nguyen <namducngu...@shaw.ca> writes:
>>>> Whatever. The fact remains Godel's work assumes we knew what we meant by
>>>> "the standard [arithmetic] model", which we actually don't. (E.g. Is GC
>>>> true in that model?)
>>> Right. For exactly the same reason, I don't know what is meant by
>>> "Barack Obama" (e.g., is the statement "Obama has an ingrown toenail"
>>> true?)
>> Would "Obama has an ingrown toenail"'s being true or false have
>> anything to do with truth or falsehood of "The US economy is very
>> bad"? Unless one works for a tabloid, the answer would be a "No".
>>
>> Your analogy is incorrect in that GC and G(PA) [or CON(PA)] are bound
>> tightly together in truth/falsehood, in computability, in the rules
>> of inference, in definition of models, ... (to name a few), while
>> the statements about Obama's toenail and US economy aren't bound by
>> any reasoning framework, and one is free to assign any truth value
>> to them all, at will! [It might be of your interest to note that
>> in his "Gödel Theorem" TF spent some effort defining "Goldbach-like"
>> statements, in conjunction the Incompleteness!].
>
> Jesse's just saying that we don't have to know every attribute of an
> object just to identity the object.
Say, you've identified a 50-mile radius meteor hurling toward the Sun,
by its *precise* brightness and location on the sky-map. Don't you think
you'd like to know if it would slam into the Earth within 6 months?
> We can identify the standard model
> of PA while not knowing whether it has the attribute of GC being true
> in it.
So, let me ask you this question: would both GC and cGC be "absolute
undecidable" formulas?
Nam Nguyen
> Jesse's just saying that we don't have to know every attribute of an
> object just to identity the object. We can identify the standard model
> of PA while not knowing whether it has the attribute of GC being true
> in it.
So, why do we care about the attributes such as G(T), CON(T) being true?
(For the record, I've said we have to know "every attribute" of the
standards models of PA; it's only a handful of them I think are important
to the foundation of FOL reasoning and we'd need to know!)
Nam Nguyen
> MoeBlee wrote:
>
>> Jesse's just saying that we don't have to know every attribute of an
>> object just to identity the object. We can identify the standard model
>> of PA while not knowing whether it has the attribute of GC being true
>> in it.
>
> So, why do we care about the attributes such as G(T), CON(T) being true?
> (For the record, I've said we have to know "every attribute" of the
I meant to say "I've never said ..."
MoeBlee
> MoeBlee wrote:
> > On Feb 7, 4:36 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
> >> Aatu Koskensilta wrote:
> >>> MoeBlee <jazzm...@hotmail.com> writes:
> >>>> If I am not mistaken, Godel's original proofs require defining 'the
> >>>> standard model of PA'.
>
> > For the record, that was a typo of omission. I meant to say:
>
> > Godel's original proofs do NOT require definining 'the standard model
> > of PA'.
>
> >>> You are mistaken. No such definition is either required or present in
> >>> Gödel's original proof. The proof of course makes use of the notion of
> >>> a natural number, being in this respect pretty much like any proof in
> >>> mathematics.
> >> First of all, had Godel not "made use" the notion of the naturals, would
> >> he have been able to prove GIT at all?
>
> > I don't see how he could have. So?
>
> So the fact he couldn't have plus a correct to my question below in the
> parenthesis would mean you and Aatu were wrong in believing that Gödel's
> original proofs don't require the standard model of PA, which is the
> natural numbers.
No. People for centuries worked with natural numbers while never even
hearing of such a thing as "the standard model of PA".
And "the natural numbers" aren't the standard model of PA, except
quite loosely speaking, though the set of natural numbers is the
universe of the standard model of PA.
> >> (And isn't it true that we'd typically
> >> regards the naturals as the standard model of arithmetics, or of PA?)
>
> Isn't-it-true here means the answer is either a Yes or No. Whatever
> you said below is not an answer to the question.
Sorry, I don't submit to interrogations here. If your question needs
to be qualified by reiterating the defintions of the terminology used,
then I may elect to do that.
> > A model of PA is requires not just a set, but rather, a set, a
> > particular member of the set, a unary operation on the set, and two
> > binary operations on the set. The standard model is a mapping from the
> > non-logical symbols of language of first order PA plus to the
> > universal quantifier, so that the universal quantifier maps to the set
> > of natural numbers, the symbol '0' to the natural number 0, the symbol
> > 'S' to the successor operation on natural numbers, and the symbols '+'
> > and '*' to the operations of addition and multiplication on natural
> > numbers, respectively.
>
> So, answer the question and you'd understand why you didn't need to ask
> "So?" above.
The answer is implicit in my response. No, the natural numbers are not
the standard model of PA. I just told you what the standard model of
PA is, and it regards not just "the naturals" but also certain
operations on them, etc., plus, more technically, a particular formal
language. Godel's proof is addressed to a particular formal theory P,
but no use is made of the notion of the standard model of PA.
> >> Secondly from the theory {AxP(x)} one can have a *syntactical* proof that
> >> ExP(x). So your notion that "any" proof would make use the notion of the
> >> naturals is wrong.
>
> > The part after 'So' follows how?
>
> So my *extremely simple* example here counters his statement that
> *any proof* would make use the notions of the naturals (which doesn't
> seem to take one a lot to understand from what he said and from my
> example!)
The theorem concerns natural numbers. So, of course, notions about
natural numbers will come in to the proof.
MoeBlee
MoeBlee
I'd like to know a lot of things. That I don't know every particular
thing about an object doesn't entail that I can't identity the
object.
> > We can identify the standard model
> > of PA while not knowing whether it has the attribute of GC being true
> > in it.
>
> So, let me ask you this question: would both GC and cGC be "absolute
> undecidable" formulas?
In this context, I know the definition of 'undecidable from a set of
formulas' (given a certain language, etc.). I don't know whether GC is
decidable from the set of formulas that is the set of axioms of PA. As
to "absolute undecidable", you'd have to give me your definition. But
please don't do that; Answering your questions, one after another
after another, is, as usual, not a very rewarding use of my time.
MoeBlee
MoeBlee
> MoeBlee wrote:
> > Jesse's just saying that we don't have to know every attribute of an
> > object just to identity the object. We can identify the standard model
> > of PA while not knowing whether it has the attribute of GC being true
> > in it.
>
> So, why do we care about the attributes such as G(T), CON(T) being true?
> (For the record, I've said we have to know "every attribute" of the
> standards models of PA; it's only a handful of them I think are important
> to the foundation of FOL reasoning and we'd need to know!)
Different people have different reasons for wanting to know various
things. Knowing that the Godel sentence of PA is is true in the
standard model of PA is interesting (at least to me) since then we
know that there are sentences that are true in the standard model of
PA but not provable from the axioms of PA.
That doesn't in the least bit disqualify my remark that we don't have
to know every attribute of an object just to identify the object; and,
in particular, we don't have whether GC is true in the standard model
of PA just to know what the standard model of PA is.
As to your notion about a handful of attributes important to the
foundation of first order reasoning, I've not made any comment.
Whatever is or is not important to the foundation of first order
reasoning (and important for whom), the standard model of PA has a
precise mathematical definition and I well understand what the
standard model of PA is even though I don't know whether GC is true in
the standard model of PA.
This has been made clear now several times. It is getting quite boring
now going over and over it with you.
MoeBlee
Nam Nguyen
> On Feb 9, 10:58 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
>> MoeBlee wrote:
>>> Jesse's just saying that we don't have to know every attribute of an
>>> object just to identity the object. We can identify the standard model
>>> of PA while not knowing whether it has the attribute of GC being true
>>> in it.
>> So, why do we care about the attributes such as G(T), CON(T) being true?
>> (For the record, I've said we have to know "every attribute" of the
>> standards models of PA; it's only a handful of them I think are important
>> to the foundation of FOL reasoning and we'd need to know!)
>
> Different people have different reasons for wanting to know various
> things. Knowing that the Godel sentence of PA is is true in the
> standard model of PA is interesting (at least to me) since then we
> know that there are sentences that are true in the standard model of
> PA but not provable from the axioms of PA.
It was explained to you before: not knowing the truth/falsehood of GC
or cGC would mean you don't know if the encoded G(T), hence the encoded
G(PA), is true or false. So, on the account of G(PA), you don't know of
any true sentence in the so called "the standard model of PA".
>
> That doesn't in the least bit disqualify my remark that we don't have
> to know every attribute of an object just to identify the object;
It does disqualify your remark. With unaided eyes, say, you look
up the sky and identify the "single" star Alpha Centauri with one of
its precise attributes (e.g. location). But is it Alpha Centauri A,
or Alpha Centauri B, (a different star) that you're looking at?
> and,
> in particular, we don't have whether GC is true in the standard model
> of PA just to know what the standard model of PA is.
In particular, it's the same argument as above.
> As to your notion about a handful of attributes important to the
> foundation of first order reasoning, I've not made any comment.
Because, e.g., you don't understand issues at foundation of FOL enough
to make any comment?
> Whatever is or is not important to the foundation of first order
> reasoning (and important for whom), the standard model of PA has a
> precise mathematical definition and I well understand what the
> standard model of PA is even though I don't know whether GC is true in
> the standard model of PA.
According to Shoenfield, the "natural numbers" is a model of PA. So,
unless you could refer to it as a non-standard model of PA, then
the standard model of PA is the natural numbers.
And no precise definition could capture all the properties of the
naturals. So you don't know precisely what the standard of model
of PA is. (Though you do know precisely what you name as "the standard
of model of PA"; but that's not the same story.)
>
> This has been made clear now several times.
To you that has been made clear to you many times, but you have either
refused or been unable to understand.
> It is getting quite boring ow going over and over it with you.
Setting aside the fact you jumped into my conversations with Aatu
or Jesse _uninvited_, your getting bored over whatsoever isn't at
all of my interest or concern.
MoeBlee
> MoeBlee wrote:
> > On Feb 9, 10:58 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
> >> MoeBlee wrote:
> >>> Jesse's just saying that we don't have to know every attribute of an
> >>> object just to identity the object. We can identify the standard model
> >>> of PA while not knowing whether it has the attribute of GC being true
> >>> in it.
> >> So, why do we care about the attributes such as G(T), CON(T) being true?
> >> (For the record, I've said we have to know "every attribute" of the
> >> standards models of PA; it's only a handful of them I think are important
> >> to the foundation of FOL reasoning and we'd need to know!)
>
> > Different people have different reasons for wanting to know various
> > things. Knowing that the Godel sentence of PA is is true in the
> > standard model of PA is interesting (at least to me) since then we
> > know that there are sentences that are true in the standard model of
> > PA but not provable from the axioms of PA.
>
> It was explained to you before: not knowing the truth/falsehood of GC
> or cGC would mean you don't know if the encoded G(T), hence the encoded
> G(PA), is true or false. So, on the account of G(PA), you don't know of
> any true sentence in the so called "the standard model of PA".
Whatever specific sentences of whatever specific language you mean by
'G(T)' and 'G(PA)', I do know what the standard model for the langauge
of PA is. It is exactly defined. The fact that there are sentences
that I don't know whether to be true in that model doesn't entail that
I don't know what that model is. And that the set of sentences that
are true in the standard model for the language of PA is not decidable
doesn't entail that the set is not defined. That the set is defined is
apparent from just seeing the DEFINITION itself.
> > That doesn't in the least bit disqualify my remark that we don't have
> > to know every attribute of an object just to identify the object;
>
> It does disqualify your remark. With unaided eyes, say, you look
> up the sky and identify the "single" star Alpha Centauri with one of
> its precise attributes (e.g. location). But is it Alpha Centauri A,
> or Alpha Centauri B, (a different star) that you're looking at?
The definition of 'the standard model for the language of PA' has a
precise definition that does not depend on sensory identification.
> > and,
> > in particular, we don't have whether GC is true in the standard model
> > of PA just to know what the standard model of PA is.
>
> In particular, it's the same argument as above.
Which is irrelvent.
> > As to your notion about a handful of attributes important to the
> > foundation of first order reasoning, I've not made any comment.
>
> Because, e.g., you don't understand issues at foundation of FOL enough
> to make any comment?
Oh, I forgot, you have a privileged understanding.
> > Whatever is or is not important to the foundation of first order
> > reasoning (and important for whom), the standard model of PA has a
> > precise mathematical definition and I well understand what the
> > standard model of PA is even though I don't know whether GC is true in
> > the standard model of PA.
>
> According to Shoenfield, the "natural numbers" is a model of PA. So,
> unless you could refer to it as a non-standard model of PA, then
> the standard model of PA is the natural numbers.
You simply skipped what I said.
> And no precise definition could capture all the properties of the
> naturals.
I didn't say anything about "all properties of the naturals". Rather,
the standard model is a certain function that has a simple definition.
> So you don't know precisely what the standard of model
> of PA is. (Though you do know precisely what you name as "the standard
> of model of PA"; but that's not the same story.)
Sure I do, it's the exact function I've mentioned already.
> > This has been made clear now several times.
>
> To you that has been made clear to you many times, but you have either
> refused or been unable to understand.
Sorry, I keep forgetting that your privileged misunderstandings are
determinative.
> > It is getting quite boring ow going over and over it with you.
>
> Setting aside the fact you jumped into my conversations with Aatu
> or Jesse _uninvited_, your getting bored over whatsoever isn't at
> all of my interest or concern.
No one needs an "invitation" to post. For that matter, YOU "jumped in
_uninvited_ in response to my post in response to david petry". You
have bizarre notions not just of logic and mathematics but of forum
interaction also. And no matter whether of your interest or concern,
it is of MY interest to mention.
MoeBlee
Nam Nguyen
> On Feb 15, 12:13 am, Nam Nguyen <namducngu...@shaw.ca> wrote:
>> MoeBlee wrote:
>>> On Feb 9, 10:58 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
>>>> MoeBlee wrote:
>>>>> Jesse's just saying that we don't have to know every attribute of an
>>>>> object just to identity the object. We can identify the standard model
>>>>> of PA while not knowing whether it has the attribute of GC being true
>>>>> in it.
>>>> So, why do we care about the attributes such as G(T), CON(T) being true?
>>>> (For the record, I've said we have to know "every attribute" of the
>>>> standards models of PA; it's only a handful of them I think are important
>>>> to the foundation of FOL reasoning and we'd need to know!)
>>> Different people have different reasons for wanting to know various
>>> things. Knowing that the Godel sentence of PA is is true in the
>>> standard model of PA is interesting (at least to me) since then we
>>> know that there are sentences that are true in the standard model of
>>> PA but not provable from the axioms of PA.
>> It was explained to you before: not knowing the truth/falsehood of GC
>> or cGC would mean you don't know if the encoded G(T), hence the encoded
>> G(PA), is true or false. So, on the account of G(PA), you don't know of
>> any true sentence in the so called "the standard model of PA".
>
> Whatever specific sentences of whatever specific language you mean by
> 'G(T)' and 'G(PA)',
This is an idiotic rambling of yours! What did you really think I meant
by G(T) or G(PA)? Hint: I asked you above "why do we care about the
attributes such as G(T), CON(T) being true?" and *your immediate response*
included "Knowing that the _Godel sentence of PA_ is true ... is interesting
(at least to me [MoeBlee])"!
> I do know what the standard model for the langauge of PA is.
> It is exactly defined. The fact that there are sentences
> that I don't know whether to be true in that model doesn't entail that
> I don't know what that model is.
I've explained it in different ways, including the relationship between
the encoded G(T)/G(PA) and GC/cGC, but you don't seem to be able to
understand it. We have been talking about Godel's results in relation
to certain *formal system* T (e.g. PA). So any model of any language
(which is simply an interpretation-structure) must be in the context
of these formal systems. By definition of a formal system T's model,
a model of a language is *not* necessarily a model of T: you have
to _demonstrate_ that such a language-model reflect all the T's theorems
as true *and* that at the same time, in meta level, T be syntactically
consistent!
Until you do that, your keeping saying you know how the language of PA
is exactly defined is irrelevant to my notion that you - or anyone else
for that matter - wouldn't know exactly what the standard model of the
*formal system* PA is, at least if you don't know whether or not GC or
cGC is true or false. For example, how would you know G(PA) isn't
(syntactically) equivalent to GC, or cGC? And you wouldn't care about
such possible equivalence?
MoeBlee
For whatever reasons, which may include my lack of recollection on the
matter, I don't know. You're free to tell me, which would facilitate
communication, though perhaps at the cost of forgiving my lack of
perfect recollection, or you an can keep going on about "idiotic
rambling", or whatever else makes you happy.
> Hint: I asked you above "why do we care about the
> attributes such as G(T), CON(T) being true?" and *your immediate response*
> included "Knowing that the _Godel sentence of PA_ is true ... is interesting
> (at least to me [MoeBlee])"!
If I recall, I didn't presume in my response that by 'T' you meant PA,
but rather, I mentioned PA specifically at least to give scope to my
OWN remark.
If by 'T' you meant 'PA' then it would be so much simpler for you just
to say, even if it were at the cost of it being a reiteration, rather
than to exercise yourself flailing against the "idiotic".
> > I do know what the standard model for the langauge of PA is.
> > It is exactly defined. The fact that there are sentences
> > that I don't know whether to be true in that model doesn't entail that
> > I don't know what that model is.
>
> I've explained it in different ways, including the relationship between
> the encoded G(T)/G(PA) and GC/cGC, but you don't seem to be able to
> understand it. We have been talking about Godel's results in relation
> to certain *formal system* T (e.g. PA). So any model of any language
> (which is simply an interpretation-structure) must be in the context
> of these formal systems. By definition of a formal system T's model,
> a model of a language is *not* necessarily a model of T:
The last thing you said is correct. A model for the language of a
theory is not necessarily a model of the theory. Yes, indeed, I've
harped on that point myself with you.
> you have
> to _demonstrate_ that such a language-model reflect all the T's theorems
> as true
Right, and we demonstrate that all the theorems of PA are true in the
standard model for the language of PA.
> *and* that at the same time, in meta level, T be syntactically
> consistent!
That PA has a model entails that PA is consistent. To prove that PA
has a model does not require at the same time proving that PA is
consistent except in the sense of redundancy: To prove that PA has a
model proves that PA is consistent. Also, I don't know what non-
redundant sense of 'syntactically consistent' you mean. We have our
definition of 'consistent'. It is syntactical. However, we also prove
that the semantical property of having a model entails the syntactical
property of consistency.
> Until you do that, your keeping saying you know how the language of PA
> is exactly defined is irrelevant to my notion that you - or anyone else
> for that matter - wouldn't know exactly what the standard model of the
> *formal system* PA is,
No, you completely misunderstand what I wrote. I didn't say just that
I know how the LANGUAGE of PA is defined (I do know that, but it's not
all I know), but rather also that I know how 'the standard model of
the language of PA' is defined.
> at least if you don't know whether or not GC or
> cGC is true or false.
The definition of 'the standard model for the language of PA' doesn't
hinge on deciding whether GC or ~GC is true in that standard model of
PA. The definition of 'X' does not require having decided everything
about, or every property of X.
> For example, how would you know G(PA) isn't
> (syntactically) equivalent to GC, or cGC?
Whether you mean logically equivalent or equivalent relative to some
axioms, I don't need to opine as to whether G(PA) and GC are
equivalent or not just to define 'the standard model for the language
of PA'. I've given you the definition. That definition does not depend
on settling such questions of equivalency as you just mentioned.
> And you wouldn't care about
> such possible equivalence?
It would be interesting for me to find that G(PA) and GC (or negations
thereof, whatever) are logically equivalent, or equivalent in PA, or
equivalent in ZFC, for example. That does not bear upon the fact that
there is a precise mathematical definition of 'the standard model of
PA'.
MoeBlee
Nam Nguyen
I didn't keep going on about that kind of "stuff" *until you initiated*
stuff like "It is getting quite boring now going over and over it with you",
while you had not responded to my key explanations, as in "As to your
notion about a handful of attributes important to the foundation of
first order reasoning, I've not made any comment".
But I'm an optimistic. Why don't we leave all that non-technical "stuff"
behind and concentrate only on technical points moving forward.
>
>> Hint: I asked you above "why do we care about the
>> attributes such as G(T), CON(T) being true?" and *your immediate response*
>> included "Knowing that the _Godel sentence of PA_ is true ... is interesting
>> (at least to me [MoeBlee])"!
>
> If I recall, I didn't presume in my response that by 'T' you meant PA,
> but rather, I mentioned PA specifically at least to give scope to my
> OWN remark.
>
> If by 'T' you meant 'PA' then it would be so much simpler for you just
> to say, even if it were at the cost of it being a reiteration, rather
> than to exercise yourself flailing against the "idiotic".
>
>>> I do know what the standard model for the langauge of PA is.
>>> It is exactly defined. The fact that there are sentences
>>> that I don't know whether to be true in that model doesn't entail that
>>> I don't know what that model is.
>> I've explained it in different ways, including the relationship between
>> the encoded G(T)/G(PA) and GC/cGC, but you don't seem to be able to
>> understand it. We have been talking about Godel's results in relation
>> to certain *formal system* T (e.g. PA). So any model of any language
>> (which is simply an interpretation-structure) must be in the context
>> of these formal systems. By definition of a formal system T's model,
>> a model of a language is *not* necessarily a model of T:
>
> The last thing you said is correct. A model for the language of a
> theory is not necessarily a model of the theory.
What about the first few things I said in the paragraph? Do you now
understand what I said about "the encoded G(T)/G(PA) and GC/cGC"?
>
>> you have to _demonstrate_ that such a language-model reflect all the
>> T's theorems as true
>
> Right, and we demonstrate that all the theorems of PA are true in the
> standard model for the language of PA.
Who among you, David, Aatu, Jesse, and I, did demonstrate that, and in what
post of the thread? Perhaps my recollection isn't that good. Would you
be able to refresh it? If not then you have *not* demonstrated so.
>
>> *and* that at the same time, in meta level, T be syntactically
>> consistent!
>
> That PA has a model entails that PA is consistent.
Again, nobody in this conversation has demonstrated PA has a model yet!
> To prove that PA
> has a model does not require at the same time proving that PA is
> consistent except in the sense of redundancy: To prove that PA has a
> model proves that PA is consistent.
But *how* do you know if PA has - or *has not* - any model, to begin
with? What criteria did you use to determine a language model is
a model of a formal system (PA in this case)?
Again, *which criteria* did you use?
> We have our definition of 'consistent'. It is syntactical.
> However, we also prove
> that the semantical property of having a model entails the syntactical
> property of consistency.
You're incorrect! How do you "prove" a _semantical_ property would
conform to a _syntactical_ definition? They're like orange and apple!
>
>> Until you do that, your keeping saying you know how the language of PA
>> is exactly defined is irrelevant to my notion that you - or anyone else
>> for that matter - wouldn't know exactly what the standard model of the
>> *formal system* PA is,
>
> No, you completely misunderstand what I wrote. I didn't say just that
> I know how the LANGUAGE of PA is defined (I do know that, but it's not
> all I know), but rather also that I know how 'the standard model of
> the language of PA' is defined.
It was a simple typo I meant "the language model of PA". (For the record,
I did use the work "language model" elsewhere in the same post).
>
>> at least if you don't know whether or not GC or
>> cGC is true or false.
>
> The definition of 'the standard model for the language of PA' doesn't
> hinge on deciding whether GC or ~GC is true in that standard model of
> PA. The definition of 'X' does not require having decided everything
> about, or every property of X.
If you don't know if either GC or ~GC is true in a *purported* model of
PA (as a formal system), then it's quite possible that PA might be actually
syntactically inconsistent! And this *purported* model of PA might just
be just a language-model *at best*, without being a PA's model.
Do you understand this?
>
>> For example, how would you know G(PA) isn't
>> (syntactically) equivalent to GC, or cGC?
>
> Whether you mean logically equivalent or equivalent relative to some
> axioms, I don't need to opine as to whether G(PA) and GC are
> equivalent or not just to define 'the standard model for the language
> of PA'.
You wouldn't opine even if G(PA) <-> GC <-> ~GC <-> cGC? The question
here is how you prove this language-model to be a PA's model?
I already stress the fact that "you - or anyone else - for that matter -
wouldn't know exactly what the standard model of the *formal system* PA is"
in the paragraph above (where I had a typo).
Do you now understand that paragraph (modulo the typo I've corrected)?
> I've given you the definition.
Again, which post in this thread/conversation did you give the definition
of the standard of the formal system PA? Perhaps shouldn't keep mentioning
thing you actually didn't say or articulate!
> That definition does not depend
> on settling such questions of equivalency as you just mentioned.
Again, I don't have any recollection of "that" definition of yours!
>
>> And you wouldn't care about
>> such possible equivalence?
>
> It would be interesting for me to find that G(PA) and GC (or negations
> thereof, whatever) are logically equivalent, or equivalent in PA, or
> equivalent in ZFC, for example.
> That does not bear upon the fact that
> there is a precise mathematical definition of 'the standard model of
> PA'.
If you don't know G(PA) <-> GC <-> ~GC <-> cGC then you don't know
what you're talking about as 'the standard model of PA'.
PHPBABY3
> slartibartfast schrieb:
>
> > what practical application has the godel result?
>
> continuing with recursion/complexity stuff, and
> further some proof stuff.
The practical mathematical result is to add one more nail in the
coffin of Hilbert's Programme to axiomatize all of Logic.
(This is beause Hilbert would require that truth, provability and
nonrefutability coincide, and any of the various proofs that two of
these sets don't suffices.)
C-B
> And last but not least, giving brain food to
> people from all kinds of adjacent areas. Even
> those that are rather not attracted.
>
> Bye
PHPBABY3
> On Feb 3, 11:54 pm, slartibartfast <tomokane2...@yahoo.ie> wrote:
>
> > Is there any statement S, -
>
> > other than the Godel statenent for a theory T,-
>
> > that is provable in T+G(T) but not provable in T?
>
> > would the statement:
>
> On standard assumptions, T doesn't prove Con(T), but T + G(T) proves
> Con(T).
>
> > what practical application has the godel result?
>
Answers existing questions.
PHPBABY3
> On Feb 4, 12:12 am, Jan Burse <janbu...@fastmail.fm> wrote:
>
> > slartibartfast schrieb:
>
> > > what practical application has the godel result?
>
> > A lot. Starting with coding/arithmetic stuff,
> > continuing with recursion/complexity stuff, and
> > further some proof stuff.
>
> The practical mathematical result is to add one more nail in the
> coffin of Hilbert's Programme to axiomatize all of Logic.
>
> (This is beause Hilbert would require that truth, provability and
> nonrefutability coincide, and any of the various proofs that two of
> these sets don't suffices.)
>
> C-B
BTW If you want practical at the highest level (physics) then each
theorem has its own lesson. E.g. Turing's Unsolvability of the
Halting Problem means that computer manufacturers should always
include a CPU restart at the power source (on/off switch).
> > And last but not least, giving brain food to
> > people from all kinds of adjacent areas. Even
> > those that are rather not attracted.
>
> > Bye- Hide quoted text -
>
> - Show quoted text -
Nam Nguyen
> On Feb 4, 12:12 am, Jan Burse <janbu...@fastmail.fm> wrote:
>> slartibartfast schrieb:
>>
>>> what practical application has the godel result?
>> A lot. Starting with coding/arithmetic stuff,
>> continuing with recursion/complexity stuff, and
>> further some proof stuff.
>
> The practical mathematical result is to add one more nail in the
> coffin of Hilbert's Programme to axiomatize all of Logic.
There's a saying that before taking revenge one should dig 2 graves.
Godel should have had 2 nails, one for his own Incompleteness.
What good is it to replace syntactical "axiomatizing _all of logic_" by
In declaring war with one devil, Godel just made with another one!
>
> (This is beause Hilbert would require that truth, provability and
> nonrefutability coincide, and any of the various proofs that two of
> these sets don't suffices.)
Hilbert's Programme has 2 major components: syntacicalization and
the one-size-catch-all PM. The later was a his mistake but the former
was (and still is) a correct component.
Nam Nguyen
> PHPBABY3 wrote:
>> On Feb 4, 12:12 am, Jan Burse <janbu...@fastmail.fm> wrote:
>>> slartibartfast schrieb:
>>>
>>>> what practical application has the godel result?
>>> A lot. Starting with coding/arithmetic stuff,
>>> continuing with recursion/complexity stuff, and
>>> further some proof stuff.
>>
>> The practical mathematical result is to add one more nail in the
>> coffin of Hilbert's Programme to axiomatize all of Logic.
>
> There's a saying that before taking revenge one should dig 2 graves.
> Godel should have had 2 nails, one for his own Incompleteness.
> What good is it to replace syntactical "axiomatizing _all of logic_" by
Sorry for a typo: I meant 'by "modeling _all of logic_"?'
MoeBlee
> MoeBlee wrote:
> > Right, and we demonstrate that all the theorems of PA are true in the
> > standard model for the language of PA.
>
> Who among you, David, Aatu, Jesse, and I, did demonstrate that, and in what
> post of the thread? Perhaps my recollection isn't that good. Would you
> be able to refresh it? If not then you have *not* demonstrated so.
I didn't say that anyone had bothered to type out the demonstration in
a Usenet post. By 'we demonstrate', I mean an ordinary sense of 'such
demonstrations can be completed by a reasonably alert student'.
We prove each of the axioms is true in the model, which, for the
single axioms is easy, and for the induction schema we perform a
straightforward by induction on formulas. I'm not going to spend my
time typing out for you what is easy enough to verify for one's self.
But if you do attempt to carry out the details but get stuck, let me
know and I'll help you out.
> But *how* do you know if PA has - or *has not* - any model, to begin
> with? What criteria did you use to determine a language model is
> a model of a formal system (PA in this case)?
By proving that each axiom of PA is true in the standard model, then
use the soundness theorem for first order logic to conclude that all
the theorems of those axioms are true in the standard model.
> > We have our definition of 'consistent'. It is syntactical.
> > However, we also prove
> > that the semantical property of having a model entails the syntactical
> > property of consistency.
>
> You're incorrect! How do you "prove" a _semantical_ property would
> conform to a _syntactical_ definition? They're like orange and apple!
If a set of sentences G has a model, then G is consistent. That is
quite easy to prove. It's about a one line argument. But I should
leave it as an exercise for you to prove. If you get stuck, I'll give
you a hint.
> If you don't know if either GC or ~GC is true in a *purported* model of
> PA (as a formal system), then it's quite possible that PA might be actually
> syntactically inconsistent!
I can prove that it is not the case that BOTH GC and ~GC are theorems
of PA while also granting that I don't know which of the two is true
in the standard model.
> And this *purported* model of PA might just
> be just a language-model *at best*, without being a PA's model.
>
> Do you understand this?
Of course I understand that simply being a model for the language of
PA does not entail that it is model of PA. But it is easy to prove
that the standard model for the language of PA is a model of PA. I
told you how to do it in this post. If you have problems completing
the excercise, then I'll help you out.
> >> For example, how would you know G(PA) isn't
> >> (syntactically) equivalent to GC, or cGC?
>
> > Whether you mean logically equivalent or equivalent relative to some
> > axioms, I don't need to opine as to whether G(PA) and GC are
> > equivalent or not just to define 'the standard model for the language
> > of PA'.
>
> You wouldn't opine even if G(PA) <-> GC <-> ~GC <-> cGC? The question
> here is how you prove this language-model to be a PA's model?
I don't need to opine on those material equivalences (and I still
don't know whether you mean them to be logical equivalences or
equivalences relative to some theory, such as PA), just to prove that
the standard model for the langauge of PA is a model of PA.
> I already stress the fact that "you - or anyone else - for that matter -
> wouldn't know exactly what the standard model of the *formal system* PA is"
> in the paragraph above (where I had a typo).
I told you what it is. I've told you about twenty times already over
the course of several threads.
> Do you now understand that paragraph (modulo the typo I've corrected)?
I understand that you keep claiming something that is not the case.
> > I've given you the definition.
>
> Again, which post in this thread/conversation did you give the definition
> of the standard of the formal system PA? Perhaps shouldn't keep mentioning
> thing you actually didn't say or articulate!
I've given it about twenty times over a period of a year or two with
you.
The standard model for the language of PA is the function that maps
the universal quantifier to w (i.e., omega, the set of natural
numbers), maps the symbol '0' to zero, maps the symbol 'S' to the
successor relation on w, maps the symbol '+' to the addition operation
on w, and maps the symbol '*' to the multiplication operation on w.
MoeBlee
Nam Nguyen
> On Feb 27, 10:22 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
>> MoeBlee wrote:
>
>>> Right, and we demonstrate that all the theorems of PA are true in the
>>> standard model for the language of PA.
>> Who among you, David, Aatu, Jesse, and I, did demonstrate that, and in what
>> post of the thread? Perhaps my recollection isn't that good. Would you
>> be able to refresh it? If not then you have *not* demonstrated so.
>
> I didn't say that anyone had bothered to type out the demonstration in
> a Usenet post. By 'we demonstrate', I mean an ordinary sense of 'such
> demonstrations can be completed by a reasonably alert student'.
Are you now in the business of writing dictionary, *trying* to define
what 'demonstrate' means? A drunk could say that much! If one didn't
do something it's one's _integrity_ to admit it and not hiding the
anonymous "we".
>
> We prove each of the axioms is true in the model, which, for the
> single axioms is easy, and for the induction schema we perform a
> straightforward by induction on formulas.
If you "prove" PA's axioms to be true in the standard model of L(PA)
using 'induction on formulas', then your proof is _circular_: induction
is impossible without a "naive" concept of the naturals; but the naturals
is the purported PA's standard model. So basically, you've just proved
PA's axioms are true in the standard model of PA is because they're ...
well ... supposed to be true in the standard model of PA!
So you still haven't successfully demonstrated that you know _precisely_
what you mean by the standard model of PA.
> I'm not going to spend my
> time typing out for you what is easy enough to verify for one's self.
> But if you do attempt to carry out the details but get stuck, let me
> know and I'll help you out.
Which are idiotic rambling of yours, hiding the fact that you didn't do
what you say you did, or didn't know how to defend your own arguments.
>
>> But *how* do you know if PA has - or *has not* - any model, to begin
>> with? What criteria did you use to determine a language model is
>> a model of a formal system (PA in this case)?
>
> By proving that each axiom of PA is true in the standard model,
By using, I suppose, induction which is circular.
> then
> use the soundness theorem for first order logic to conclude that all
> the theorems of those axioms are true in the standard model.
>
>>> We have our definition of 'consistent'. It is syntactical.
>>> However, we also prove
>>> that the semantical property of having a model entails the syntactical
>>> property of consistency.
>> You're incorrect! How do you "prove" a _semantical_ property would
>> conform to a _syntactical_ definition? They're like orange and apple!
>
> If a set of sentences G has a model, then G is consistent.
But how do you know if a set of sentences G has a model? Because
G is (syntactically) consistent? (Hint: a set of sentences never
"has" a model on its own! _Models are interpretation done by human_
and human have made mistake from time to time: mismatching models
and formal systems! (Have you have heard of Quinne's ML theory?)
You've very much border-lined being a dishonest person, MoeBlee.
My question is specific about "which post in this thread/conversation".
You either did or didn't, and either citing where in the thread you did,
or "I didn't do it" would have been a _straightforward_ answer.
No one would remember the context about whatever you might, or might have
not, said a year or 2 ago!
>
> The standard model for the language of PA is the function that maps
> the universal quantifier to w (i.e., omega, the set of natural
> numbers), maps the symbol '0' to zero, maps the symbol 'S' to the
> successor relation on w, maps the symbol '+' to the addition operation
> on w, and maps the symbol '*' to the multiplication operation on w.
Assuming by "w" you meant the set of naturals, then I've refuted this
above with a circularity.
In summary, *you still have failed* to demonstrate precisely what you meant
by "the standard model" of the formal system PA. (Seriously, MoeBlee, even
Godel himself never tried to "precisely define" it - the naturals. He only
_assumed_ some knowledge about it. If he couldn't, what makes you think you
could?)
MoeBlee
> MoeBlee wrote:
> > On Feb 27, 10:22 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
> >> MoeBlee wrote:
>
> >>> Right, and we demonstrate that all the theorems of PA are true in the
> >>> standard model for the language of PA.
> >> Who among you, David, Aatu, Jesse, and I, did demonstrate that, and in what
> >> post of the thread? Perhaps my recollection isn't that good. Would you
> >> be able to refresh it? If not then you have *not* demonstrated so.
>
> > I didn't say that anyone had bothered to type out the demonstration in
> > a Usenet post. By 'we demonstrate', I mean an ordinary sense of 'such
> > demonstrations can be completed by a reasonably alert student'.
>
> Are you now in the business of writing dictionary, *trying* to define
> what 'demonstrate' means? A drunk could say that much! If one didn't
> do something it's one's _integrity_ to admit it and not hiding the
> anonymous "we".
Oh please, you're really scavenging at the bottom now. Use of
editorial 'we' is not "hiding". And my purpose was just to tell you
what I meant by 'we demonstrate'; I didn't give any impression of
defining any words for a dictionary. When a person tells you what he
meant by use of a particular phrase in a particular context, I'd think
you'd accept that as simply what it is - an attempt at communication
with you; not a presumption of "writing a dictionary". You're
ridiculous. I should say "What I meant by "the weather looks bad
today" is "there are a lot of dark clouds in the sky moving in our
direction." Then you say, "Are you now in the business of writing a
dictionary, trying to define what 'bad weather' means".
> > We prove each of the axioms is true in the model, which, for the
> > single axioms is easy, and for the induction schema we perform a
> > straightforward by induction on formulas.
>
> If you "prove" PA's axioms to be true in the standard model of L(PA)
> using 'induction on formulas', then your proof is _circular_: induction
> is impossible without a "naive" concept of the naturals;
I prove it in a formal theory (say, Z set theory) in which the
informal concept of natural numbers is formalized. Of course, if we
demand formalizing each meta theory, going up ad infinitum, we get an
infinite escalation. Or, we may cut at a certain point to see that
each escalation is in the manner of the previous one. Or, we may
simply take certain notions as natural number as informal and use our
informal notions to talk about theories in which our notions are
formalized. In that regard, of course, I recognize a certain sense of
circularity. However, that seems to come with ANY project in
formalization. It does not seem to be a problem unique to discussing
formal first order PA or proving its consistency. If your argument
finally rests on the difficulty in this regard, then I don't contest
your argument other than to ask, "Okay, so show me a formalization of
mathematics that doesn't also have this difficulty".
> but the naturals
> is the purported PA's standard model.
The set of natural numbers is the universe of the standard model.
> So basically, you've just proved
> PA's axioms are true in the standard model of PA is because they're ...
> well ... supposed to be true in the standard model of PA!
Nope. In in proving in, say Z set theory, that the axioms of PA are
true in the standard model I don't use any premise that can be
rendered as "the axioms of PA are true in the standard model". Rather,
it is from the DEFINITION of the standard model that the triviality of
the proof comes.
> So you still haven't successfully demonstrated that you know _precisely_
> what you mean by the standard model of PA.
I gave you the definition.
> > I'm not going to spend my
> > time typing out for you what is easy enough to verify for one's self.
> > But if you do attempt to carry out the details but get stuck, let me
> > know and I'll help you out.
>
> Which are idiotic rambling of yours, hiding the fact that you didn't do
> what you say you did, or didn't know how to defend your own arguments.
Sorry, no, a refusal to be your clerk and to type out for you trivial
proofs is not "hiding".
> >> But *how* do you know if PA has - or *has not* - any model, to begin
> >> with? What criteria did you use to determine a language model is
> >> a model of a formal system (PA in this case)?
>
> > By proving that each axiom of PA is true in the standard model,
>
> By using, I suppose, induction which is circular.
We prove the induction theorems in, say Z theory, that are used to
justify proofs by induction on such things as formulas in languages.
> > then
> > use the soundness theorem for first order logic to conclude that all
> > the theorems of those axioms are true in the standard model.
>
> >>> We have our definition of 'consistent'. It is syntactical.
> >>> However, we also prove
> >>> that the semantical property of having a model entails the syntactical
> >>> property of consistency.
> >> You're incorrect! How do you "prove" a _semantical_ property would
> >> conform to a _syntactical_ definition? They're like orange and apple!
>
> > If a set of sentences G has a model, then G is consistent.
>
> But how do you know if a set of sentences G has a model?
That is another matter. At least recognize here that I have answered
your "orange and apple" objection. It is trivial to prove that if a
set of sentences G has a model then G is consistent.
> Because
> G is (syntactically) consistent?
You are correct that if G is consistent then G has a model.
> (Hint: a set of sentences never
> "has" a model on its own! _Models are interpretation done by human_
> and human have made mistake from time to time: mismatching models
> and formal systems! (Have you have heard of Quinne's ML theory?)
Whatever your view of the status of mathematical objects vis-a-vis
human action, it is still a theorem that a set of sentences G has a a
model iff G is consistent (where 'has a model' means 'there exists a
model for the language of G in which every member of G is true).
I didn't say you hadn't mentioned THIS thread. And I didn't answer as
if I were answering about THIS thread. Rather, INSTEAD, I mentioned
that I have given you the definition in PREVIOUS threads. That doesn't
make me "borderline dishonest". Really, you have BIZARRE notions not
just about mathematics but about simple posting interaction as well.
As to context of a year or two ago, I've given you the definition over
and over and over in posts to YOU DURING the past few years. I'm not
interested in doing the clerical work of looking up dates and posts
numbers for you. If you prefer then to think that I've never given you
the definition, then I can't stop you from that. The purpose of my
time is not post something over and over and over only then to somehow
obligate myself to research the posting record to satisfy you.
> > The standard model for the language of PA is the function that maps
> > the universal quantifier to w (i.e., omega, the set of natural
> > numbers), maps the symbol '0' to zero, maps the symbol 'S' to the
> > successor relation on w, maps the symbol '+' to the addition operation
> > on w, and maps the symbol '*' to the multiplication operation on w.
>
> Assuming by "w" you meant the set of naturals, then I've refuted this
> above with a circularity.
I've addressed in this post the question of circularity. However,
there is no special circularity regarding 'w'. In, say Z set theory,
we define 'w'. That occurs in Z set theory, not in first order PA.
> In summary, *you still have failed* to demonstrate precisely what you meant
> by "the standard model" of the formal system PA. (Seriously, MoeBlee, even
> Godel himself never tried to "precisely define" it - the naturals. He only
> _assumed_ some knowledge about it. If he couldn't, what makes you think you
> could?)
If by referring to the set of natural numbers I am thereby doomed to
your charge of circularity then there's all kinds of ordinary
mathematics that refers to the set of natural numbers that would be
also doomed.
As to Godel, his incompleteness theorem was given in informal
mathematics, but he mentioned that his arguments could also be
formalized. Moreover, the methods of formalization have been greatly
refined since 1930. I'm not presumptuous ("what makes you think you
could") simply by the fact that I work with such ordinary definitions
as those in Z set theory.
MoeBlee