Nam Nguyen wrote:
> >> H: T = T1 + T2 + T3 + ....
> >> where each Ti is in a collection K of formal systems (K isn't
> >> necessarily finite).
> >> .
> >> C1: Inconsistent(T) <=> (There exists a Ti: Inconsistent(Ti)).
> >> C2: Consistent(T) <=> (For _any given_ Ti: Consistent(Ti)).
> >> Proof: The proof for C1 or C2 is trivial and taken for granted here.
I tell you what, since the proofs are trivial write them down anyway.
And when you done that, you can deal with the long outstanding issues of
the interpretation of '=', the number of $\in$'s that set theory needs,
'x = x' always being an axiom of FOL= theories, and so on.
Not that this has anything to do with G\"odel's proof of G\"odel's
theorem, which you've never read, or any other version of G\"odel's
theorem.
-- When a true genius appears in the world, you may know him by this sign, that the dunces are all in confederacy against him.
Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting
> On Nov 14, 5:11 am, Nam Nguyen <namducngu...@shaw.ca> wrote:
>> On 13/11/2012 8:04 AM, Rupert wrote:
>>> On Nov 12, 9:31 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
>>>> One of my assumptions here is that in today terminology a Godel's
>>>> object T (or P) is just a FOL= formal system, where x=x is a logical
>>>> axiom and where _any_ T would be an extension of T0, with T0 being
>>>> the formal system that has no non-logical (including contingent)
>>>> axioms.
>>> The system P is a many-sorted first-order theory. But equality is a
>>> defined notion, and x=x is not a logical axiom. You would have to be
>>> clear about which of the axioms of P you consider to be logical
>>> axioms. Presumably you don't want to count the comprehension axioms as
>>> logical.
>> So GIT would not be applicable to today FOL= systems such as the
>> familiar arithmetic systems of Q and PA; and whoever says G(PA)
>> is true but undecidable in PA wouldn't know what he's talking about
>> because the system would be outside the scope of Godel Incompleteness?
>> Please confirm.
> Of course not. The theorem applies to any first-order theory in a
> language in which the first-order language of arithmetic can be
> interpreted, such that the theory is recursively enumerable,
> consistent, and strongly represents all the primitive recursive
> functions. In particular this is the case whenever the arithmetical
> part of the theory is a recursively enumerable consistent extension of
> Q.
So modern formal systems such as Q, PA, ZF, ZFC, ... aren't outside
the scope of GIT which is what I assumed above.
>>>> Now, let's prove a few meta theorems (MT) that we'd use later, no
>>>> matter how trivial they might be.
>>>> --------------------> Mt0 - Impossibility of Consistency Proof.
>>>> H: T is a formal system of FOL=; Consistent(T) is true.
>>>> (Note that "true" here is a meta level syntactical-factually
>>>> truth, _NOT_ a language-structure theoretical truth, or truth
>>>> in the naturals).
>>>> C: It's impossible know, to verify, it so using the knowledge of
>>>> FOL proof via rules of inference.
>>>> Proof: By definition, we'd have NEG(T|- CON(T)). But rules of inference
>>>> could _only_ yield a finite proof-string of the form (T |- F), _never_
>>>> of the form NOT(T |- F), by definitions of FOL proof. Hence, it's
>>>> impossible to prove T is consistent if it is so. QED.
>>> A proof in the theory T is not going to be a consistency proof for T.
>> Right: that's really what MT0 is saying!
> Well, there do exist consistent recursively enumerable theories in the
> first-order language of arithmetic which can prove their own
> consistency sentences, in some appropriate sense of "consistency
> sentence". Of necessity they are not extensions of Q. Are you
> disputing this result?
You seem to contradict yourself from one moment to the other on MT0!
>> (It's given that C here should
>> include the phrase "in T" somewhere). Iow, MT0 says that there's _no_
>> _proof in T_ for T's consistency if it is so, unlike the fact that
>> there's a proof for T's inconsistency if it is so. For any T.
>> That's really all MT0 says.
>>> That doesn't mean that you can't have a consistency proof for T in
>>> some metatheory. For example, you can prove that Q is consistent if
>>> you use PRA as your metatheory.
>> Before we could go further talking about the "proving" PRA which, iirc,
>> you said is a good old FOL= formal system, I would like you to confirm
>> that MT0 is correct in asserting that it's impossible to prove the
>> consistency of T, in T itself, (if it so is consistent).
> Well, there do exist some recursively enumerable consistent theories
> in the first-order language of arithmetic which can prove their own
> consistency sentences, in some appropriate sense of "consistency
> sentence". Of necessity they are not extensions of Q.
Then you're not confirming that MT0 is correct, or refuting it.
So I don't see how you'd be ale to see my explanation as to why
GIT is a logically invalid assertion.
Let me categorically say this: _There is NO logical sense_ in proving
a consistency of a T 'in some appropriate sense of "consistency
sentence"'.
The definition of inconsistency or consistency is _within proving in T_
_using rules of inference_ .
Let L1(<) and L2(e) are 2 languages each with a 2-ary predicate symbol.
Let:
T2a = {Axy[x e y] /\ ~Axy[x e y]}
T2b = {Axy[x e y] \/ ~Axy[x e y]}
If you'd like to prove T1a is inconsistent, you'd prove that in T1a,
_not_ in T2a, whether or not you could prove it so; and in this case
you could.
If you'd like to prove T1b is consistent, you'd prove that in T1b,
_not_ in T2b, whether or not you could prove it so; and in this case
you could _NOT_ .
To say that you could prove the undecidability of G(PA) in PRA is as
not conforming to the definition of consistency and as not logical
as to prove T1a is inconsistent using a proof in T2a!
In summary, you either clearly acknowledge MT0 as true or refute it.
Otherwise you'd not understand my proof that GIT is in invalid inference
in meta level.
-- ----------------------------------------------------
There is no remainder in the mathematics of infinity.
>>>> H: T = T1 + T2 + T3 + ....
>>>> where each Ti is in a collection K of formal systems (K isn't
>>>> necessarily finite).
>>>> .
>>>> C1: Inconsistent(T) <=> (There exists a Ti: Inconsistent(Ti)).
>>>> C2: Consistent(T) <=> (For _any given_ Ti: Consistent(Ti)).
>>>> Proof: The proof for C1 or C2 is trivial and taken for granted here.
> I tell you what, since the proofs are trivial write them down anyway.
No need for me to respond further until you admit you were wrong
on the following.
You yourself voluntarily accused my expression:
"x > the greatest counter example of the Goldbach conjecture"
as "is not well-formed"
then I explained to you that it's a well-formed expression, using the
well-formed formula below to define it:
~cGC /\ Ay[~GC(y) -> (y < x)]
But then you didn't see that I've correctly answered your "is not
well-formed" assertion.
So, until you acknowledge that you were wrong - and ignorant of the
matter - and that my:
~cGC /\ Ay[~GC(y) -> (y < x)]
correctly expresses "x > the greatest counter example of the Goldbach
conjecture", there's no point to answer another question of yours.
There got to be closure from a question of yours, before we could
move to another one.
So, acknowledge that you were wrong before, with your "is not
well-formed" assertion, if you'd like to hear further answer
from me on any technical questions.
-- ----------------------------------------------------
There is no remainder in the mathematics of infinity.
>> x > the greatest counter example of the Goldbach conjecture
>> can be defined as:
>> ~cGC /\ Ay[~GC(y) -> (y < x)]
>> Where GC(e) <-> even(e) -> "e is a sum of 2 primes".
> Yes, you could define "x > the greatest counter example of the Goldbach
> conjecture" that way if you didn't mind "the greatest counter example of
> the Goldbach conjecture" losing its usual meaning. Why anyone would use
> plain English when the plain English meaning was not intended, I don't
> know.
But you haven't technically explained to the ng as to _why_
~cGC /\ Ay[~GC(y) -> (y < x)] would make "x > the greatest counterexample of the Goldbach conjecture" lose "its usual meaning"!
Again, _WHY_ ?
> An honest person would say something like:
> I see that "x > the greatest counter example of the Goldbach conjecture"
> will not do, and I shall replace it with '~cGC /\ Ay[~GC(y) -> (y < x)],
> where GC(e) <-> even(e) -> "e is a sum of 2 primes"'.
Just because you're an idiot and are technically incompetent to
understand ~cGC /\ Ay[~GC(y) -> (y < x)] would expresses:
"x > the greatest counter example of the Goldbach conjecture"
doesn't make my explanation wrong at all.
-- ----------------------------------------------------
There is no remainder in the mathematics of infinity.
> On 14/11/2012 3:48 AM, Rupert wrote:
>> On Nov 14, 5:11 am, Nam Nguyen <namducngu...@shaw.ca> wrote:
>>> On 13/11/2012 8:04 AM, Rupert wrote:
>>>> On Nov 12, 9:31 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
>>>>> One of my assumptions here is that in today terminology a Godel's
>>>>> object T (or P) is just a FOL= formal system, where x=x is a logical
>>>>> axiom and where _any_ T would be an extension of T0, with T0 being
>>>>> the formal system that has no non-logical (including contingent)
>>>>> axioms.
>>>> The system P is a many-sorted first-order theory. But equality is a
>>>> defined notion, and x=x is not a logical axiom. You would have to be
>>>> clear about which of the axioms of P you consider to be logical
>>>> axioms. Presumably you don't want to count the comprehension axioms as
>>>> logical.
>>> So GIT would not be applicable to today FOL= systems such as the
>>> familiar arithmetic systems of Q and PA; and whoever says G(PA)
>>> is true but undecidable in PA wouldn't know what he's talking about
>>> because the system would be outside the scope of Godel Incompleteness?
>>> Please confirm.
>> Of course not. The theorem applies to any first-order theory in a
>> language in which the first-order language of arithmetic can be
>> interpreted, such that the theory is recursively enumerable,
>> consistent, and strongly represents all the primitive recursive
>> functions. In particular this is the case whenever the arithmetical
>> part of the theory is a recursively enumerable consistent extension of
>> Q.
> So modern formal systems such as Q, PA, ZF, ZFC, ... aren't outside
> the scope of GIT which is what I assumed above.
>>>>> Now, let's prove a few meta theorems (MT) that we'd use later, no
>>>>> matter how trivial they might be.
>>>>> --------------------> Mt0 - Impossibility of Consistency Proof.
>>>>> H: T is a formal system of FOL=; Consistent(T) is true.
>>>>> (Note that "true" here is a meta level syntactical-factually
>>>>> truth, _NOT_ a language-structure theoretical truth, or truth
>>>>> in the naturals).
>>>>> C: It's impossible know, to verify, it so using the knowledge of
>>>>> FOL proof via rules of inference.
>>>>> Proof: By definition, we'd have NEG(T|- CON(T)). But rules of
>>>>> inference
>>>>> could _only_ yield a finite proof-string of the form (T |- F), _never_
>>>>> of the form NOT(T |- F), by definitions of FOL proof. Hence, it's
>>>>> impossible to prove T is consistent if it is so. QED.
>>>> A proof in the theory T is not going to be a consistency proof for T.
>>> Right: that's really what MT0 is saying!
>> Well, there do exist consistent recursively enumerable theories in the
>> first-order language of arithmetic which can prove their own
>> consistency sentences, in some appropriate sense of "consistency
>> sentence". Of necessity they are not extensions of Q. Are you
>> disputing this result?
> You seem to contradict yourself from one moment to the other on MT0!
Then again, it seems you and I aren't talking about the same thing.
I'm saying consistency of T means T can _not_ prove certain formulas,
while you're talking about T can prove some formulas, as in "prove
[...] own consistency sentences".
How would T's proving some formulas _conform_ with the consistency-
requirement that T can _not_ prove certain formulas?
-- ----------------------------------------------------
There is no remainder in the mathematics of infinity.
> > On Nov 14, 5:11 am, Nam Nguyen <namducngu...@shaw.ca> wrote:
> >> On 13/11/2012 8:04 AM, Rupert wrote:
> >>> On Nov 12, 9:31 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
> >>>> One of my assumptions here is that in today terminology a Godel's
> >>>> object T (or P) is just a FOL= formal system, where x=x is a logical
> >>>> axiom and where _any_ T would be an extension of T0, with T0 being
> >>>> the formal system that has no non-logical (including contingent)
> >>>> axioms.
> >>> The system P is a many-sorted first-order theory. But equality is a
> >>> defined notion, and x=x is not a logical axiom. You would have to be
> >>> clear about which of the axioms of P you consider to be logical
> >>> axioms. Presumably you don't want to count the comprehension axioms as
> >>> logical.
> >> So GIT would not be applicable to today FOL= systems such as the
> >> familiar arithmetic systems of Q and PA; and whoever says G(PA)
> >> is true but undecidable in PA wouldn't know what he's talking about
> >> because the system would be outside the scope of Godel Incompleteness?
> >> Please confirm.
> > Of course not. The theorem applies to any first-order theory in a
> > language in which the first-order language of arithmetic can be
> > interpreted, such that the theory is recursively enumerable,
> > consistent, and strongly represents all the primitive recursive
> > functions. In particular this is the case whenever the arithmetical
> > part of the theory is a recursively enumerable consistent extension of
> > Q.
> So modern formal systems such as Q, PA, ZF, ZFC, ... aren't outside
> the scope of GIT which is what I assumed above.
> >>>> Now, let's prove a few meta theorems (MT) that we'd use later, no
> >>>> matter how trivial they might be.
> >>>> --------------------> Mt0 - Impossibility of Consistency Proof.
> >>>> H: T is a formal system of FOL=; Consistent(T) is true.
> >>>> (Note that "true" here is a meta level syntactical-factually
> >>>> truth, _NOT_ a language-structure theoretical truth, or truth
> >>>> in the naturals).
> >>>> C: It's impossible know, to verify, it so using the knowledge of
> >>>> FOL proof via rules of inference.
> >>>> Proof: By definition, we'd have NEG(T|- CON(T)). But rules of inference
> >>>> could _only_ yield a finite proof-string of the form (T |- F), _never_
> >>>> of the form NOT(T |- F), by definitions of FOL proof. Hence, it's
> >>>> impossible to prove T is consistent if it is so. QED.
> >>> A proof in the theory T is not going to be a consistency proof for T.
> >> Right: that's really what MT0 is saying!
> > Well, there do exist consistent recursively enumerable theories in the
> > first-order language of arithmetic which can prove their own
> > consistency sentences, in some appropriate sense of "consistency
> > sentence". Of necessity they are not extensions of Q. Are you
> > disputing this result?
> You seem to contradict yourself from one moment to the other on MT0!
I should have been a bit more careful. I apologize.
I thought that what was going on was that you were confusing the
object theory and the metatheory. But you can have a situation where
the metatheory and the object theory are in fact equal, and a proof in
the metatheory is a proof of a consistency sentence for the object
theory.
> >> (It's given that C here should
> >> include the phrase "in T" somewhere). Iow, MT0 says that there's _no_
> >> _proof in T_ for T's consistency if it is so, unlike the fact that
> >> there's a proof for T's inconsistency if it is so. For any T.
> >> That's really all MT0 says.
> >>> That doesn't mean that you can't have a consistency proof for T in
> >>> some metatheory. For example, you can prove that Q is consistent if
> >>> you use PRA as your metatheory.
> >> Before we could go further talking about the "proving" PRA which, iirc,
> >> you said is a good old FOL= formal system, I would like you to confirm
> >> that MT0 is correct in asserting that it's impossible to prove the
> >> consistency of T, in T itself, (if it so is consistent).
> > Well, there do exist some recursively enumerable consistent theories
> > in the first-order language of arithmetic which can prove their own
> > consistency sentences, in some appropriate sense of "consistency
> > sentence". Of necessity they are not extensions of Q.
> Then you're not confirming that MT0 is correct, or refuting it.
> So I don't see how you'd be ale to see my explanation as to why
> GIT is a logically invalid assertion.
> Let me categorically say this: _There is NO logical sense_ in proving
> a consistency of a T 'in some appropriate sense of "consistency
> sentence"'.
> T2a = {Axy[x e y] /\ ~Axy[x e y]}
> T2b = {Axy[x e y] \/ ~Axy[x e y]}
> If you'd like to prove T1a is inconsistent, you'd prove that in T1a,
> _not_ in T2a, whether or not you could prove it so; and in this case
> you could.
> If you'd like to prove T1b is consistent, you'd prove that in T1b,
> _not_ in T2b, whether or not you could prove it so; and in this case
> you could _NOT_ .
> To say that you could prove the undecidability of G(PA) in PRA is as
> not conforming to the definition of consistency and as not logical
> as to prove T1a is inconsistent using a proof in T2a!
It's not.
> In summary, you either clearly acknowledge MT0 as true or refute it.
It's wrong. You can prove the consistency of a first-order theory. You
can prove the consistency of Q in PRA, for example. You can read about
that in Shoenfield.
> > On 14/11/2012 3:48 AM, Rupert wrote:
> >> On Nov 14, 5:11 am, Nam Nguyen <namducngu...@shaw.ca> wrote:
> >>> On 13/11/2012 8:04 AM, Rupert wrote:
> >>>> On Nov 12, 9:31 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
> >>>>> One of my assumptions here is that in today terminology a Godel's
> >>>>> object T (or P) is just a FOL= formal system, where x=x is a logical
> >>>>> axiom and where _any_ T would be an extension of T0, with T0 being
> >>>>> the formal system that has no non-logical (including contingent)
> >>>>> axioms.
> >>>> The system P is a many-sorted first-order theory. But equality is a
> >>>> defined notion, and x=x is not a logical axiom. You would have to be
> >>>> clear about which of the axioms of P you consider to be logical
> >>>> axioms. Presumably you don't want to count the comprehension axioms as
> >>>> logical.
> >>> So GIT would not be applicable to today FOL= systems such as the
> >>> familiar arithmetic systems of Q and PA; and whoever says G(PA)
> >>> is true but undecidable in PA wouldn't know what he's talking about
> >>> because the system would be outside the scope of Godel Incompleteness?
> >>> Please confirm.
> >> Of course not. The theorem applies to any first-order theory in a
> >> language in which the first-order language of arithmetic can be
> >> interpreted, such that the theory is recursively enumerable,
> >> consistent, and strongly represents all the primitive recursive
> >> functions. In particular this is the case whenever the arithmetical
> >> part of the theory is a recursively enumerable consistent extension of
> >> Q.
> > So modern formal systems such as Q, PA, ZF, ZFC, ... aren't outside
> > the scope of GIT which is what I assumed above.
> >>>>> Now, let's prove a few meta theorems (MT) that we'd use later, no
> >>>>> matter how trivial they might be.
> >>>>> --------------------> Mt0 - Impossibility of Consistency Proof.
> >>>>> H: T is a formal system of FOL=; Consistent(T) is true.
> >>>>> (Note that "true" here is a meta level syntactical-factually
> >>>>> truth, _NOT_ a language-structure theoretical truth, or truth
> >>>>> in the naturals).
> >>>>> C: It's impossible know, to verify, it so using the knowledge of
> >>>>> FOL proof via rules of inference.
> >>>>> Proof: By definition, we'd have NEG(T|- CON(T)). But rules of
> >>>>> inference
> >>>>> could _only_ yield a finite proof-string of the form (T |- F), _never_
> >>>>> of the form NOT(T |- F), by definitions of FOL proof. Hence, it's
> >>>>> impossible to prove T is consistent if it is so. QED.
> >>>> A proof in the theory T is not going to be a consistency proof for T.
> >>> Right: that's really what MT0 is saying!
> >> Well, there do exist consistent recursively enumerable theories in the
> >> first-order language of arithmetic which can prove their own
> >> consistency sentences, in some appropriate sense of "consistency
> >> sentence". Of necessity they are not extensions of Q. Are you
> >> disputing this result?
> > You seem to contradict yourself from one moment to the other on MT0!
> Then again, it seems you and I aren't talking about the same thing.
> I'm saying consistency of T means T can _not_ prove certain formulas,
> while you're talking about T can prove some formulas, as in "prove
> [...] own consistency sentences".
> How would T's proving some formulas _conform_ with the consistency-
> requirement that T can _not_ prove certain formulas?
Take the example of PRA proving the consistency of Q. For Q to be
consistent is for it to fail to prove certain formulas. But there is a
formula in the language of PRA which expresses the assertion that Q is
consistent. And PRA can prove this formula.
Nam Nguyen wrote:
> But you haven't technically explained to the ng as to _why_
> ~cGC /\ Ay[~GC(y) -> (y < x)] would make "x > the greatest
> counterexample of the Goldbach conjecture" lose "its usual meaning"!
> Again, _WHY_ ?
Don't nag. You sound like a fishwife.
In "x > the greatest counterexample of the Goldbach conjecture" what
logic governs that "the"? There are various ways of dealing with
definite descriptions, which do you use?
-- When a true genius appears in the world, you may know him by this sign, that the dunces are all in confederacy against him.
Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting
> On 14/11/2012 6:46 AM, Frederick Williams wrote:
> > Nam Nguyen wrote:
> >>>> H: T = T1 + T2 + T3 + ....
> >>>> where each Ti is in a collection K of formal systems (K isn't
> >>>> necessarily finite).
> >>>> .
> >>>> C1: Inconsistent(T) <=> (There exists a Ti: Inconsistent(Ti)).
> >>>> C2: Consistent(T) <=> (For _any given_ Ti: Consistent(Ti)).
> >>>> Proof: The proof for C1 or C2 is trivial and taken for granted here.
> > I tell you what, since the proofs are trivial write them down anyway.
> No need for me to respond further until you admit you were wrong
> on the following.
In other words you can't prove it. Have you stopped to wonder why you
can't prove it.
-- When a true genius appears in the world, you may know him by this sign, that the dunces are all in confederacy against him.
Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting
You're thick, aren't you? I don't just mean that you're ignorant of
logic, that's obvious. You're general-purpose, all-round thick. Did
you really do a four year college degree in mathematics?
-- When a true genius appears in the world, you may know him by this sign, that the dunces are all in confederacy against him.
Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting
>> But you haven't technically explained to the ng as to _why_
>> ~cGC /\ Ay[~GC(y) -> (y < x)] would make "x > the greatest
>> counterexample of the Goldbach conjecture" lose "its usual meaning"!
>> Again, _WHY_ ?
> Don't nag. You sound like a fishwife.
> In "x > the greatest counterexample of the Goldbach conjecture" what
> logic governs that "the"? There are various ways of dealing with
> definite descriptions, which do you use?
You're really incapable of understanding simple mathematical
expression, using L(PA).
Can you _express_ :
x > the greatest even prime
_without even knowing_ if there's the greatest even prime?
-- ----------------------------------------------------
There is no remainder in the mathematics of infinity.
> On Nov 17, 7:45 am, Nam Nguyen <namducngu...@shaw.ca> wrote:
>> On 16/11/2012 9:18 PM, Nam Nguyen wrote:
>>> On 14/11/2012 3:48 AM, Rupert wrote:
>>>> On Nov 14, 5:11 am, Nam Nguyen <namducngu...@shaw.ca> wrote:
>>>>> On 13/11/2012 8:04 AM, Rupert wrote:
>>>>>> On Nov 12, 9:31 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
>>>>>>> One of my assumptions here is that in today terminology a Godel's
>>>>>>> object T (or P) is just a FOL= formal system, where x=x is a logical
>>>>>>> axiom and where _any_ T would be an extension of T0, with T0 being
>>>>>>> the formal system that has no non-logical (including contingent)
>>>>>>> axioms.
>>>>>> The system P is a many-sorted first-order theory. But equality is a
>>>>>> defined notion, and x=x is not a logical axiom. You would have to be
>>>>>> clear about which of the axioms of P you consider to be logical
>>>>>> axioms. Presumably you don't want to count the comprehension axioms as
>>>>>> logical.
>>>>> So GIT would not be applicable to today FOL= systems such as the
>>>>> familiar arithmetic systems of Q and PA; and whoever says G(PA)
>>>>> is true but undecidable in PA wouldn't know what he's talking about
>>>>> because the system would be outside the scope of Godel Incompleteness?
>>>>> Please confirm.
>>>> Of course not. The theorem applies to any first-order theory in a
>>>> language in which the first-order language of arithmetic can be
>>>> interpreted, such that the theory is recursively enumerable,
>>>> consistent, and strongly represents all the primitive recursive
>>>> functions. In particular this is the case whenever the arithmetical
>>>> part of the theory is a recursively enumerable consistent extension of
>>>> Q.
>>> So modern formal systems such as Q, PA, ZF, ZFC, ... aren't outside
>>> the scope of GIT which is what I assumed above.
>>>>>>> Now, let's prove a few meta theorems (MT) that we'd use later, no
>>>>>>> matter how trivial they might be.
>>>>>>> --------------------> Mt0 - Impossibility of Consistency Proof.
>>>>>>> H: T is a formal system of FOL=; Consistent(T) is true.
>>>>>>> (Note that "true" here is a meta level syntactical-factually
>>>>>>> truth, _NOT_ a language-structure theoretical truth, or truth
>>>>>>> in the naturals).
>>>>>>> C: It's impossible know, to verify, it so using the knowledge of
>>>>>>> FOL proof via rules of inference.
>>>>>>> Proof: By definition, we'd have NEG(T|- CON(T)). But rules of
>>>>>>> inference
>>>>>>> could _only_ yield a finite proof-string of the form (T |- F), _never_
>>>>>>> of the form NOT(T |- F), by definitions of FOL proof. Hence, it's
>>>>>>> impossible to prove T is consistent if it is so. QED.
>>>>>> A proof in the theory T is not going to be a consistency proof for T.
>>>>> Right: that's really what MT0 is saying!
>>>> Well, there do exist consistent recursively enumerable theories in the
>>>> first-order language of arithmetic which can prove their own
>>>> consistency sentences, in some appropriate sense of "consistency
>>>> sentence". Of necessity they are not extensions of Q. Are you
>>>> disputing this result?
>>> You seem to contradict yourself from one moment to the other on MT0!
>> Then again, it seems you and I aren't talking about the same thing.
>> I'm saying consistency of T means T can _not_ prove certain formulas,
>> while you're talking about T can prove some formulas, as in "prove
>> [...] own consistency sentences".
>> How would T's proving some formulas _conform_ with the consistency-
>> requirement that T can _not_ prove certain formulas?
> Take the example of PRA proving the consistency of Q. For Q to be
> consistent is for it to fail to prove certain formulas. But there is a
> formula in the language of PRA which expresses the assertion that Q is
> consistent. And PRA can prove this formula.
> What's the problem?
There are 2 problems that for various reasons you seem to have refused
to acknowledge; and I've already explained these 2 problems.
----------------> 1st problem.
A language expressing an assertion does _NOT equate_ to the
assertion being true or false, logically speaking.
For instance, in the thread where I defined cGC, you can
certainly use the same technique to define a similarly formed
formula that would express "There are infinitely many even primes",
whether or not there _actually_ are infinitely many even primes!
And I have already explained this viz-a-viz non-standard
interpretation of formula expression-truth. Would you
understand what I said there?
So a formula expressing "the assertion that Q is consistent" can
_NOT_ be equated to Q being _actually_ consistent, _if_ Q is so.
Logically speaking.
Formula semantic and formula semantic-truth are not (even) the same!
Is alive(Kennedy_Spirit) true or false?
----------------> 2nd problem.
I've already explained it: the problem is the FOL definition of
inconsistency, consistency of a T is _absolutely agnostic_ about
any theory other than T!
If you use a method to come up with what you'd call "proof" of
consistency but the method doesn't conform with FOL definition
of consistency then for sure that's a logically invalid method,
however well intended.
Why can't you acknowledge that simple fact?
-- ----------------------------------------------------
There is no remainder in the mathematics of infinity.
> On 17/11/2012 7:06 AM, Frederick Williams wrote:
> > Nam Nguyen wrote:
> >> But you haven't technically explained to the ng as to _why_
> >> ~cGC /\ Ay[~GC(y) -> (y < x)] would make "x > the greatest
> >> counterexample of the Goldbach conjecture" lose "its usual meaning"!
> >> Again, _WHY_ ?
> > Don't nag. You sound like a fishwife.
> > In "x > the greatest counterexample of the Goldbach conjecture" what
> > logic governs that "the"? There are various ways of dealing with
> > definite descriptions, which do you use?
> You're really incapable of understanding simple mathematical
> expression, using L(PA).
> Can you _express_ :
> x > the greatest even prime
> _without even knowing_ if there's the greatest even prime?
Express in what formal language? Definite descriptions are handled in
different ways by different authors.
Meanwhile:
You:
> H: T = T1 + T2 + T3 + ....
> where each Ti is in a collection K of formal systems (K isn't
> necessarily finite).
> .
> C1: Inconsistent(T) <=> (There exists a Ti: Inconsistent(Ti)).
> C2: Consistent(T) <=> (For _any given_ Ti: Consistent(Ti)).
> Proof: The proof for C1 or C2 is trivial and taken for granted here.
Me:
Really? If T = T1 + T2 means that the predicates (etc) of T is the
union of those of T1 and T2 and the axioms of T is the union of those of
T1 and T2, and T is closed under logical consequence; then it's obvious
that T can be inconsistent though both T1 and T2 are consistent. If
that's not what you mean by +, you need to say so.
And when you done that, you can deal with the long outstanding issues of
the interpretation of '=', the number of $\in$'s that set theory needs,
'x = x' always being an axiom of FOL= theories, and so on.
Do you think that these things go away just because you ignore them? Do
you think that just because you insist ("Again, _WHY_ ?"), like a
termagant, on having your points addressed (which they are) that others
will forget all the points that you have left unanswered?
Have you read G\"odel's paper yet? No. Or any other account of
G\"odel's incompleteness theorem? No. And will you continue to hold
forth about it nonetheless? Yes.
-- When a true genius appears in the world, you may know him by this sign, that the dunces are all in confederacy against him.
Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting
Nam Nguyen wrote:
> I've already explained it: the problem is the FOL definition of
> inconsistency, consistency of a T is _absolutely agnostic_ about
> any theory other than T!
> If you use a method to come up with what you'd call "proof" of
> consistency but the method doesn't conform with FOL definition
> of consistency then for sure that's a logically invalid method,
> however well intended.
How do you express that a theory T is consistent in a first order
language? What symbols does the first order language have, and what
symbols does T have?
> Why can't you acknowledge that simple fact?
It's a mystery.
-- When a true genius appears in the world, you may know him by this sign, that the dunces are all in confederacy against him.
Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting
>> On 17/11/2012 7:06 AM, Frederick Williams wrote:
>>> Nam Nguyen wrote:
>>>> But you haven't technically explained to the ng as to _why_
>>>> ~cGC /\ Ay[~GC(y) -> (y < x)] would make "x > the greatest
>>>> counterexample of the Goldbach conjecture" lose "its usual meaning"!
>>>> Again, _WHY_ ?
>>> Don't nag. You sound like a fishwife.
>>> In "x > the greatest counterexample of the Goldbach conjecture" what
>>> logic governs that "the"? There are various ways of dealing with
>>> definite descriptions, which do you use?
>> You're really incapable of understanding simple mathematical
>> expression, using L(PA).
>> Can you _express_ :
>> x > the greatest even prime
>> _without even knowing_ if there's the greatest even prime?
> Express in what formal language? Definite descriptions are handled in
> different ways by different authors.
Didn't I just say " using L(PA)"?
> Meanwhile:
There's no "Meanwhile:", until we have a closure and that you admit
you were technically wrong in not acknowledging that:
~cGC /\ Ay[~GC(y) -> (y < x)]
would correctly express:
"x > the greatest counterexample of the Goldbach conjecture".
-- ----------------------------------------------------
There is no remainder in the mathematics of infinity.
> On 17/11/2012 9:48 AM, Frederick Williams wrote:
> > Nam Nguyen wrote:
> >> On 17/11/2012 7:06 AM, Frederick Williams wrote:
> >>> Nam Nguyen wrote:
> >>>> But you haven't technically explained to the ng as to _why_
> >>>> ~cGC /\ Ay[~GC(y) -> (y < x)] would make "x > the greatest
> >>>> counterexample of the Goldbach conjecture" lose "its usual meaning"!
> >>>> Again, _WHY_ ?
> >>> Don't nag. You sound like a fishwife.
> >>> In "x > the greatest counterexample of the Goldbach conjecture" what
> >>> logic governs that "the"? There are various ways of dealing with
> >>> definite descriptions, which do you use?
> >> You're really incapable of understanding simple mathematical
> >> expression, using L(PA).
> >> Can you _express_ :
> >> x > the greatest even prime
> >> _without even knowing_ if there's the greatest even prime?
> > Express in what formal language? Definite descriptions are handled in
> > different ways by different authors.
> Didn't I just say " using L(PA)"?
PA is expressed in various languages. I'm not sure that I've met one
which had definite descriptions. That is not to say there isn't such a
language, but in the one that you have in mind what is the truth value
of
the x such that phi
if there is no x such that phi, or if there are a number of x's such
that phi? You'll really have to tell me because I don't know. One
possibility is that
the x such that phi
is never used in a formula unless it is first established that
there is just one x such that phi,
but that may not be a good idea since the theory of arithmetic being
considered is recursively undecidable.
> > Meanwhile:
> There's no "Meanwhile:", until we have a closure and that you admit
> you were technically wrong in not acknowledging that:
> ~cGC /\ Ay[~GC(y) -> (y < x)]
> would correctly express:
> "x > the greatest counterexample of the Goldbach conjecture".
If "~cGC /\ Ay[~GC(y) -> (y < x)]" has a definite truth value and "x >
the greatest counterexample of the Goldbach conjecture" doesn't, then
one can't correctly express the other. I don't now if "x > the greatest
counterexample of the Goldbach conjecture" has a definite truth value
until you explain how you're handing definite descriptions.
When you've done that, you might want to address:
You:
> H: T = T1 + T2 + T3 + ....
> where each Ti is in a collection K of formal systems (K isn't
> necessarily finite).
> .
> C1: Inconsistent(T) <=> (There exists a Ti: Inconsistent(Ti)).
> C2: Consistent(T) <=> (For _any given_ Ti: Consistent(Ti)).
> Proof: The proof for C1 or C2 is trivial and taken for granted here.
Me:
Really? If T = T1 + T2 means that the predicates (etc) of T is the
union of those of T1 and T2 and the axioms of T is the union of those of
T1 and T2, and T is closed under logical consequence; then it's obvious
that T can be inconsistent though both T1 and T2 are consistent. If
that's not what you mean by +, you need to say so.
And when you done that, you can deal with the long outstanding issues of
the interpretation of '=', the number of $\in$'s that set theory needs,
'x = x' always being an axiom of FOL= theories, and so on.
Do you think that these things go away just because you ignore them? Do
you think that just because you insist ("Again, _WHY_ ?"), like a
termagant, on having your points addressed (which they are) that others
will forget all the points that you have left unanswered?
Have you read G\"odel's paper yet? No. Or any other account of
G\"odel's incompleteness theorem? No. And will you continue to hold
forth about it nonetheless? Yes.
-- When a true genius appears in the world, you may know him by this sign, that the dunces are all in confederacy against him.
Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting
Frederick Williams wrote:
> until you explain how you're handing definite descriptions.
handling, sorry.
-- When a true genius appears in the world, you may know him by this sign, that the dunces are all in confederacy against him.
Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting
>> On 17/11/2012 9:48 AM, Frederick Williams wrote:
>>> Nam Nguyen wrote:
>>>> On 17/11/2012 7:06 AM, Frederick Williams wrote:
>>>>> Nam Nguyen wrote:
>>>>>> But you haven't technically explained to the ng as to _why_
>>>>>> ~cGC /\ Ay[~GC(y) -> (y < x)] would make "x > the greatest
>>>>>> counterexample of the Goldbach conjecture" lose "its usual meaning"!
>>>>>> Again, _WHY_ ?
>>>>> Don't nag. You sound like a fishwife.
>>>>> In "x > the greatest counterexample of the Goldbach conjecture" what
>>>>> logic governs that "the"? There are various ways of dealing with
>>>>> definite descriptions, which do you use?
>>>> You're really incapable of understanding simple mathematical
>>>> expression, using L(PA).
>>>> Can you _express_ :
>>>> x > the greatest even prime
>>>> _without even knowing_ if there's the greatest even prime?
>>> Express in what formal language? Definite descriptions are handled in
>>> different ways by different authors.
>> Didn't I just say " using L(PA)"?
> PA is expressed in various languages. I'm not sure that I've met one
> which had definite descriptions. That is not to say there isn't such a
> language,
L(PA) in this case is L(0,S,<,+,*).
> but in the one that you have in mind what is the truth value
> of
> the x such that phi
> if there is no x such that phi, or if there are a number of x's such
> that phi? You'll really have to tell me because I don't know.
That's why you were wrong: you were confused between semantic and truth.
_Truth is NOT required_ here; we're talking about semantics, expression
of the L(PA) language, to express say "x > the greatest even prime"
using formulas.
Until you're clear of this semantic vs. truth confusion, you'd not
be able to understand and admit you're wrong here in believing that:
>>>>>> ~cGC /\ Ay[~GC(y) -> (y < x)] would make "x > the greatest
>>>>>> counterexample of the Goldbach conjecture" lose "its usual
>>>>>> meaning"
-- ----------------------------------------------------
There is no remainder in the mathematics of infinity.
> On 17/11/2012 10:42 AM, Frederick Williams wrote:
> > PA is expressed in various languages. I'm not sure that I've met one
> > which had definite descriptions. That is not to say there isn't such a
> > language,
> L(PA) in this case is L(0,S,<,+,*).
> > but in the one that you have in mind what is the truth value
> > of
> > the x such that phi
> > if there is no x such that phi, or if there are a number of x's such
> > that phi? You'll really have to tell me because I don't know.
> That's why you were wrong: you were confused between semantic and truth.
> _Truth is NOT required_ here; we're talking about semantics, expression
> of the L(PA) language, to express say "x > the greatest even prime"
> using formulas.
> Until you're clear of this semantic vs. truth confusion, you'd not
> be able to understand and admit you're wrong here in believing that:
Since there's a "the" there, your language has a logical constant for
definite descriptions (typically "iota x F(x)" is read as "the x such
that F(x)". What logic governs your iota? Specifically, what if there
is no x such that F(x), or more than one?
Meanwhile don't forget:
You:
> H: T = T1 + T2 + T3 + ....
> where each Ti is in a collection K of formal systems (K isn't
> necessarily finite).
> .
> C1: Inconsistent(T) <=> (There exists a Ti: Inconsistent(Ti)).
> C2: Consistent(T) <=> (For _any given_ Ti: Consistent(Ti)).
> Proof: The proof for C1 or C2 is trivial and taken for granted here.
Me:
Really? If T = T1 + T2 means that the predicates (etc) of T is the
union of those of T1 and T2 and the axioms of T is the union of those of
T1 and T2, and T is closed under logical consequence; then it's obvious
that T can be inconsistent though both T1 and T2 are consistent. If
that's not what you mean by +, you need to say so.
And when you done that, you can deal with the long outstanding issues of
the interpretation of '=', the number of $\in$'s that set theory needs,
'x = x' always being an axiom of FOL= theories, and so on.
Do you think that these things go away just because you ignore them? Do
you think that just because you insist ("Again, _WHY_ ?"), like a
termagant, on having your points addressed (which they are) that others
will forget all the points that you have left unanswered?
Have you read G\"odel's paper yet? No. Or any other account of
G\"odel's incompleteness theorem? No. And will you continue to hold
forth about it nonetheless? Yes.
And don't pretend that I'm obliged to deal with your questions (which I
do) and that you are not obliged to deal with your backlog. Shall I
tell you how easy it is to deal with it? You just say: "I wish to
withdraw my claims about T = T1 + T2, the interpretation of '=', the
number of $\in$'s that set theory needs, 'x = x' being an axiom, etc,
etc." Unfortunately you are too devious and dishonest to do so. Do you
think people haven't noticed, or will just forget?
-- When a true genius appears in the world, you may know him by this sign, that the dunces are all in confederacy against him.
Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting
>> On 17/11/2012 10:42 AM, Frederick Williams wrote:
>>> PA is expressed in various languages. I'm not sure that I've met one
>>> which had definite descriptions. That is not to say there isn't such a
>>> language,
>> L(PA) in this case is L(0,S,<,+,*).
>>> but in the one that you have in mind what is the truth value
>>> of
>>> the x such that phi
>>> if there is no x such that phi, or if there are a number of x's such
>>> that phi? You'll really have to tell me because I don't know.
>> That's why you were wrong: you were confused between semantic and truth.
>> _Truth is NOT required_ here; we're talking about semantics, expression
>> of the L(PA) language, to express say "x > the greatest even prime"
>> using formulas.
>> Until you're clear of this semantic vs. truth confusion, you'd not
>> be able to understand and admit you're wrong here in believing that:
> Since there's a "the" there, your language has a logical constant for
> definite descriptions (typically "iota x F(x)" is read as "the x such
> that F(x)". What logic governs your iota? Specifically, what if there
> is no x such that F(x), or more than one?
So that's the source of your technical ignorance of the matter: you
don't seem to realize there's such thing as logical equivalence
of 2 (syntactically) different formulas or expressions.
For your lack of information:
x > the greatest even prime
is equivalent to:
There are finitely many even primes each of which is less than x.
See: there is _NO_ "the" there.
_One expression could be inferred from the other and vice versa_ .
Actually, have you ever heard of logical equivalence of formulas
or expressions?
> Meanwhile
As said before, there's _NO_ "Meanwhile" _UNTIL you acknowledge_
_you are technically wrong in the matter here_ .
-- ----------------------------------------------------
There is no remainder in the mathematics of infinity.
> On 17/11/2012 3:26 PM, Frederick Williams wrote:
> > Nam Nguyen wrote:
> >> On 17/11/2012 10:42 AM, Frederick Williams wrote:
> >>> PA is expressed in various languages. I'm not sure that I've met one
> >>> which had definite descriptions. That is not to say there isn't such a
> >>> language,
> >> L(PA) in this case is L(0,S,<,+,*).
> >>> but in the one that you have in mind what is the truth value
> >>> of
> >>> the x such that phi
> >>> if there is no x such that phi, or if there are a number of x's such
> >>> that phi? You'll really have to tell me because I don't know.
> >> That's why you were wrong: you were confused between semantic and truth.
> >> _Truth is NOT required_ here; we're talking about semantics, expression
> >> of the L(PA) language, to express say "x > the greatest even prime"
> >> using formulas.
> >> Until you're clear of this semantic vs. truth confusion, you'd not
> >> be able to understand and admit you're wrong here in believing that:
> > Since there's a "the" there, your language has a logical constant for
> > definite descriptions (typically "iota x F(x)" is read as "the x such
> > that F(x)". What logic governs your iota? Specifically, what if there
> > is no x such that F(x), or more than one?
> So that's the source of your technical ignorance of the matter: you
> don't seem to realize there's such thing as logical equivalence
> of 2 (syntactically) different formulas or expressions.
> For your lack of information:
> x > the greatest even prime
> is equivalent to:
> There are finitely many even primes each of which is less than x.
Your example is not interesting. "x > the greatest even prime" is just
the same as "x > 2".
More interesting is
"x > the greatest counterexample of the Goldbach conjecture"
which is after all what you wrote. Are you trying to disown it
already? How do you formalize it? What axioms and rules govern your
formalization?
Note that there is no known numeral n such that
"x > the greatest counterexample of the Goldbach conjecture"
is equivalent to
"x > n".
If the Goldbach conjecture is true there is no such n, known or
unknown. And if there are infinitely many counterexamples to the
Goldbach conjecture, what does "the greatest counterexample" mean in
that case?
Note that "x > the greatest even prime" is quite unproblematic, because
it is just the same as "x > 2" come what may. I suspect you realize
that "x > the greatest counterexample of the Goldbach conjecture" is
problematic, and you have introduced "x > the greatest even prime"
because it isn't problematic, and you hope the Goldbach conjecture
problem will go away. Actually, you can make it go away by saying:
"I admit that the 'the' in 'x > the greatest counterexample of the
Goldbach conjecture' is problematic, and my knowledge of logic is too
inadequate for me to know how to deal with it. Therefore, whatever I
was saying about 'x > the greatest counterexample of the Goldbach
conjecture' I now withdraw."
Or you could say:
"I admit that the 'the' in 'x > the greatest counterexample of the
Goldbach conjecture' is problematic, and I shall learn about definite
descriptions and how to deal with it. I may then return to the fray."
When you've dealt with that, you might want to address:
You:
> H: T = T1 + T2 + T3 + ....
> where each Ti is in a collection K of formal systems (K isn't
> necessarily finite).
> .
> C1: Inconsistent(T) <=> (There exists a Ti: Inconsistent(Ti)).
> C2: Consistent(T) <=> (For _any given_ Ti: Consistent(Ti)).
> Proof: The proof for C1 or C2 is trivial and taken for granted here.
Me:
Really? If T = T1 + T2 means that the predicates (etc) of T is the
union of those of T1 and T2 and the axioms of T is the union of those of
T1 and T2, and T is closed under logical consequence; then it's obvious
that T can be inconsistent though both T1 and T2 are consistent. If
that's not what you mean by +, you need to say so.
And when you done that, you can deal with the long outstanding issues of
the interpretation of '=', the number of $\in$'s that set theory needs,
'x = x' always being an axiom of FOL= theories, and so on.
Do you think that these things go away just because you ignore them? Do
you think that just because you insist ("Again, _WHY_ ?"), like a
termagant, on having your points addressed (which they are) that others
will forget all the points that you have left unanswered?
Have you read G\"odel's paper yet? No. Or any other account of
G\"odel's incompleteness theorem? No. And will you continue to hold
forth about it nonetheless? Yes.
You are truly stupid. There is an easy way out of your difficulties. First, admit that you know nothing about the formalization of "the" and
wish to give up on it. Then just say: "I also wish to withdraw my
claims about T = T1 + T2, the interpretation of '=', the number of
$\in$'s that set theory needs, 'x = x' being an axiom, etc, etc." Unfortunately you are too devious and dishonest to do so. Do you think
people haven't noticed, or will just forget?
-- When a true genius appears in the world, you may know him by this sign, that the dunces are all in confederacy against him.
Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting
>> On 17/11/2012 3:26 PM, Frederick Williams wrote:
>>> Nam Nguyen wrote:
>>>> On 17/11/2012 10:42 AM, Frederick Williams wrote:
>>>>> PA is expressed in various languages. I'm not sure that I've met one
>>>>> which had definite descriptions. That is not to say there isn't such a
>>>>> language,
>>>> L(PA) in this case is L(0,S,<,+,*).
>>>>> but in the one that you have in mind what is the truth value
>>>>> of
>>>>> the x such that phi
>>>>> if there is no x such that phi, or if there are a number of x's such
>>>>> that phi? You'll really have to tell me because I don't know.
>>>> That's why you were wrong: you were confused between semantic and truth.
>>>> _Truth is NOT required_ here; we're talking about semantics, expression
>>>> of the L(PA) language, to express say "x > the greatest even prime"
>>>> using formulas.
>>>> Until you're clear of this semantic vs. truth confusion, you'd not
>>>> be able to understand and admit you're wrong here in believing that:
>>> Since there's a "the" there, your language has a logical constant for
>>> definite descriptions (typically "iota x F(x)" is read as "the x such
>>> that F(x)". What logic governs your iota? Specifically, what if there
>>> is no x such that F(x), or more than one?
>> So that's the source of your technical ignorance of the matter: you
>> don't seem to realize there's such thing as logical equivalence
>> of 2 (syntactically) different formulas or expressions.
>> For your lack of information:
>> x > the greatest even prime
>> is equivalent to:
>> There are finitely many even primes each of which is less than x.
> Your example is not interesting. "x > the greatest even prime" is just
> the same as "x > 2".
You're technically incompetent.
There are ways to express "x > the greatest even prime" without
requiring SS0 to be present in the formula. I already explain how
to do it: you're just not capable of understanding that _basic fact_ .
> More interesting is
> "x > the greatest counterexample of the Goldbach conjecture"
> which is after all what you wrote. Are you trying to disown it
> already?
Of course not. I always maintain that ~cGC /\ Ay[~GC(y) -> (y < x)]
would express "x > the greatest counterexample of the Goldbach
conjecture". My example about "There are finitely many even primes
each of which is less than x" was meant to help you to understand
you error of a _basic fact_ .
> How do you formalize it? What axioms and rules govern your
> formalization?
This is your 2nd utterly confusion: formula semantics expression
has nothing to do with _formalization needing axioms_ !
> Note that there is no known numeral n such that
> "x > the greatest counterexample of the Goldbach conjecture"
> is equivalent to
> "x > n".
You're really hopeless with all that idiotic rambling, while
not knowing what a language _formula expression_ is.
Sorry that I can't help you much more, until you admit you
were wrong.
-- ----------------------------------------------------
There is no remainder in the mathematics of infinity.
> On 17/11/2012 6:05 PM, Frederick Williams wrote:
>> Nam Nguyen wrote:
>>> On 17/11/2012 3:26 PM, Frederick Williams wrote:
>>>> Nam Nguyen wrote:
>>>>> On 17/11/2012 10:42 AM, Frederick Williams wrote:
>>>>>> PA is expressed in various languages. I'm not sure that I've met one
>>>>>> which had definite descriptions. That is not to say there isn't
>>>>>> such a
>>>>>> language,
>>>>> L(PA) in this case is L(0,S,<,+,*).
>>>>>> but in the one that you have in mind what is the truth value
>>>>>> of
>>>>>> the x such that phi
>>>>>> if there is no x such that phi, or if there are a number of x's such
>>>>>> that phi? You'll really have to tell me because I don't know.
>>>>> That's why you were wrong: you were confused between semantic and
>>>>> truth.
>>>>> _Truth is NOT required_ here; we're talking about semantics,
>>>>> expression
>>>>> of the L(PA) language, to express say "x > the greatest even prime"
>>>>> using formulas.
>>>>> Until you're clear of this semantic vs. truth confusion, you'd not
>>>>> be able to understand and admit you're wrong here in believing that:
>>>> Since there's a "the" there, your language has a logical constant for
>>>> definite descriptions (typically "iota x F(x)" is read as "the x such
>>>> that F(x)". What logic governs your iota? Specifically, what if there
>>>> is no x such that F(x), or more than one?
>>> So that's the source of your technical ignorance of the matter: you
>>> don't seem to realize there's such thing as logical equivalence
>>> of 2 (syntactically) different formulas or expressions.
>>> For your lack of information:
>>> x > the greatest even prime
>>> is equivalent to:
>>> There are finitely many even primes each of which is less than x.
>> Your example is not interesting. "x > the greatest even prime" is just
>> the same as "x > 2".
> You're technically incompetent.
> There are ways to express "x > the greatest even prime" without
> requiring SS0 to be present in the formula. I already explain how
> to do it: you're just not capable of understanding that _basic fact_ .
>> More interesting is
>> "x > the greatest counterexample of the Goldbach conjecture"
>> which is after all what you wrote. Are you trying to disown it
>> already?
> Of course not. I always maintain that ~cGC /\ Ay[~GC(y) -> (y < x)]
> would express "x > the greatest counterexample of the Goldbach
> conjecture". My example about "There are finitely many even primes
> each of which is less than x" was meant to help you to understand
> you error of a _basic fact_ .
>> How do you formalize it? What axioms and rules govern your
>> formalization?
> This is your 2nd utterly confusion: formula semantics expression
> has nothing to do with _formalization needing axioms_ !
>> Note that there is no known numeral n such that
>> "x > the greatest counterexample of the Goldbach conjecture"
>> is equivalent to
>> "x > n".
> You're really hopeless with all that idiotic rambling, while
> not knowing what a language _formula expression_ is.
> Sorry that I can't help you much more, until you admit you
> were wrong.
Seriously, Frederick. Why don't you bring my definition of cGC
and ask an informed poster or a Professor that you could talk to,
to see if my claim that ~cGC /\ Ay[~GC(y) -> (y < x)] would express
"x > the greatest counterexample of the Goldbach conjecture" is wrong,
and if so bring back their explanation and present it here for people
to see.
Until then you've shown you don't know a _very basic fact_ of
mathematical reasoning.
-- ----------------------------------------------------
There is no remainder in the mathematics of infinity.
> > On Nov 17, 7:45 am, Nam Nguyen <namducngu...@shaw.ca> wrote:
> >> On 16/11/2012 9:18 PM, Nam Nguyen wrote:
> >>> On 14/11/2012 3:48 AM, Rupert wrote:
> >>>> On Nov 14, 5:11 am, Nam Nguyen <namducngu...@shaw.ca> wrote:
> >>>>> On 13/11/2012 8:04 AM, Rupert wrote:
> >>>>>> On Nov 12, 9:31 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
> >>>>>>> One of my assumptions here is that in today terminology a Godel's
> >>>>>>> object T (or P) is just a FOL= formal system, where x=x is a logical
> >>>>>>> axiom and where _any_ T would be an extension of T0, with T0 being
> >>>>>>> the formal system that has no non-logical (including contingent)
> >>>>>>> axioms.
> >>>>>> The system P is a many-sorted first-order theory. But equality is a
> >>>>>> defined notion, and x=x is not a logical axiom. You would have to be
> >>>>>> clear about which of the axioms of P you consider to be logical
> >>>>>> axioms. Presumably you don't want to count the comprehension axioms as
> >>>>>> logical.
> >>>>> So GIT would not be applicable to today FOL= systems such as the
> >>>>> familiar arithmetic systems of Q and PA; and whoever says G(PA)
> >>>>> is true but undecidable in PA wouldn't know what he's talking about
> >>>>> because the system would be outside the scope of Godel Incompleteness?
> >>>>> Please confirm.
> >>>> Of course not. The theorem applies to any first-order theory in a
> >>>> language in which the first-order language of arithmetic can be
> >>>> interpreted, such that the theory is recursively enumerable,
> >>>> consistent, and strongly represents all the primitive recursive
> >>>> functions. In particular this is the case whenever the arithmetical
> >>>> part of the theory is a recursively enumerable consistent extension of
> >>>> Q.
> >>> So modern formal systems such as Q, PA, ZF, ZFC, ... aren't outside
> >>> the scope of GIT which is what I assumed above.
> >>>>>>> Now, let's prove a few meta theorems (MT) that we'd use later, no
> >>>>>>> matter how trivial they might be.
> >>>>>>> --------------------> Mt0 - Impossibility of Consistency Proof.
> >>>>>>> H: T is a formal system of FOL=; Consistent(T) is true.
> >>>>>>> (Note that "true" here is a meta level syntactical-factually
> >>>>>>> truth, _NOT_ a language-structure theoretical truth, or truth
> >>>>>>> in the naturals).
> >>>>>>> C: It's impossible know, to verify, it so using the knowledge of
> >>>>>>> FOL proof via rules of inference.
> >>>>>>> Proof: By definition, we'd have NEG(T|- CON(T)). But rules of
> >>>>>>> inference
> >>>>>>> could _only_ yield a finite proof-string of the form (T |- F), _never_
> >>>>>>> of the form NOT(T |- F), by definitions of FOL proof. Hence, it's
> >>>>>>> impossible to prove T is consistent if it is so. QED.
> >>>>>> A proof in the theory T is not going to be a consistency proof for T.
> >>>>> Right: that's really what MT0 is saying!
> >>>> Well, there do exist consistent recursively enumerable theories in the
> >>>> first-order language of arithmetic which can prove their own
> >>>> consistency sentences, in some appropriate sense of "consistency
> >>>> sentence". Of necessity they are not extensions of Q. Are you
> >>>> disputing this result?
> >>> You seem to contradict yourself from one moment to the other on MT0!
> >> Then again, it seems you and I aren't talking about the same thing.
> >> I'm saying consistency of T means T can _not_ prove certain formulas,
> >> while you're talking about T can prove some formulas, as in "prove
> >> [...] own consistency sentences".
> >> How would T's proving some formulas _conform_ with the consistency-
> >> requirement that T can _not_ prove certain formulas?
> > Take the example of PRA proving the consistency of Q. For Q to be
> > consistent is for it to fail to prove certain formulas. But there is a
> > formula in the language of PRA which expresses the assertion that Q is
> > consistent. And PRA can prove this formula.
> > What's the problem?
> There are 2 problems that for various reasons you seem to have refused
> to acknowledge; and I've already explained these 2 problems.
> ----------------> 1st problem.
> A language expressing an assertion does _NOT equate_ to the
> assertion being true or false, logically speaking.
> For instance, in the thread where I defined cGC, you can
> certainly use the same technique to define a similarly formed
> formula that would express "There are infinitely many even primes",
> whether or not there _actually_ are infinitely many even primes!
> And I have already explained this viz-a-viz non-standard
> interpretation of formula expression-truth. Would you
> understand what I said there?
> So a formula expressing "the assertion that Q is consistent" can
> _NOT_ be equated to Q being _actually_ consistent, _if_ Q is so.
> Logically speaking.
> Formula semantic and formula semantic-truth are not (even) the same!
> Is alive(Kennedy_Spirit) true or false?
I don't understand your point. Just because I can write down a formula
doesn't mean it is true, no. But many would feel that if I can prove
it in PRA that's a pretty good reason for thinking it true. You may
perhaps feel differently, but in that case the onus is on you to tell
us which systems you do trust.
> ----------------> 2nd problem.
> I've already explained it: the problem is the FOL definition of
> inconsistency, consistency of a T is _absolutely agnostic_ about
> any theory other than T!
> If you use a method to come up with what you'd call "proof" of
> consistency but the method doesn't conform with FOL definition
> of consistency then for sure that's a logically invalid method,
> however well intended.
> Why can't you acknowledge that simple fact?
Again, I just don't get what your point is. As far as I can tell you
are talking incoherent nonsense.
> On 17/11/2012 6:05 PM, Frederick Williams wrote:
> > Nam Nguyen wrote:
> >> On 17/11/2012 3:26 PM, Frederick Williams wrote:
> >>> Nam Nguyen wrote:
> >>>> On 17/11/2012 10:42 AM, Frederick Williams wrote:
> >>>>> PA is expressed in various languages. I'm not sure that I've met one
> >>>>> which had definite descriptions. That is not to say there isn't such a
> >>>>> language,
> >>>> L(PA) in this case is L(0,S,<,+,*).
> >>>>> but in the one that you have in mind what is the truth value
> >>>>> of
> >>>>> the x such that phi
> >>>>> if there is no x such that phi, or if there are a number of x's such
> >>>>> that phi? You'll really have to tell me because I don't know.
> >>>> That's why you were wrong: you were confused between semantic and truth.
> >>>> _Truth is NOT required_ here; we're talking about semantics, expression
> >>>> of the L(PA) language, to express say "x > the greatest even prime"
> >>>> using formulas.
> >>>> Until you're clear of this semantic vs. truth confusion, you'd not
> >>>> be able to understand and admit you're wrong here in believing that:
> >>> Since there's a "the" there, your language has a logical constant for
> >>> definite descriptions (typically "iota x F(x)" is read as "the x such
> >>> that F(x)". What logic governs your iota? Specifically, what if there
> >>> is no x such that F(x), or more than one?
> >> So that's the source of your technical ignorance of the matter: you
> >> don't seem to realize there's such thing as logical equivalence
> >> of 2 (syntactically) different formulas or expressions.
> >> For your lack of information:
> >> x > the greatest even prime
> >> is equivalent to:
> >> There are finitely many even primes each of which is less than x.
> > Your example is not interesting. "x > the greatest even prime" is just
> > the same as "x > 2".
> You're technically incompetent.
> There are ways to express "x > the greatest even prime" without
> requiring SS0 to be present in the formula. I already explain how
> to do it: you're just not capable of understanding that _basic fact_ .
The example "x > the greatest even prime" is of no interest. You are
just trying to divert attention away from your failure to understand
that "the" is a logical constant that is regulated by axioms and/or
rules, and you need to say what those axioms and/or rules are. What is
"the x s.t. F(x)" in the cases where no x Fs or more than one x Fs?
> > [...]
> Sorry that I can't help you much more, until you admit you
> were wrong.
Do you think that by snipping what follows I will forget about it? Why
are such a cowardly, lying, devious little cunt?
You:
> H: T = T1 + T2 + T3 + ....
> where each Ti is in a collection K of formal systems (K isn't
> necessarily finite).
> .
> C1: Inconsistent(T) <=> (There exists a Ti: Inconsistent(Ti)).
> C2: Consistent(T) <=> (For _any given_ Ti: Consistent(Ti)).
> Proof: The proof for C1 or C2 is trivial and taken for granted here.
Me:
Really? If T = T1 + T2 means that the predicates (etc) of T is the
union of those of T1 and T2 and the axioms of T is the union of those of
T1 and T2, and T is closed under logical consequence; then it's obvious
that T can be inconsistent though both T1 and T2 are consistent. If
that's not what you mean by +, you need to say so.
And when you done that, you can deal with the long outstanding issues of
the interpretation of '=', the number of $\in$'s that set theory needs,
'x = x' always being an axiom of FOL= theories, and so on.
Do you think that these things go away just because you ignore them? Do
you think that just because you insist ("Again, _WHY_ ?"), like a
termagant, on having your points addressed (which they are) that others
will forget all the points that you have left unanswered?
Have you read G\"odel's paper yet? No. Or any other account of
G\"odel's incompleteness theorem? No. And will you continue to hold
forth about it nonetheless? Yes.
You are truly stupid. There is an easy way out of your difficulties. First, admit that you know nothing about the formalization of "the" and
wish to give up on it. Then just say: "I also wish to withdraw my
claims about T = T1 + T2, the interpretation of '=', the number of
$\in$'s that set theory needs, 'x = x' being an axiom, etc, etc." Unfortunately you are too devious and dishonest to do so. Do you think
people haven't noticed, or will just forget?
-- When a true genius appears in the world, you may know him by this sign, that the dunces are all in confederacy against him.
Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting
