Godel's theorem is invalid?
LordBeotian
Torkel Franzen
> I would like to read some opinions about this:
Why?
LordBeotian
"Torkel Franzen" <tor...@sm.luth.se> ha scritto
>> I would like to read some opinions about this:
>
> Why?
Because I need someone who looks competent to say me that the author of the
article is wrong and Godel's theorem is safe...
Torkel Franzen
> Because I need someone who looks competent to say me that the author of the
> article is wrong and Godel's theorem is safe...
You can conclude for yourself, for all sorts of reasons, that the
article is standard crank stuff.
LordBeotian
"Torkel Franzen" <tor...@sm.luth.se> ha scritto
>> Because I need someone who looks competent to say me that the author of
>> the
>> article is wrong and Godel's theorem is safe...
>
> You can conclude for yourself, for all sorts of reasons, that the
> article is standard crank stuff.
I was trying to look for a reason to make such a conclusion.
Torkel Franzen
> I was trying to look for a reason to make such a conclusion.
You can't think of any?
Daryl McCullough
<pokispy76@[CANCELLA QUESTO]yahoo.it> says...
Here's an opinion: The author is mathematically incompetent.
For one thing, he claims that the axiom schema of natural
induction is unnecessary. His argument basically boils down
to this:
If P(0) is true, and for all x, (P(x) -> P(x+1)), then
we can prove P(n) for each natural number n.
That's true. But there is a big difference between
(1) For all n, we can prove P(n).
and
(2) We can prove "for all n, P(n)"
You need induction to be able to prove the universally
quantified formula. If you ignore the difference between
(1) and (2) then of course Godel's theorem leads to an
inconsistency.
--
Daryl McCullough
Ithaca, NY
LordBeotian
"Torkel Franzen" <tor...@sm.luth.se> ha scritto
>> I was trying to look for a reason to make such a conclusion.
>
> You can't think of any?
Ok, I have read it more carefully and it in fact seems meaningless...
The author seems to think that every axiomatic system is complete.
george
Daryl McCullough wrote:
> Here's an opinion: The author is mathematically incompetent.
Well, sure, but all incompetence is not created equal.
> For one thing, he claims that the axiom schema of natural
> induction is unnecessary. His argument basically boils down
> to this:
>
> If P(0) is true, and for all x, (P(x) -> P(x+1)), then
> we can prove P(n) for each natural number n.
>
> That's true.
Well, in that case, he thinks he wins.
I mean, it really is important to understand WHY he thinks
that's enough. Mere dismissals of incompetence are
uncharitable to put it charitably. There is a PERFECTLY GOOD
REASON why people who DO have at least a LITTLE bit of
mathematical competence might think they had won, here.
> But there is a big difference between
>
> (1) For all n, we can prove P(n).
>
> and
>
> (2) We can prove "for all n, P(n)"
Not if you don't know that there can be models
with things other than natural numbers in them, there
isn't. In the first place, (1) is in the meta-language rather
than the object language, so the domain of quantification is
obscure. In the second place, the possibility of the object-
language's (2) "for all n" ranging over things that might be
greater than all natural numbers simply has not occurred
to BUNCHES of people.
All I'm trying to say is that unawareness of non-
standard models of PA and supernatural numbers is not
the same thing as mathematical incompetence.
A stronger defense for anyone who would make this
mistake is that in the usual presentation of PA, there
are no terms in the object-language that refer to anything
other than natural numbers. Nonstandard models can
have supernatural numbers, but in the usual language, all
the higher numbers are "anonymous". That is a good reason
why even a (minimally) competent person might fail to
intuit the possibility of their existence.
> You need induction to be able to prove the universally
> quantified formula. If you ignore the difference between
> (1) and (2) then
then you are just sort of subconsciously circumscribing
your possible universe to and only to the things you have
names for. This is not a cardinal sin (OK, maybe it
*is* a *cardinal*, AND an ordinal, sin, but it is not evidence
by itself of unusual stupidity; at worst it is a station along
the way to understanding).
Torkel Franzen
> > But there is a big difference between
> >
> > (1) For all n, we can prove P(n).
> >
> > and
> >
> > (2) We can prove "for all n, P(n)"
>
> Not if you don't know that there can be models
> with things other than natural numbers in them, there
> isn't.
Why drag in the model theory of first order logic? It's irrelevant
to the distinction between (1) and (2).
george
> > >
> > > (1) For all n, we can prove P(n).
> > >
> > > and
> > >
> > > (2) We can prove "for all n, P(n)"
I replied,
> > Not if you don't know that there can be models
> > with things other than natural numbers in them, there
> > isn't.
Torkel Franzen wrote:
> Why drag in the model theory of first order logic?
Because of the completeness theorem. The issue is
whether being able to prove (1) is or isn't "like"
being able to prove (2). In order for them not to be "alike",
one of them has to be non-provable "in the same context"
where the other is provable. Here, that context is the
prior known truth of the premises P(n), for each n.
"for all n, P(n)" IS true in ALL models where these n's
are all&only the things there are. Since they are all and
only the things that are named, in the usual language,
thinking that they are all there are should be forgiveable,
for newbies anyway; they have a right to be SHOWN more,
before they have to know about more.
> It's irrelevant to the distinction between (1) and (2).
Well, the compactness theorem says it's irrelevant.
But the compactness theorem is non-trivial because it
is in some sense surprising, because it might not have
been intuited if it hadn't been proven.
What the compactness theorem makes relevant is that
FOL is finitary in some deep sense; everything inferred via
FOL *must* logically follow from only a finite number of
premises. Obviously, if your infinite domain is properly
interpreted under an infinite language, where the x in
Ax[P(x)] can legitimately be instantiated to every term
in the domain, yielding a sentence, then Ax[P(x)] would HAVE
to be true whenever all-this-infinity-of-premises were, and my point is
that it is reasonable for people to be (originally/naively)
SURPRISED that this is not a valid inference.
Its non-validity DOES depend both on the compactness theorem
and on the existence of non-named elements.
Perhaps the naive are more confused by the outlawing
of infinitary inference rules, especially when the infinity-being-
inferred-from is the smallest one and the infinite things are as
"homogeneous" as all natural numbers. They think surely you
OUGHT to be able to infer, that, regardless of whether it's "provable",
truth MUST in fact follow, in this case. The most pedagogically
effective
way to show that it doesn't is by exhibiting a nonstandard model.
But I already regret trying this because the article is in fact flawed
in even deeper ways that Daryl suggested. The article's author
seems to allege that "the induction principle" is ABOUT natural
numbers,
inherently. That it is about first-order formulas and models
generally,
which might or might NOT be natural (and might or might not be
inductive,
and that the two questions are partly independent), was completely lost
on
the original author.
sradhakr
In classical logic, it is assumed that (1) is a meaningful assertion
with n taken successively as 0, 1, 2, etc., i.e., it is assumed that
one can run through "all" the natural numbers by thinking about them
one (or finitely many) at a time. I submit that this is false. A simple
induction shows that one cannot exhaust the natural numbers by listing
them one at a time.
In fact we can *know* that (1) is true, and hence a meaningful
assertion, if and only if:
(3) We can prove P(n) for an *arbitrary* natural number n, where
n is left in symbolic form.
Hence there *should* really be no distinction between (1) and (3), a
distinction which classical first-order logic dubiously tries to
maintain. By the rule of universal generalization, (3) *should be* the
same as (2), and so nonstandard models of arithmetic *ought* not to
exist. This is exactly what my proposed logic NAFL asserts. See the
following links:
http://philsci-archive.pitt.edu/archive/00001666
http://philsci-archive.pitt.edu/archive/00001923
http://arxiv.org/abs/math.LO/0506475
I am still wating for a substantial response from the academic
community on the above work.
Regards, RS
Torkel Franzen
> Well, the compactness theorem says it's irrelevant.
Why go on about the compactness theorem and completeness theorem for
first order logic? Let's just forget about first order logic and ask
ourselves whether (1) and (2) are equivalent. (2) clearly implies (1)
(in an idealized sense of "can prove"). What about the other
direction? If Goldbach's conjecture is true, we can prove (again in an
idealized sense) "2n+2 is the sum of two primes" for any given natural
number n. Does it follow that if Goldbach's conjecture is true, we can
prove "For every n, 2n+2 is the sum of two primes"? Not in any obvious
way. (2), it seems, is a stronger statement than (1).
sradhakr
sradhakr
"Goldbach's conjecture is true" is a meaningful statement if and only
if one can prove it for an arbitrary natural number n, where n is in
symbolic form. Hence there is no distinction between (1), "Goldbach's
conjecture is true", (2) and
(3) We can prove Goldbach's conjecture for an arbitrary natural number
n, where n is in symbolic form.
Regards, RS
David C. Ullrich
That's silly. Look at any book on logic - they all contain
proofs of Godel's theorem. Do you have some suspicion that
_all_ those authors are incompetent? (But although you
don't believe _any_ of those many authors you're going to
believe the opinion you read here on sci.logic???)
>
************************
David C. Ullrich
Daryl McCullough
>> > > But there is a big difference between
>> > >
>> > > (1) For all n, we can prove P(n).
>> > >
>> > > and
>> > >
>> > > (2) We can prove "for all n, P(n)"
>In fact we can *know* that (1) is true, and hence a meaningful
>assertion, if and only if:
>
> (3) We can prove P(n) for an *arbitrary* natural number n, where
>n is left in symbolic form.
Yes, you're right. The only way to know that (1) is true is if
something like (2) holds. However, that wasn't the point. The
point is that (1) can be *true* without our *knowing* that it
is true.
Imagine that you have some formula Phi(n). For example, n might
be the statement "2*n is a number that can be expressed as the
sum of two prime numbers". You can check to see that Phi(0) is
true, you can check to see that Phi(1) is true. For any n, you
can check whether Phi(n) is true. Yet there may be no known way to
check whether "forall n, Phi(n)" is true.
>Hence there *should* really be no distinction between (1) and (3), a
>distinction which classical first-order logic dubiously tries to
>maintain.
Why do you say that there should be no distinction?
Daryl McCullough
>"Goldbach's conjecture is true" is a meaningful statement if and only
>if one can prove it for an arbitrary natural number n, where n is in
>symbolic form.
Why do you say that? Why must a statement be provable to be meaningful?
That's an odd notion of "meaning".
sradhakr
> sradhakr says...
>
> >> > > But there is a big difference between
> >> > >
> >> > > (1) For all n, we can prove P(n).
> >> > >
> >> > > and
> >> > >
> >> > > (2) We can prove "for all n, P(n)"
>
> >In fact we can *know* that (1) is true, and hence a meaningful
> >assertion, if and only if:
> >
> > (3) We can prove P(n) for an *arbitrary* natural number n, where
> >n is left in symbolic form.
>
> Yes, you're right. The only way to know that (1) is true is if
> something like (2) holds. However, that wasn't the point. The
> point is that (1) can be *true* without our *knowing* that it
> is true.
This is the classical viewpoint, which has been disputed. For example,
I believe that Neil Tennant has asserted in his book ("The taming of
the true"?) that all truths are knowable. The issue here is whether one
can meaningfully talk of "truths" that are *in principle* unknowable.
Please see my argument below.
>
> Imagine that you have some formula Phi(n). For example, n might
> be the statement "2*n is a number that can be expressed as the
> sum of two prime numbers". You can check to see that Phi(0) is
> true, you can check to see that Phi(1) is true. For any n, you
> can check whether Phi(n) is true.
Let us take this real slow. What exactly do you mean by the assertion
that
"For any n, you can check whether Phi(n) is true"?
What you obviously mean here is that you can check Phi(n) for
infinitely many *instances* of n, taken one at a time and exhaust the
class N of natural numbers. This is precisely what I am saying is
meaningless. Every time you check Phi(n) for a given instance, there
are infinitely many instances remaining to be checked; so you really
cannot exhaust N this way. So I don't accept that you can meaningfully
make the above assertion, unless and until what you say below is false:
>Yet there may be no known way to check whether "forall n, Phi(n)" is true.
In other words, I dispute your assertion that "If one continues
checking the truth of Phi(n), for each instance of n, taken one at a
time, one will never ever find a counter-example" is meaningful; it is
an infinite process that can never be completed *in principle*; what
you are really saying is that we have some means of checking (proving)
the truth of Phi(n) for an *arbitrary* (unspecified) n, which would
obviously prove "For all n Phi(n)". Of course, classical logicians (you
included) will not agree with me, but what I have said above actually
holds in my proposed logic NAFL and follows from its postulates.
>
> >Hence there *should* really be no distinction between (1) and (3), a
> >distinction which classical first-order logic dubiously tries to
> >maintain.
>
> Why do you say that there should be no distinction?
Please see my explanation above. Actually the nonexsitence of
nonstandard models follows in my proposed logic NAFL from other
considerations than what I have argued above, which may be thought of
as an intuitive justification for the postulate(s) of NAFL.
Regards, RS
sradhakr
To be precise, I am saying that that the *truth* of a fprmal
arithmetical proposition (such as, Goldbach's conjecture, GC) can only
be meaningfully asserted if one has a proof of that proposition in
Peano Arithmetic. This follows in my proposed logic NAFL, since NAFL
does not admit nonstandard models of PA. In this specific case, the
assertion that "GC is true" means that each instance of GC is true,
which means that each instance of GC is provable in PA. So for the rest
of the argument, please see my other reply.
Regards, RS
Daryl McCullough
>
>Daryl McCullough wrote:
>> Yes, you're right. The only way to know that (1) is true is if
>> something like (2) holds. However, that wasn't the point. The
>> point is that (1) can be *true* without our *knowing* that it
>> is true.
>
>This is the classical viewpoint, which has been disputed. For example,
>I believe that Neil Tennant has asserted in his book ("The taming of
>the true"?) that all truths are knowable. The issue here is whether one
>can meaningfully talk of "truths" that are *in principle* unknowable.
You can certainly meaningfully talk about it. People certainly
have talked about it.
Daryl McCullough
>
>Daryl McCullough wrote:
>> sradhakr says...
>>
>> >"Goldbach's conjecture is true" is a meaningful statement if and only
>> >if one can prove it for an arbitrary natural number n, where n is in
>> >symbolic form.
>>
>> Why do you say that? Why must a statement be provable to be meaningful?
>> That's an odd notion of "meaning".
>To be precise, I am saying that that the *truth* of a fprmal
>arithmetical proposition (such as, Goldbach's conjecture, GC) can only
>be meaningfully asserted if one has a proof of that proposition in
>Peano Arithmetic.
I know you are saying that, but I'm saying that you are wrong.
We can meaningfully talk about the truth of sentences that are
not provable.
george
Torkel Franzen wrote:
> "george" <gre...@cs.unc.edu> writes:
>
> > Well, the compactness theorem says it's irrelevant.
>
> Why go on about the compactness theorem and completeness theorem for
> first order logic?
Because it's relevant and it's the answer, that's why.
> Let's just forget about first order logic and ask
> ourselves whether (1) and (2) are equivalent. (2) clearly implies (1)
> (in an idealized sense of "can prove"). What about the other
> direction?
Exactly. I figured this out later while falling asleep.
Yours is the correct short introduction (obviously you're
way way better at explaining things than I am, you just
aren't always willing). So, following you, lemme try again--
The newbie mistakenly thinks (1) and (2) are equivalent.
That (2)->(1) is obvious.
Thus, the newbie mistakenly thinks that (1)->(2).
Why is this a mistake?
My point stands that if the things mentioned in (1)
are all and only the things there are, then the inference
(1)->(2) IS valid. Unfortunately, you can't make it valid
at first order because you can't say "these {0,s(0),s(s(0)),...}
are all & only the things there are" at first order. In order for
(1)->(2) to fail, there have to be more things (than just the
things quantified over in the meta-language in (1))
in the universe.
The newbie's first defense will be to insist that the axioms
show that there CAN'T be more things than natural numbers
in the universe. You have to actually exhibit a
non-standard model to convince him otherwise.
Before you go that far, though, you could just add a whole
new language with some new terms and say "even without
the rest of PA, if you just have P(0), P(s(0)), etc., it should be
obvious that you can't infer P(t(t(t0))), where t is some other
functor completely unrelated to s". That is sort of an indirect
appeal to the compactness theorem (explaining why an infinity
of premises doesn't "help" you infer "more" theorems).
Charlie-Boo
LordBeotian wrote:
> I would like to read some opinions about this:
>
> http://front.math.ucdavis.edu/math.GM/0510469
The fallacy is on page 5, statement (12):
(all X)BEW(P(X)) => Bew( (all X)P(X) ) (12)
i.e. call it:
[ (all X)|-P(X) ] => [ |- (all X)P(X) ]
and note that it is equivalent to (substitute <=> for =>):
[ (all X)|-P(X) ] <=> [ |- (all X)P(X) ] (12.1)
Godel's proof (using w-consistency) is actualy proving that (12.1) is
false, which it is. (all X)|-P(X) is not a recursively enumerable
relation over the Godel number of wff P. If it were true, then the set
of Turing Machines that always halt yes would be recursively enumerable
(making the Halting Problem solvable.)
Similar to Godel's proof: The relation "X is not a proof of Y" is
recursive. Call it NPR. If (12.1) were true, then substitute NPR(X,Y)
for P(X) where Y is any constant that we wish to use, to produce:
(all X) |- NPR(X,Y) iff |- (all X) NPR(X,Y)
Since (all X) |- NPR(X,Y) means that Y is not provable, and |- (all X)
NPR(X,Y) is recursively enumerable (there is a wff that is provable iff
the given relation is true) since |- w for any wff w is, the above
would mean that the set of unprovable statements is recursively
eumerable (which of course is not true.)
C-B
Charlie-Boo
> Daryl McCullough wrote:
> > But there is a big difference between
> >
> > (1) For all n, we can prove P(n).
> >
> > and
> >
> > (2) We can prove "for all n, P(n)"
>
> Not if you don't know that there can be models
> with things other than natural numbers in them, there
> isn't. . . . the possibility of the object-
> language's (2) "for all n" ranging over things that might be
> greater than all natural numbers simply has not occurred
> to BUNCHES of people.
They're not talking about that shit. They're talking about the natural
numbers as Godel was.
People like to take a perfecly good result and then claim that some
really stupid weird thing trumps them. Like when I made a comment
about Turing Machines and Frege makes some stupid-ass remark (as usual)
about some jerk-off who tried to redefine what the term "Turing
Machine" means.
IT DOESN'T HAVE ANYTHING TO DO WITH NON-NATURAL NUMBERS
C-B
> All I'm trying to say is that unawareness of non-
> standard models of PA and supernatural numbers is not
> the same thing as mathematical incompetence.
Nor is unawareness of Peskini's Theorem of Thermal Dynamic
Oscillations.
Charlie-Boo
Bravo!
(Because he thinks it makes him look smart. Only to stupid people.)
C-B
Charlie-Boo
> On Wed, 9 Nov 2005 18:26:29 +0100, "LordBeotian" <pokispy76@[CANCELLA
> QUESTO]yahoo.it> wrote:
> >I need someone who looks competent to say me that the author of the
> >article is wrong and Godel's theorem is safe...
>
> That's silly. Look at any book on logic - they all contain
> proofs of Godel's theorem. Do you have some suspicion that
> _all_ those authors are incompetent?
They're all just repeating (more or less) the same argument, so it
doesn't matter how many do it. It's not like they actually thought
through the argument themselves. How many of them even address the
theorem itself, but rather stick to the lesser result given in the
introduction, requiring soundness, rather than the stronger result
based on w-consistency? And fewer even mention Rosser's improvement,
much less explain or understand that.
> (But although you
> don't believe _any_ of those many authors you're going to
> believe the opinion you read here on sci.logic???)
Do you still believe Hilbert?
"In questions of science, the authority of a thousand is not worth the
humble reasoning of a single individual." - Galileo Galilei
As usual, you add no mathematical content and appeal only to people's
emotions and prejudices.
(See my formal refutation elsewhere.)
C-B
> ************************
>
> David C. Ullrich
george
sradhakr wrote:
> To be precise, I am saying that that the *truth* of a fprmal
> arithmetical proposition (such as, Goldbach's conjecture, GC) can only
> be meaningfully asserted if one has a proof of that proposition in
> Peano Arithmetic.
That's ridiculous.
> This follows in my proposed logic NAFL,
Please. The rest of us are talking about proving things
from PA in classical FOL. If you are going to use a whole
new logic then obviously other things are going to be possible.
The universal quantifer already has a semantics.
If you want to use a different one then you are going to need
a lot of luck defining it, articulating it, and conveying it.
> since NAFL
> does not admit nonstandard models of PA.
So you allege.
> In this specific case, the
> assertion that "GC is true" means that each instance of GC is true,
> which means that each instance of GC is provable in PA.
If GC was true, then each instance of it was ALREADY
provable in PA, EVEN if GC itself was NOT!
Truth and provability are inherently different.
Your calling them the same is not going to make
them the same. Where we come from, proofs HAVE to be
finitary. If you are going to allow infinitary proofs, then fine,
but that is not necessarily new or interesting (other people
are already working on infinitary logic). It is not going to
give anybody any insight into what remains accomplishable
with finitary inference rules, nor is it going to de-legitimize
any non-standard model that is already constructed.
Charlie-Boo
More carefully? Seems to think? Since the title says that he is
refuting Godel's claim that incomplete systems exist, how long does it
take one to conclude that he thinks that all systems are complete?
GAD!!!
C-B
Let me guess: You thought that you could make a vague unsubstantiated
statement that sounds smart, and if anyone challenges it you just give
some more vague nonspecific reasons and if you are finally proven wrong
you can fall back on the final qualifier, "seems to think" and say you
were only wrong abut reading his mind?
Wait a second - I think I have it. You figured you'd throw out some
known result that may be related to the subject matter, of course
without showing how it actually results in anything. So you grabbed
Godel's theorem, forgetting that the issue you were addressing was the
futation of that very theorem! HA!!
One last guess: You're a professor, right? Huh? Huh?
george
sradhakr wrote:
> "Goldbach's conjecture is true" is a meaningful statement if and only
> if one can prove it for an arbitrary natural number n,
That's WAY confused.
If Goldbach's conjecture is true, then ONE CERTAINLY
CAN prove it for EVERY natural number n.
But one still may be UNable to prove GC *itself*!
All THAT is true completely IRrespective of whether ANYthing
is meaningful.
> where n is in symbolic form.
I hope you mean something like "Peano Normal" form
(i.e. unary, or n represented by n applications of the same functor).
> Hence there is no distinction between (1), "Goldbach's
> conjecture is true", (2) and
>
> (3) We can prove Goldbach's conjecture for an arbitrary natural number
> n, where n is in symbolic form.
There is inherently, necessarily, a distinction.
1 involves 1 proof of 1 finite generalization;
3 involves infinitely many different proofs.
Torkel Franzen
> My point stands that if the things mentioned in (1)
> are all and only the things there are, then the inference
> (1)->(2) IS valid.
(1) only says that we can prove A(n) for every n, not that it is
in any sense whatever provable that we can prove A(n) for every n.
Charlie-Boo
> > > > But there is a big difference between
> > > >
> > > > (1) For all n, we can prove P(n).
> > > >
> > > > and
> > > >
> > > > (2) We can prove "for all n, P(n)"
> The issue is
> whether being able to prove (1) is or isn't "like"
> being able to prove (2).
No, it's whether (1) is true iff (2) is true.
C-B
Charlie-Boo
> "george" <gre...@cs.unc.edu> writes:
>
> > Well, the compactness theorem says it's irrelevant.
>
> Why go on about the compactness theorem and completeness theorem for
> prove "For every n, 2n+2 is the sum of two primes"? Not in any obvious
> way. (2), it seems, is a stronger statement than (1).
Why go on about Goldbach's conjecture - do you think that the set of
Turing Machines that halt on all inputs is recursively enumerable?
C-B
Charlie-Boo
> sradhakr says...
> >> > > (1) For all n, we can prove P(n).
> >> > >
> >> > > and
> >> > >
> >> > > (2) We can prove "for all n, P(n)"
> The
> point is that (1) can be *true* without our *knowing* that it
> is true.
Not without "knowing" it meaning |-(aA)|-P(A) but without "knowing"
that |-(aA)P(A) but maybe you were assuming . . .
C-B
Charlie-Boo
I think he's referring to the fact that if it is true of a given number
then it is provable of that number because it is a recursively
enumerable (even recursive) predicate.
C-B
LordBeotian
"Charlie-Boo" <ch...@aol.com> ha scritto
>> Ok, I have read it more carefully and it in fact seems meaningless...
>> The author seems to think that every axiomatic system is complete.
>
> More carefully? Seems to think? Since the title says that he is
> refuting Godel's claim that incomplete systems exist,
Not exactly: he just says that the *proof* is invalid, not the conclusion.
> how long does it
> take one to conclude that he thinks that all systems are complete?
The problem is that his argument rests on the "a priori" assumption that
every system is complete (formalized into his version of "induction").
> Let me guess: You thought that you could make a vague unsubstantiated
> statement that sounds smart, and if anyone challenges it you just give
> some more vague nonspecific reasons and if you are finally proven wrong
> you can fall back on the final qualifier, "seems to think" and say you
> were only wrong abut reading his mind?
Oh my God!! What's your problem? Do you usually need to prove to be smarter
then other people?
Charlie-Boo
> sradhakr wrote:
> > In this specific case, the
> > assertion that "GC is true" means that each instance of GC is true,
> > which means that each instance of GC is provable in PA.
>
> If GC was true, then each instance of it was ALREADY
> provable in PA, EVEN if GC itself was NOT!
> Truth and provability are inherently different.
> Your calling them the same is not going to make
> them the same.
He didn't say that GC had to be provable, only premising that it is
true.
C-B
george
Torkel Franzen wrote:
> "george" <gre...@cs.unc.edu> writes:
>
> > My point stands that if the things mentioned in (1)
> > are all and only the things there are, then the inference
> > (1)->(2) IS valid.
>
> (1) only says that we can prove A(n) for every n,
Of course.
> not that it is in any sense whatever
Be careful. If you say "any sense whatever" then
I get to supply MY OWN senses, tailored for my arguments.
> provable that we can prove A(n) for every n.
Obviously it IS provable IF you allow infinitary inference rules.
It is even more provable if it in fact IS provable, REGARDLESS
of whether (1) SAYS so!
Let's try this with an upper bound of 7, as opposed to w:
Then it becomes "(1) says that we can prove A(n) for
every n less than 7, not that it is in any sense whatever
provable that we can prove A(n) for every n less than 7."
The ludicrosity of that is obvious; the 7 proofs of A(n)
for every n less than 7 jointly CONSTITUTE a proof that
it is provable for that we can prove A(n) for every
n less than 7. Every proof of everything constitutes a proof
that that thing is provable. That's just basic existential
generalization.
The question remains as to why this doesn't apply for w, when
it DOES apply for all lesser ordinals. One available answer is
"because proofs have to be finite, because I said so."
The other is that the counter-models are ontologically committed
to more things than
the particular infinity from which the premises are drawn.
Charlie-Boo
> sradhakr wrote:
> > "Goldbach's conjecture is true" is a meaningful statement if and only
> > if one can prove it for an arbitrary natural number n,
>
> That's WAY confused.
> If Goldbach's conjecture is true, then ONE CERTAINLY
> CAN prove it for EVERY natural number n.
> But one still may be UNable to prove GC *itself*!
He didn't say anythng about proving GC itself.
> > Hence there is no distinction between (1), "Goldbach's
> > conjecture is true" . . . and
> > (3) We can prove Goldbach's conjecture for an arbitrary natural number
> > n, where n is in symbolic form.
>
> There is inherently, necessarily, a distinction.
> 1 involves 1 proof of 1 finite generalization;
> 3 involves infinitely many different proofs.
Who says that makes any difference? (1) is certainly equivalent to
(3), don't you think?
C-B
george
> > sradhakr wrote:
> > > In this specific case, the
> > > assertion that "GC is true" means that each instance of GC is true,
> > > which means that each instance of GC is provable in PA.
> >
> > If GC was true, then each instance of it was ALREADY
> > provable in PA, EVEN if GC itself was NOT!
> > Truth and provability are inherently different.
> > Your calling them the same is not going to make
> > them the same.
Charlie-Boo wrote:
> He didn't say that GC had to be provable,
Yes, he did.
> only premising that it is true.
Yes, but he then EQUATED truth and provability:
> > > each instance of GC is true,
> > > which means that each instance of GC is provable in PA.
More to the point, you can't say he is "only premising" that a
thing is "true", when he is REDEFINING "true"! In the dictionary
in use by the rest of us, a thing's being "true" does NOT entail
its being provable in PA!
>
> C-B
george
Charlie-Boo wrote:
> Why go on about Goldbach's conjecture - do you think that the set of
> Turing Machines that halt on all inputs is recursively enumerable?
That's not at all relevant; that's a much easier problem,
easier in the sense that the answer is already known.
Charlie-Boo
> "Charlie-Boo" <ch...@aol.com> ha scritto
>
> >> Ok, I have read it more carefully and it in fact seems meaningless...
> >> The author seems to think that every axiomatic system is complete.
> > More carefully? Seems to think? Since the title says that he is
> > refuting Godel's claim that incomplete systems exist,
>
> Not exactly: he just says that the *proof* is invalid, not the conclusion.
Interesting distinction, but if he thinks he has an alternate proof
then the title is VERY misleading (which refers to the theorem and not
the proof - HA!) and is underselling his own discovery or whatever
makes him think that the theorem is true for a different reason.
> > Let me guess: You thought that you could make a vague unsubstantiated
> > statement that sounds smart, and if anyone challenges it you just give
> > some more vague nonspecific reasons and if you are finally proven wrong
> > you can fall back on the final qualifier, "seems to think" and say you
> > were only wrong abut reading his mind?
>
> Oh my God!! What's your problem? Do you usually need to prove to be smarter
> then other people?
No, I druther prove the true proposition that many people obfuscate an
issue by dragging in irrelevant principles. Great example: You solve a
problem in a very nice short way and submit a paper. The referee (a
member of the Old Boy's Club) sees you're not affiliated with a famous
school, and so says that you left out talking about some BS term that
someone else wrote about - even though you solved the problem without
it and Occam's Razor, of course.
If that really wasn't your intention, then my apologies. But lots of
people do that e.g. the earlier references to non-standard models of
Peano's Axioms.
C-B
Charlie-Boo
That's the point. It's easier and it answers the question
(speculation) as to whether (1) and (2) are equivalent. They aren't.
(A solution is certainly relevant to a problem.)
C-B
george
sradhakr wrote:
> This is the classical viewpoint, which has been disputed.
Never coherently.
> For example,
> I believe that Neil Tennant has asserted in his book ("The taming of
> the true"?) that all truths are knowable.
Then his definition of truth simply does not match the classical one.
> The issue here is whether one
> can meaningfully talk of "truths" that are *in principle* unknowable.
I don't know that anyone has ever alleged the existence of
such a truth. Truths that are in principle unprovable in PA or
ZFC are always going to be provable in some stronger system.
So if knowledge is being linked to provability then there simply
ARE NO truths that are "in principle unknowable". The first thing
you have to do to motivate THIS argument is find someone foolish
enough to allege that some particular mathematical thing is
"in principle unknowable". We can then thereAFTER dispute whether
there is any sense in which that "thing" could ALSO be a "truth".
> Please see my argument below.
> >
> > Imagine that you have some formula Phi(n). For example, n might
> > be the statement "2*n is a number that can be expressed as the
> > sum of two prime numbers". You can check to see that Phi(0) is
> > true, you can check to see that Phi(1) is true. For any n, you
> > can check whether Phi(n) is true.
>
> Let us take this real slow. What exactly do you mean by the assertion
> that
>
> "For any n, you can check whether Phi(n) is true"?
I mean that if you give me an n, I can check it, and it
doesn't matter what n you give me. And I won't need
infinite resources to check it, either. Some finite amount
of resources r(n) will ALWAYS suffice. r(.) might not
be recursive, though.
> What you obviously mean here is that you can check Phi(n) for
> infinitely many *instances* of n, taken one at a time
Right.
> and exhaust the class N of natural numbers.
Wrong. I am not claiming the ability to exhaust anything.
I am just saying that the "I can handle any 1" challenge
applies (exhaustively) throughout n. This does NOT mean
that *I* can exhaust anything.
> This is precisely what I am saying is meaningless.
Well, I'm sorry, you CAN'T SAY that.
The components have meanings. They combine
properly. I can in fact DO exactly this, so you can't claim
that the words that describe what I'm doing are meaningless;
I'm refuting claim *BY DOING* it.
> Every time you check Phi(n) for a given instance, there
> are infinitely many instances remaining to be checked; so you really
> cannot exhaust N this way.
Of course I can't, and I conceded that, and that has
NOT ONE IOTA OF IMPACT on the TRUTH or the MEANING
of what I originally said.
> So I don't accept that you can meaningfully
> make the above assertion,
Whether you accept it or not is irrelevant;
neither the meaning nor the satisfaction of the assertion
depends on your acceptance.
> unless and until what you say below is false:
>
> >Yet there may be no known way to check whether "forall n, Phi(n)" is true.
Classically there is simply NO inferential connection between
these two things. All the instances, for all n, even if we COULD
exhaust
them, could all come up true, and "forall n, Phi(n)" could STILL come
up
FALSE, because the "forall" INside the object language ranges over
MORE things than the "for all" out in the meta-language did.
> In other words, I dispute your assertion that "If one continues
> checking the truth of Phi(n), for each instance of n, taken one at a
> time, one will never ever find a counter-example" is meaningful;
Please; if it is false, that IS meaningful, it is just false,
and it is false IN VIRTUE OF its meaning, so you can't say
it doesn't have a meaning. The only possible way you could
be right in asserting that it is meaningless via THIS objection
is IF IT IS TRUE, but the fact that we are CONFIRMING ITS TRUTH
in this case means that it MUST have a meaning IN THIS CASE AS WELL.
So Shut Up.
> it is
> an infinite process that can never be completed *in principle*;
NO, it is NOT a process AT ALL, because we ARE NOT alleging
any TIME or any degree of difficulty in ANY of this. ALL the true
instances are true ALREADY, REGARDLESS of whether anybody
has "confirmed" or checked ANY of them YET. All the valid
proofs of all the instances exist ALREADY. No further process
IS REQUIRED. This boneheaded invocation of physicalist metaphors
is just that. We live in a place without time or place. DEAL WITH
THAT.
> what
> you are really saying is that we have some means of checking (proving)
> the truth of Phi(n) for an *arbitrary* (unspecified) n,
NO, I am NOT saying that. I am CONTRASTING this with that.
> which would
> obviously prove "For all n Phi(n)".
Obviously, but that's NOT WHAT WE'RE SAYING.
> Of course, classical logicians (you
> included) will not agree with me, but what I have said above actually
> holds in my proposed logic NAFL
Please. It CAN'T POSSIBLY hold.
It's not soomething you get to have an opinion about.
We are already talking about OUR logic.
Our logic actually has the property that infinitely many
instances of something can be individually provable.
If yours doesn't then good for you, but even there,
you can't deny that the concept is MEANINGFUL.
> and follows from its postulates.
I doubt that, frankly.
> > >Hence there *should* really be no distinction between (1) and (3),
Even if there shouldn't, there is.
If you know how to eliminate it, please
concentrate on coherently explaining YOUR
alternative logic instead of irrelevantly alleging
shortcomings in classical. Classical just is the way
it is; no value judgments required, and ESPECIALLY
no rewriting of the English dictionary ALLOWED.
> > > a
> > >distinction which classical first-order logic dubiously tries to
> > >maintain.
"dubiously" my butt.
Charlie-Boo
> sradhakr wrote:
> > The issue here is whether one
> > can meaningfully talk of "truths" that are *in principle* unknowable.
>
> I don't know that anyone has ever alleged the existence of
> such a truth. Truths that are in principle unprovable in PA or
> ZFC are always going to be provable in some stronger system.
> So if knowledge is being linked to provability then there simply
> ARE NO truths that are "in principle unknowable".
"George does not believe this." is true but you'll never know it. (I
know it, however.)
"Nobody believes this." is true and nobody will ever know it.
So there!
C-B
Torkel Franzen
> The other is that the counter-models are ontologically committed
> to more things than
> the particular infinity from which the premises are drawn.
Models of first order theories are not at issue. It's just a simple
observation that there is no apparent way of concluding from "for
every n, we can prove A(n)" to "we can prove that for every n, A(n)",
and so the latter statement is on the face of it a stronger one.
sradhakr
george wrote:
> sradhakr wrote:
> > To be precise, I am saying that that the *truth* of a fprmal
> > arithmetical proposition (such as, Goldbach's conjecture, GC) can only
> > be meaningfully asserted if one has a proof of that proposition in
> > Peano Arithmetic.
>
> That's ridiculous.
Why?
>
> > This follows in my proposed logic NAFL,
>
> Please. The rest of us are talking about proving things
> from PA in classical FOL. If you are going to use a whole
> new logic then obviously other things are going to be possible.
> The universal quantifer already has a semantics.
> If you want to use a different one then you are going to need
> a lot of luck defining it, articulating it, and conveying it.
Well, I have already done it. It is upto the academic community to
respond. And sci.logic is not just about discussion of FOL.
>
> > since NAFL
> > does not admit nonstandard models of PA.
>
> So you allege.
>
Take a look at my work and refute it or criticize it. E.g. take a look
at:
http://arxiv.org/abs/math.LO/0506475
> > In this specific case, the
> > assertion that "GC is true" means that each instance of GC is true,
> > which means that each instance of GC is provable in PA.
>
> If GC was true, then each instance of it was ALREADY
> provable in PA, EVEN if GC itself was NOT!
> Truth and provability are inherently different.
> Your calling them the same is not going to make
> them the same. Where we come from, proofs HAVE to be
> finitary. If you are going to allow infinitary proofs, then fine,
> but that is not necessarily new or interesting (other people
> are already working on infinitary logic). It is not going to
> give anybody any insight into what remains accomplishable
> with finitary inference rules, nor is it going to de-legitimize
> any non-standard model that is already constructed.
Of course all proofs have to be finitary. All I'm saying here is if
each instance of GC were to be provable in PA, then there must exist a
proof of GC in PA. Any argument that alleges the contrary (e.g. Goedel,
Turing) is inherently part of infinitary reasoning (not acceptable in
NAFL) and leads to infinitary conclusions (nonstandard models, infinite
sets) that are not acceptable in NAFL. Take or leave it or refute it or
deal with it.
Regards, RS
sradhakr
> sradhakr wrote:
> > "Goldbach's conjecture is true" is a meaningful statement if and only
> > if one can prove it for an arbitrary natural number n,
>
> That's WAY confused.
> If Goldbach's conjecture is true, then ONE CERTAINLY
> CAN prove it for EVERY natural number n.
> But one still may be UNable to prove GC *itself*!
> All THAT is true completely IRrespective of whether ANYthing
> is meaningful.
>
> > where n is in symbolic form.
>
> I hope you mean something like "Peano Normal" form
> (i.e. unary, or n represented by n applications of the same functor).
>
as a free variable (or an "arbitrary, but fixed" constant, which is the
same thing) in the open formula expressing GC (without the
quantifiers). Once you have proved that formula, then by universal
generalization, GC has been proved. For example, I can prove the
formula
(n+1)^2=n^2+2n+1
in PA, where n is arbitrary, unspecified natural number. By universal
generalization, this formula must hold for all natural numbers n. What
you are alleging is that the open formula expressing GC in terms of n
(without the quantifiers) may not be provable in PA, but `all'
instances of it for n=0,1,2,.. may be provable. This is what I claim is
a purely infnitary assertion that would require infinitary reasoning to
prove. It is open to me to reject it and formulate an alternative
system that uses strctly finitary methods and arrives at strictly
finitary conclusions.
> > Hence there is no distinction between (1), "Goldbach's
> > conjecture is true", (2) and
> >
> > (3) We can prove Goldbach's conjecture for an arbitrary natural number
> > n, where n is in symbolic form.
>
> There is inherently, necessarily, a distinction.
> 1 involves 1 proof of 1 finite generalization;
> 3 involves infinitely many different proofs.
No, 3 involves just one finitary proof -- see above.
Regards, RS
Charlie-Boo
> sradhakr wrote:
> > This follows in my proposed logic NAFL,
> Please. The rest of us are talking about proving things
> from PA in classical FOL. . . .
> If you want to use a different one then you are going to need
> a lot of luck defining it, articulating it, and conveying it.
Why stand in the way of progress? Let him introduce his improvement
over conventional wisdom. Don't you know that all scientific beliefs
are eventually replaced by more detailed and refined ones that show
that the previous beliefs weren't completely correct? The earth is not
at the center of our solar system and there is an ignoramibus.
> > since NAFL
> > does not admit nonstandard models of PA.
>
> So you allege.
As do you.
> If you are going to allow infinitary proofs, then fine,
> but that is not necessarily new or interesting (other people
> are already working on infinitary logic).
If other people are working on it, then it is still not solved. So why
not let him work on it too, and perhaps provide the solution? If
people are working on it, then why should he be excluded? That is a
reason for him to be allowed, not a reason to artificially exclude his
improvement.
> It is not going to
> give anybody any insight into what remains accomplishable
> with finitary inference rules, nor is it going to de-legitimize
> any non-standard model that is already constructed.
How do you know? You haven't even seen it yet.
C-B
sradhakr
george wrote:
> sradhakr wrote:
> > This is the classical viewpoint, which has been disputed.
>
> Never coherently.
>
> > For example,
> > I believe that Neil Tennant has asserted in his book ("The taming of
> > the true"?) that all truths are knowable.
>
> Then his definition of truth simply does not match the classical one.
>
on Dummett's version of intuitionism, which I don't really agree with).
> > The issue here is whether one
> > can meaningfully talk of "truths" that are *in principle* unknowable.
>
> I don't know that anyone has ever alleged the existence of
> such a truth. Truths that are in principle unprovable in PA or
> ZFC are always going to be provable in some stronger system.
So how do you know that the "stronger system" is consistent? E.g. let
us take FLT, which Wiles has proved in a stronger system than PA
(something intermediate between PA and ZF). Has he established the
truth of FLT, i.e., can we be certain that counter-examples to FLT will
never be found? Certainly not. If FLT is undecidable in PA (which is
something that we cannot prove with certainty) then we can never ever
know whether it is true or false. As far as my logic NAFL is concerned,
your alleged "stronger system" doesn't exist and it is a requirement of
the consistency of PA that PA-undecidable propositions (which lead to
nonstandard models of PA) do not exist.
> So if knowledge is being linked to provability then there simply
> ARE NO truths that are "in principle unknowable". The first thing
> you have to do to motivate THIS argument is find someone foolish
> enough to allege that some particular mathematical thing is
> "in principle unknowable". We can then thereAFTER dispute whether
> there is any sense in which that "thing" could ALSO be a "truth".
See my argument above.
>
> > Please see my argument below.
> > >
> > > Imagine that you have some formula Phi(n). For example, n might
> > > be the statement "2*n is a number that can be expressed as the
> > > sum of two prime numbers". You can check to see that Phi(0) is
> > > true, you can check to see that Phi(1) is true. For any n, you
> > > can check whether Phi(n) is true.
> >
> > Let us take this real slow. What exactly do you mean by the assertion
> > that
> >
> > "For any n, you can check whether Phi(n) is true"?
>
> I mean that if you give me an n, I can check it, and it
> doesn't matter what n you give me. And I won't need
> infinite resources to check it, either. Some finite amount
> of resources r(n) will ALWAYS suffice. r(.) might not
> be recursive, though.
Nope. If "I give you an n" and you check the truth of Phi(n), that is
just one instance of the above assertion, which therefore has not been
established. I can repeat this any number of times, and the above
assertion still has not established, It has been established "after" I
have given you "all" possible values of n, this is what I am saying is
not a meaningful assertion.
>
>
> > What you obviously mean here is that you can check Phi(n) for
> > infinitely many *instances* of n, taken one at a time
>
> Right.
>
> > and exhaust the class N of natural numbers.
>
> Wrong. I am not claiming the ability to exhaust anything.
> I am just saying that the "I can handle any 1" challenge
> applies (exhaustively) throughout n. This does NOT mean
> that *I* can exhaust anything.
No, the above assertion is estabilshed as true only "after" N has been
exhausted.
>
> > This is precisely what I am saying is meaningless.
>
> Well, I'm sorry, you CAN'T SAY that.
> The components have meanings. They combine
> properly. I can in fact DO exactly this, so you can't claim
> that the words that describe what I'm doing are meaningless;
> I'm refuting claim *BY DOING* it.
No, you are not.
>
> > Every time you check Phi(n) for a given instance, there
> > are infinitely many instances remaining to be checked; so you really
> > cannot exhaust N this way.
>
> Of course I can't, and I conceded that, and that has
> NOT ONE IOTA OF IMPACT on the TRUTH or the MEANING
> of what I originally said.
Nope. What you originally said, namely that "if I give you an n", you
can check the truth of Phi(n), is established only "after" I have
"given you" infinitely many values of n, one at a time, and exhausted
N.
>
> > So I don't accept that you can meaningfully
> > make the above assertion,
>
> Whether you accept it or not is irrelevant;
> neither the meaning nor the satisfaction of the assertion
> depends on your acceptance.
>
> > unless and until what you say below is false:
> >
> > >Yet there may be no known way to check whether "forall n, Phi(n)" is true.
>
> Classically there is simply NO inferential connection between
> these two things. All the instances, for all n, even if we COULD
> exhaust
> them, could all come up true, and "forall n, Phi(n)" could STILL come
> up
> FALSE, because the "forall" INside the object language ranges over
> MORE things than the "for all" out in the meta-language did.
This presumes the existence of nonstandard models and the infinitary
reasoning (e.g. infinite sets) needed to establish their existence.
>
> > In other words, I dispute your assertion that "If one continues
> > checking the truth of Phi(n), for each instance of n, taken one at a
> > time, one will never ever find a counter-example" is meaningful;
>
> Please; if it is false, that IS meaningful, it is just false,
> and it is false IN VIRTUE OF its meaning, so you can't say
> it doesn't have a meaning. The only possible way you could
> be right in asserting that it is meaningless via THIS objection
> is IF IT IS TRUE, but the fact that we are CONFIRMING ITS TRUTH
> in this case means that it MUST have a meaning IN THIS CASE AS WELL.
>
> So Shut Up.
>
> > it is
> > an infinite process that can never be completed *in principle*;
>
> NO, it is NOT a process AT ALL, because we ARE NOT alleging
> any TIME or any degree of difficulty in ANY of this. ALL the true
> instances are true ALREADY, REGARDLESS of whether anybody
> has "confirmed" or checked ANY of them YET. All the valid
> proofs of all the instances exist ALREADY. No further process
> IS REQUIRED. This boneheaded invocation of physicalist metaphors
> is just that. We live in a place without time or place. DEAL WITH
> THAT.
>
No, what I am saying is not "physicalist". Instead it conforms with the
idea that mathematical truth consists of axiomatic declarations in the
human mind, as in my logic NAFL. What YOU are saying amounts to
Platonism, i.e., infinitely many natural numbers already "exist" in a
Platonic world, in which "all" instances of Phi(n) will turn up true,
regardless of our ability to establish such a truth. THAT is what I
reject. Deal with it, or delve into my stuff and refute what I am
saying. If you can take a serious look at my work on NAFL, you would do
me an enormous favour, for you obviously know a lot more about
"formalization" than I do.
> > what
> > you are really saying is that we have some means of checking (proving)
> > the truth of Phi(n) for an *arbitrary* (unspecified) n,
>
> NO, I am NOT saying that. I am CONTRASTING this with that.
>
> > which would
> > obviously prove "For all n Phi(n)".
>
> Obviously, but that's NOT WHAT WE'RE SAYING.
>
> > Of course, classical logicians (you
> > included) will not agree with me, but what I have said above actually
> > holds in my proposed logic NAFL
>
> Please. It CAN'T POSSIBLY hold.
> It's not soomething you get to have an opinion about.
> We are already talking about OUR logic.
> Our logic actually has the property that infinitely many
> instances of something can be individually provable.
> If yours doesn't then good for you, but even there,
> you can't deny that the concept is MEANINGFUL.
>
> > and follows from its postulates.
>
> I doubt that, frankly.
>
Well, then. Take a look at
http://arxiv.org/abs/math.LO/0506475
and let me know the shortcomings in my work. I am obviously very
enthusiastic about my work, but unlike you folks I am not emotionally
bonded to it. If my work turns out to be flawed, and you point it out,
I would be grateful to you and be the first to abandon my line of
thinking.
Regards, RS
Charlie-Boo
> The first thing
> you have to do to motivate THIS argument is find someone foolish
> enough to allege that some particular mathematical thing is
> "in principle unknowable".
False assertions are unknowable.
> > So I don't accept that you can meaningfully
> > make the above assertion,
>
> Whether you accept it or not is irrelevant;
> neither the meaning nor the satisfaction of the assertion
> depends on your acceptance.
It is relevant to communicating his position.
> All the instances, for all n, even if we
> . . . come up true, . . . "forall n, Phi(n)" could STILL come
> up
> FALSE, because the "forall" INside the object language ranges over
> MORE things than the "for all" out in the meta-language did.
The metalanguage must include the object language.
> It's not soomething you get to have an opinion about.
Only other people are allowed?
C-B
Charlie-Boo
> george wrote:
> > sradhakr wrote:
> > > This follows in my proposed logic NAFL,
> >
> > Please. The rest of us are talking about proving things
> > from PA in classical FOL.
> > The universal quantifer already has a semantics.
If one formalizes FOL as functions, the universal quantifier becomes
not a new primitive as you describe, but rather an expression composed
of two existing primitives: equality between sets and the universal
set. (all X)P(X) is P(a)=TRUE(a) for any given P where
P(a)=(P(a)^TRUE(a)) for any given P.
> > If you want to use a different one then you are going to need
> > a lot of luck defining it, articulating it, and conveying it.
> Well, I have already done it. It is upto the academic community to
> respond. And sci.logic is not just about discussion of FOL.
> Take a look at my work and refute it or criticize it. E.g. take a look
> at http://arxiv.org/abs/math.LO/0506475
He's not interested in investigating the truth or improvements. He's
only interested in reinforcing his existing beliefs.
"It is as fatal as it is cowardly to blink facts because they are not
to our taste." - John Tyndall
C-B
LordBeotian
"Charlie-Boo" <ch...@aol.com> ha scritto
> If that really wasn't your intention, then my apologies. But lots of
> people do that e.g. the earlier references to non-standard models of
> Peano's Axioms.
I accept your apologies.
sradhakr
thereabouts) is the complete intolerance to dissent, probably because
of the instituitionalization of research. Once research becomes a
profession and a career, people inevitably become career-minded and
start to protect turf rather than investigate the truth impartially.
What I don't understand here is the complete hostility to those who
would not accept the mainstream viewpoint. Surely if my work on NAFL is
correct, people can comment on it, cite it, etc. without feeling
threatened in some way? Surely I ought to be able to develop this logic
and further my own career, without in any way harming others? On the
other hand, the present attitude of intolerance has already had tragic
consequences upon generations of researchers, who were more or less
forced to tow the line (or else face ex-communication, denial of
funding, ridicule, etc.).
Regards, RS
sradhakr
Daryl McCullough
>If FLT is undecidable in PA (which is
>something that we cannot prove with certainty) then we can never ever
>know whether it is true or false.
Why do you say that? What is sacred about PA that makes it
the final arbiter of what is true and what is false?
>As far as my logic NAFL is concerned, your alleged "stronger
>system" doesn't exist
Of course it exists.
>and it is a requirement of the consistency of PA that
>PA-undecidable propositions (which lead to
>nonstandard models of PA) do not exist.
No. The consistency of PA implies no such thing.
>Nope. If "I give you an n" and you check the truth of Phi(n), that is
>just one instance of the above assertion, which therefore has not been
>established. I can repeat this any number of times, and the above
>assertion still has not established, It has been established "after" I
>have given you "all" possible values of n, this is what I am saying is
>not a meaningful assertion.
Well, the notion of being "meaningful" is not objective.
What's meaningful to me may not be meaningful to you,
or to my 7-year-old.
Charlie-Boo
> thereabouts) is the complete intolerance to dissent, probably because
> of the instituitionalization of research.
"Astronomer Halton C. Arp, described by colleagues as one of the
world's best observers, was barred from major observatories when he
came up with a radical theory about mysterious objects in space called
quasars. Thomas Gold, a distinguished astronomer who had a novel
theory about pulsars and neutron stars, was denied the right to speak
at an important meeting. Lynn Margulis, a biologists whose theories on
how living cells evolved are now universally accepted, was turned down
repeatedly for research funding and told never to apply again." -
David L. Chandler, The Boston Globe
> Once research becomes a
> profession and a career, people inevitably become career-minded and
> start to protect turf rather than investigate the truth impartially.
"I know that most men, including those at ease with problems of the
greatest complexity, can seldom accept even the simplest and most
obvious truth if it be such as would oblige them to admit the falsity
of conclusions which they have delighted in explaining to colleagues,
which they have proudly taught to others, and which they have woven,
thread by thread, into the fabric of their lives." - Leo Tolstoy
> What I don't understand here is the complete hostility to those who
> would not accept the mainstream viewpoint.
"New opinions are always suspected, and usually opposed, without any
other reason but because they are not already common." - John Locke
"It is hard to let old beliefs go. They are familiar. We are
comfortable with them and have spent years building systems and
developing habits that depend on them." - Kenichi Ohmae
"The innovator has for enemies all those who have done well under the
old conditions." - Niccolo Machiavelli
"New and stirring things are belittled because if they are not
belittled, the humiliating question arises, 'Why then are you not
taking part in them?' " - H. G. Wells
> Surely if my work on NAFL is
> correct, people can comment on it, cite it, etc. without feeling
> threatened in some way?
"Illness strikes men when they are exposed to change." - Herodotus
of Halicarnassus
"The vast majority of human beings dislike and even dread all notions
with which they are not familiar. Hence it comes about that at their
first appearance innovators have always been derided as fools and
madmen." - Aldous Huxley
"The soft-minded man always fears change. He feels security in the
status quo, and he has an almost morbid fear of the new. For him, the
greatest pain is the pain of a new idea." - Dr. Martin Luther King Jr.
> Surely I ought to be able to develop this logic
> and further my own career, without in any way harming others?
"Innovation - any new idea - by definition will not be accepted at
first. It takes repeated attempts, endless demonstrations, and
monotonous rehearsals before innovation can be accepted. This requires
courageous patience." - Warren Bennis
"A new scientific truth does not triumph by convincing its opponents
and making them see the light, but rather because its opponents
eventually die and a new generation grows up that is familiar with it.
Science advances funeral by funeral." - Max Planck
> On the
> other hand, the present attitude of intolerance has already had tragic
> consequences upon generations of researchers, who were more or less
> forced to tow the line (or else face ex-communication, denial of
> funding, ridicule, etc.).
"These are not perpetrators of scientific fraud, nor were they
scientists whose credentials, experience or capabilities were in
question. Their only crime - for which they were sentenced to work
without funding or facilities or denied the opportunity to communicate
with their peers - was that they disagreed with the prevailing
scientific mainstream." - David L. Chandler, The Boston Globe
> Regards, RS
"Inventions rarely come from people within an industry, but, instead
come from people on the outside who aren't under the same limiting
beliefs & habitual thinking that forms within any organization or
industry. - Dr. James Asher, San Jose State University, "On Advanced
Learning"
C-B
sradhakr
Daryl McCullough wrote:
> sradhakr says...
>
> >If FLT is undecidable in PA (which is
> >something that we cannot prove with certainty) then we can never ever
> >know whether it is true or false.
>
> Why do you say that? What is sacred about PA that makes it
> the final arbiter of what is true and what is false?
To prove FLT in the above situation, you need a stronger system. But we
can never be sure of the consistency of the stronger system (see the
paper by Daniel Velleman in the Math Intelligencer, maybe in 1998,
titled something like "Fermat's last theorem and Hilbert's program").
So we still can't be sure of the truth of FLT from such a proof. How
about a proof in PA? I would argue that that would be much more
convincing. A counter-example in the face of such a proof would force
us into ultra-finitism and reject even Euclidean Geometry. I don't
really believe that is likely.
>
> >As far as my logic NAFL is concerned, your alleged "stronger
> >system" doesn't exist
>
> Of course it exists.
>
Are you saying that my arguments to the contrary are wrong? Please feel
free to criticize my work in detail.
> >and it is a requirement of the consistency of PA that
> >PA-undecidable propositions (which lead to
> >nonstandard models of PA) do not exist.
>
> No. The consistency of PA implies no such thing.
That is what you claim for classical PA. I am talking about the NAFL
version of PA. Again, please feel free to look into my arguments and
refute them.
>
> >Nope. If "I give you an n" and you check the truth of Phi(n), that is
> >just one instance of the above assertion, which therefore has not been
> >established. I can repeat this any number of times, and the above
> >assertion still has not established, It has been established "after" I
> >have given you "all" possible values of n, this is what I am saying is
> >not a meaningful assertion.
>
> Well, the notion of being "meaningful" is not objective.
> What's meaningful to me may not be meaningful to you,
> or to my 7-year-old.
>
Let me make one clarificatoin. I am talking about a situation in which
you want to assert the truth or provability of Phi(n) for each natural
number n, without being able to prove "For all n Phi(n)". That is what
I find to be problematic, for it requires the Platonic existence of
infinitely many natural numbers n.
Regards, RS
Charlie-Boo
> it is a requirement of
> the consistency of PA that PA-undecidable propositions (which lead to
> nonstandard models of PA) do not exist.
No, if PA is consistent then PA-undecidable propositions do exist.
Suppose otherwise. Then PA is consistent and there are no
PA-undecidable propositions - all PA propositions are PA-decidable.
Then provable propositions are not refutable (by consistency), and
unprovable propositions are refutable (by decidability.) Then
unprovable propositions coincide with refutable ones. But the set of
refutable propositions is recursively enumerable, while the set of
unprovable propositions is not, and there is a contradiction.
(I didn't think of this proof/theorem. My formal system generated
it:
PR = provable
REF = refutable
CONS = consistent (nothing is both provable and refutable) PR => ~REF
DEC = decidable (all are provable or refutable) ~PR => REF
Thm: CONS => ~DEC is (PR => ~REF) => ~(~PR => REF)
Proof:
1. ~ ( (PR => ~REF) => ~(~PR => REF) ) Negate Theorem
2. PR => ~REF ~(A=>B) left rule
3. ~~(~PR => REF) ~(A=>B) right rule
4. ~PR => REF 3 Double Negative
5. ~~REF => ~PR 2 Contrapositive
6. REF => ~PR 5 Double Negative
7. ~PR = REF 4 6 Definition
8. REF(x) THM: The set of refutable propositions is r.e.
9. ~PR(x) 7 8 Substitution: The set of unprovable propositions is
r.e.
10. -~PR(x) Axiom: The set of unprovable propositions is not r.e.
11. False 9 10
qed
REF(x) in line 8 can be proven formally and -~PR(x) in line 10 can
likewise by creating definitions using rules, if there is interest in
that last bit of detail.
Now who has ever given a rigorous proof of MetaMathematical results
such as this?)
C-B
> Regards, RS
sradhakr
the "set of unprovable propositions" or "the set of refutable
propositons" or even "provability" cannot be formalized in NAFL. And
many classical results of recursion theory/computability theory fail in
NAFL. See my paper math.LO/0506475 in the arXiv. Its a different ball
game altogether in NAFL, and I really would like some experts in logic
to go into my work in detail. And thanks a lot for you rother
encouraging posts. I really do need to lift my flagging spirits at this
point of time.
Regards, RS
Charlie-Boo
All of the above concerns PA, not NAFL.
C-B
> Regards, RS
sradhakr
> "george" <gre...@cs.unc.edu> writes:
>
> > My point stands that if the things mentioned in (1)
> > are all and only the things there are, then the inference
> > (1)->(2) IS valid.
>
> in any sense whatever provable that we can prove A(n) for every n.
Ok, let us check this out. Take provability below to mean provability
in PA. Start with:
(1) For every natural number n (We can prove A(n))
I claim that (1) is equivalent to
We can prove A(0), and
For all n (We can prove A(n) --> We can prove A(n+1))
Do you agree? Or do you require that "For all n" be replaced by "For
all standard n", (i.e., For all natural numbers n)?
If you agree, (1) is equivalent to an assertion of (the truth of):
For all n A(n)
Do you agree? Or do you require that "For all n" be replaced as in the
previous remark?
If you do agree, then no counter-example to "For all n A(n)" can exist,
which means that it must be provable in PA (by the completeness
theorem).
If you disagree, then you have really *presumed* the existence of
nonstandard models and the validity of Godel's theorems, but that is
what I am questioning in the first place. Alternatively, you will claim
that (1) simply expresses a Platonic truth about a pre-existing
totality of natural numbers, not formalizable in classical PA.
Regards, RS
sradhakr
Fine. If you look at my original assertion, it was about the NAFL
version of PA (which I should give a different name). At some point,
this got snipped.
Regards, RS
Torkel Franzen
> If you do agree, then no counter-example to "For all n A(n)" can exist,
> which means that it must be provable in PA (by the completeness
> theorem).
In the NAFL sense, using the NAFL-completeness theorem, this may
well be the case. However, there is little point in presenting your
NAFL thinking as though it could be sensibly discussed on the basis of
old PA thinking. You need, rather, to put forward NAFL and NAFL
concepts on their own merits.
Daryl McCullough
>Ok, let us check this out. Take provability below to mean provability
>in PA. Start with:
>
>(1) For every natural number n (We can prove A(n))
...
>If you agree, (1) is equivalent to an assertion of (the truth of):
>
>For all n A(n)
>
>Do you agree?
Let's assume for now that all the axioms of PA are true, and
by "prove" we mean "prove in PA". Then we have the following
two statements:
S1: forall n, A(n)
S2: forall n, we can prove A(n)
S3: we can prove S2
S4: we can prove S1
If all the axioms of PA are true,
then
S4 implies S3
S3 implies S2
S2 implies S1
On the other hand
S1 does not necessarily imply S2
S2 does not necessarily imply S3
S3 does not necessarily imply S4
Daryl McCullough
>Daryl McCullough wrote:
>> sradhakr says...
>>
>> >If FLT is undecidable in PA (which is
>> >something that we cannot prove with certainty) then we can never ever
>> >know whether it is true or false.
>>
>> Why do you say that? What is sacred about PA that makes it
>> the final arbiter of what is true and what is false?
>
>To prove FLT in the above situation, you need a stronger system. But we
>can never be sure of the consistency of the stronger system (see the
>paper by Daniel Velleman in the Math Intelligencer, maybe in 1998,
>titled something like "Fermat's last theorem and Hilbert's program").
Why do you say that? There is nothing to prevent us from being sure
of the consistency of a stronger system than PA. People are just as
sure of the consistency of
PA + "PA is consistent"
as they are of the consistency of PA.
Chris Menzel
> ...
> Ok, let us check this out. Take provability below to mean provability
> in PA. Start with:
>
> (1) For every natural number n (We can prove A(n))
>
> I claim that (1) is equivalent to
>
> We can prove A(0), and
> For all n (We can prove A(n) --> We can prove A(n+1))
>
> Do you agree? Or do you require that "For all n" be replaced by "For
> all standard n", (i.e., For all natural numbers n)?
>
> If you agree, (1) is equivalent to an assertion of (the truth of):
>
> For all n A(n)
But it isn't. Part of the problem seems to be that you are using "n"
carelessly. The proper statement of (1) is
(1') For every natural number n (We can prove "A([n])"),
where "n" is a metavariable ranging over the numbers and "[n]" is the
*numeral* in the language of PA that standardly denotes the value of
"n". In PA, this would be the term consisting of n occurrences of the
successor function symbol "s" followed by an occurrence of the numeral
"0". And (changing the object language variable to "x" to avoid
confusion) the claim is now that from (1') it does not follow that
(2') We can prove "(x)A(x)".
A pretty obvious reason for this is that the actual syntactic structure
of the numeral [n] might (and typically does) play a critical role in
the proof of any instance of "A([n])". But that structure is
unavailable when you try to prove simply "A(x)" (and thence "(x)A(x)",
by UG) for any arbitrary value of "x".
sradhakr
somewhere. You are probably right in the sense that someone might claim
to be as confident of ZFC as I am of PA and Euclidean Geometry. There
is no end to that argument (unless of course, someone can come up with
a demonstrable inconsistency in ZFC).
Regards, RS
David C. Ullrich
<cme...@remove-this.tamu.edu> wrote:
Thanks - I've been considering whether to try to say exactly
that, now I don't have to.
Not that it's going to help anyone's confusion about this, of
course. But it's exactly the point that had me totally
puzzled about a lot of things I read many years ago - I
saw statements in proofs in books of the form "for every
n, <something about [n]>", and I was very confused about
why the proofs didn't prove much more than they claimed
because I was missing the distinction you clarify above.
Years later I realized what I was missing...
************************
David C. Ullrich
Daryl McCullough
>> Why do you say that? There is nothing to prevent us from being sure
>> of the consistency of a stronger system than PA. People are just as
>> sure of the consistency of
>>
>> PA + "PA is consistent"
>>
>> as they are of the consistency of PA.
>>
>The only answer to that question is: We have to draw the line
>somewhere. You are probably right in the sense that someone might claim
>to be as confident of ZFC as I am of PA and Euclidean Geometry. There
>is no end to that argument (unless of course, someone can come up with
>a demonstrable inconsistency in ZFC).
Well, for someone who is certain of ZFC, it is trivial to show that
there are formulas Phi(x) of PA such that
For each natural number n, PA proves Phi(n)
but
PA does not prove "forall x, Phi(x)"
sradhakr
I fully understand what you are saying. What I intended to convey is
that from
(1) For every natural number n (PA proves "A([n])")
if follows that
(2) There does not exist a natural number n for which PA does not prove
"A([n])"
The issue then is whether we can conclude from (2) that:
(3) (x) A(x) will turn out "true" in every model of PA, and therefore
from the completeness theorem, we can infer the existence of a PA-proof
of (x)A(x).
Note: I am not claiming that from purely syntactic considerations, it
follows from (1) that there must exist a PA-proof of (x)A(x). As you
say, this is certainly not obvious. In fact (3) only infers the
existence of said proof non-construcitvely, without giving a clue of
how we might go about finding such a proof. On the other hand, if
(x)A(x) is equivalent to the Godel sentence for PA, you would be making
the following stronger claim (via Godel's theorem):
(4) It follows from purely syntactic considerations that PA does not
prove (x)A(x) despite the truth of (1).
Therefore you would not agree with (3), and instead use the
completeness theorem to infer the existence of nonstandard models of PA
in which (x)A(x) is false.
I, on the other hand, would rule out the existence of nonstandard
models of PA (in my logic NAFL) and insist that (3) does follow from
(1).
An intuitive reason for my viewpoint is as follows. We know that
PA proves the existence of 0
PA proves the existence of S(0)
PA proves the existence of S(S(0)), and so on.
It follows in NAFL that from the above infinitely many PA-proofs, that
PA proves the existence and uniqueness of the universal class
U = {0, S(0), S(S(0)), ...}
U contains all and only the standard natural numbers. Intuitively, the
only way infinitely many natural numbers 0, S(0), S(S(0)), ... can be
thought of as existing in a "completed" sense is via their
*simultaneous* existence in the infinite class U. There is no way to
conceive of 'all' natural numbers, taken one at a time, because such a
process cannot be completed ( by induction). The axiom of
extensionality for classes is inherent in the above identifiation of an
infinite class by all of its elements and ensures the uniqueness of U,
and hence rules out nonstandard models of PA.
The above is obviously very different from the classical picture, and
so it is not surprising that NAFL clashes with Godel.
Regards, RS
sradhakr
This is certainly an inspirational post, one that deserves to be saved
to my computer. You can be sure that I will be getting back to it from
time to time, whenever I need a morale-booster.
Regards, RS
柑\/b
can you do an example of that? can you find a proposition phi() ?
thank you
Barb Knox
"sradhakr" <srad...@in.ibm.com> wrote:
And here's another quote, for when you need a reality-check:
"But the fact that some geniuses were laughed at does not imply that all
who are laughed at are geniuses. They laughed at Columbus, they laughed
at Fulton, they laughed at the Wright brothers. But they also laughed
at Bozo the Clown." -- Carl Sagan
>Regards, RS
--
---------------------------
| BBB b \ Barbara at LivingHistory stop co stop uk
| B B aa rrr b |
| BBB a a r bbb | Quidquid latine dictum sit,
| B B a a r b b | altum viditur.
| BBB aa a r bbb |
-----------------------------
sradhakr
feel free to explain your rationale in straghtforward technical terms.
In particular, tell me what YOU think of
http://arxiv.org/abs/math.LO/0506475
This paper outlines a new method for doing real analysis in a newl
logic NAFL. Give YOUR technical opinion, don't waste your time and
energy in laughing or making vague derogatory comments. As I have said
elsewhere, I will abandon my work if someone refutes it and convinces
me tha NAFL is not a logic. In the meanwhile, I will assume that the
creation of a new logic, such as, NAFL, is a non-trivial intellectual
achievement that is noteworthy regardless of the fact that the acadmic
community has chosen to ignore it.
Regards, RS
Barb Knox
"sradhakr" <srad...@in.ibm.com> wrote:
Chill, dude. I just provided a quote, hopefully to balance the
self-righteous tone of the other list of quotes. It's not personal;
it's just that most people who think they are revolutionary geniuses are
in fact crackpots. The truth of this observation can be readily seen on
Usenet, including this ng. If in fact you are a revolutionary then
congratulations, but /a priori/ the odds are slim.
>If so, please
>feel free to explain your rationale in straghtforward technical terms.
>In particular, tell me what YOU think of
>http://arxiv.org/abs/math.LO/0506475
I had a look at an earlier draft, and it didn't hold my attention.
>This paper outlines a new method for doing real analysis in a newl
>logic NAFL. Give YOUR technical opinion, don't waste your time and
>energy in laughing or making vague derogatory comments.
Where in my reply did I engage in laughing or making vague derogatory
comments? But if the shoe fits....
>As I have said
>elsewhere, I will abandon my work if someone refutes it and convinces
>me tha NAFL is not a logic. In the meanwhile, I will assume that the
>creation of a new logic, such as, NAFL, is a non-trivial intellectual
>achievement that is noteworthy regardless of the fact that the acadmic
>community has chosen to ignore it.
You might find the reviewers' comments instructive. Care to share them
with us?
sradhakr
its worth. I have also come to the conclusion that the "odds are slim"
that my work will be recognized or even acknowledged. But not for
reasons that you imagine.
>
> >If so, please
> >feel free to explain your rationale in straghtforward technical terms.
> >In particular, tell me what YOU think of
> >http://arxiv.org/abs/math.LO/0506475
>
> I had a look at an earlier draft, and it didn't hold my attention.
>
Vague. In any case a new, detailed version is posted to the arXiv. And
it is written much better. Take a look at it and post your comments to
the arXiv, or let me know what you think. Sec. 4 on real analysis can
be read independently of other sections (you only need to assume that
infinite sets are not permitted and quantification over proper classes
are banned. You don't need NAFL to understand how the real analysis
part is formulated, if you accept these two assumptions).
> >This paper outlines a new method for doing real analysis in a newl
> >logic NAFL. Give YOUR technical opinion, don't waste your time and
> >energy in laughing or making vague derogatory comments.
>
> Where in my reply did I engage in laughing or making vague derogatory
> comments? But if the shoe fits....
>
> >As I have said
> >elsewhere, I will abandon my work if someone refutes it and convinces
> >me tha NAFL is not a logic. In the meanwhile, I will assume that the
> >creation of a new logic, such as, NAFL, is a non-trivial intellectual
> >achievement that is noteworthy regardless of the fact that the acadmic
> >community has chosen to ignore it.
>
> You might find the reviewers' comments instructive. Care to share them
> with us?
>
Sure. Two of the referees claimed not to understand NAFL from the
description given in the paper (restricted to 12 pages by the FSTTCS
conference). A third one said something to the effect that if the
inference rules are classical, he fails to understand how Pv~P can fail
to be a theorem of an NAFL theory T in which P is undecidable in T. One
of the referees also said he failed to understand how the NAFL truth
definiton (The Main Postulate) can influence the logic for in his view
truth definitions are not part of logic. In short, the referees cannot
get out of their classical mind-set. Sadly, they never got to the real
analysis part of the paper (which can be read more or less
independently of other sections of the paper). I have invited them to
post their criticisms if any, of the revised version, to the arXiv. In
the meanwhile, don't hold yourself back. Feel free to rip into my work
and post your criticisms to the arXiv. It is a respectable resource and
my work deserves to be there if and only if it is correct (and I am
betting it is correct).
Regards, RS
David C. Ullrich
Are the _theorems_ of your PA the same as the theorems of
standard PA? Hint: the answer is yes or no.
If no: Then you're changing the subject - you should call
your PA something other than "PA", to prevent confusion.
If yes: Then the fact that you rule out the existence of
nonstandard models in your logic is irrelevant. If I
use a nonstandard model to give a counterexample to
(3) then it _is_ a counterexample in your PA as well;
the only difference is that you're possibly unable to
prove that it's a counterexample.
>An intuitive reason for my viewpoint is as follows. We know that
>
>PA proves the existence of 0
>
>PA proves the existence of S(0)
>
>PA proves the existence of S(S(0)), and so on.
>
>It follows in NAFL that from the above infinitely many PA-proofs, that
>PA proves the existence and uniqueness of the universal class
>
>U = {0, S(0), S(S(0)), ...}
>
>U contains all and only the standard natural numbers. Intuitively, the
>only way infinitely many natural numbers 0, S(0), S(S(0)), ... can be
>thought of as existing in a "completed" sense is via their
>*simultaneous* existence in the infinite class U. There is no way to
>conceive of 'all' natural numbers, taken one at a time, because such a
>process cannot be completed ( by induction). The axiom of
>extensionality for classes is inherent in the above identifiation of an
>infinite class by all of its elements and ensures the uniqueness of U,
>and hence rules out nonstandard models of PA.
>
>The above is obviously very different from the classical picture, and
>so it is not surprising that NAFL clashes with Godel.
>
>Regards, RS
************************
David C. Ullrich
sradhakr
>
> Are the _theorems_ of your PA the same as the theorems of
> standard PA? Hint: the answer is yes or no.
>
> If no: Then you're changing the subject - you should call
> your PA something other than "PA", to prevent confusion.
You are right. The theorems are not quite the same, and I have indeed
called my PA as NPA in my paper (NPA proves the existence of infinite
(proper) classes of natural numbers, unlike the classical PA).
What I intended to debate with my previous argument was *why* we do not
have the freedom to say in classical PA that there is only one "all",
and that defines all (standard) natural numbers.
Indeed, there is nothing infinitary about either the axioms of PA or
the assertion below:
For every natural number n (PA proves "A([n])"),
But Godel's conclusion that PA does not prove (x)A(x) despite the above
(when (x)A(x) is the Godel sentence) is infinitary and *forces* us to
admit nonstandard models of PA.
Godel's incompleteness theorems force seemingly unwanted infinities
upon us (via nonstandard models) when it seems that the axioms of PA
are perfectly finitary and do not require us to admit such infinities.
Is is possible that Godel's theorems use some infinitary reasoning
(say, via diagonalization) that is not inherent in PA and comes to an
infinitary conclusion?
Regards, RS
Daryl McCullough
>(Daryl McCullough) wrote:
>>Well, for someone who is certain of ZFC, it is trivial to show that
>>there are formulas Phi(x) of PA such that
>>
>> For each natural number n, PA proves Phi(n)
>>
>>but
>>
>> PA does not prove "forall x, Phi(x)"
>
>can you do an example of that? can you find a proposition phi() ?
>thank you
Godel's theorem shows us one such formula. Godel shows how to
code statements and proofs as natural numbers, and he constructs
a formula Pr(x,y) such that
Pr(x,y) holds if and only if y is a code for some sentence
in PA, and x is a code for a proof of that sentence.
Letting c be a code for some provably false statement in PA, such
as "0=1", then we can show in ZFC that
forall n, PA proves: not Pr(n,c)
but
PA does not prove: forall n, not Pr(n,c)
There are other examples that don't rely on coding, but instead
rely on computations that can be proved in ZFC to halt in for every
input, but cannot be proved to always halt in PA. Such a computation
leads to a formula
Phi(x)
which holds if and only if the computation halts on input x. If
the computation halts on all inputs, but PA cannot prove this,
then we have:
forall n, PA proves Phi(n)
but
PA does not prove forall n, Phi(n)
sradhakr
however, the following question:
How can one isolate strictly finitary reasoning and what are
scientifically useful conclusions (say in theoretical physics or
computer science) that one can draw from such reasoning.
Therefore I need something at least as strong as PA, but I don't want
the infinitary conclusions of Godel or that of stronger theories like
ZFC.
Regards, RS
Chris Menzel
>> You might find the reviewers' comments instructive. Care to share
>> them with us?
>>
> inference rules are classical, he fails to understand how Pv~P can fail
> to be a theorem of an NAFL theory T in which P is undecidable in T.
But that is a completely legitimate criticism -- assuming (as the
referee's comment would suggest) you claim the inference rules are
classical, as Pv~P is a classical theorem. Did you so claim? If not,
what could have led to the referee to make such a remark out of the
blue?
sradhakr
Well, here is a slightly long-winded explanation. I request you to read
it patiently (it is not *too* long).
The classical rules of inference are certainly present in NAFL, with
certain additional non-classical features (such as, the provability of
the existence of infinite classes). So the referee's question was, if
Pv~P is deducible by the classical rules of inference, how can it be
fail to be a theorem of an NAFL theory T in which P is undecidable?
Actually, I thought this was explained in some detail in my paper, even
with the page restrictions. To answer the question, in NAFL theories,
there are two levels of syntax, namely, the "proof syntax" and the
"theory syntax". Suppose P is an undecidable proposition of an NAFL
theory T. Then Pv~P is certainly deducible in the proof syntax of T and
will appear there as a deduction. However, I, nevertheless do not admit
Pv~P as a legitimate proposition of T in its theory syntax, which
determines the theorems of T (and admits the legitimate undecidable
propositions of T, which satisfy the Main Postulate of NAFL). Since
Pv~P is not in the theory syntax of T, it is not a theorem (or even a
legitimate undecidable proposition) of T and so this allows me to
assert that there must exist a non-classical model of T in which P&~P
is the case, as required by Proposition 1 of my paper
(math.LO/0506475).
You, like the referee, might be suspicious of this formulation and find
it "vague", "not clear", etc. That is why I specifically considered the
case of the theory T0 consisting of the null set of axioms in my paper.
Let P be a legitimate proposition of T0. Then I assert that P is
undecidable in T0, i.e., every legitimate proposition of T0 is
undecidable in T0. So in this case Pv~P is still a legitimate deduction
in the proof syntax of T0, but it is not a legitimate proposition of T0
in its theory syntax. Hence there will be a non-classical model of T0
in which P&~P is the case. How can this be, you may ask. Well, P&~P
contradicts only Pv~P, which is not in the theory syntax of T0. So P&~P
does not contradict any theorem of T0 (which has no theorems). I
interpret P&~P in this non-classical model as: 'P' in P&~P denotes that
"~P is not provable in the interpretation T0* of T0" and "~P" denotes
that "P is not provable in the interpretation T0* of T0 where the
meaning of T0* is explained in my paper: T0* is an "interpretation" of
T0 that the human mind determines out of its free will, and is also an
axiomatic NAFL theory that "generates" the model of T0 (T0* can vary in
time according to the free will of the human mind, and different human
minds can have different T0*'s in mind at a given time for a given
theory T0). So, P&~P is true in this non-classical sense. On the other
hand, the classical (with respect to the proposition "P") models of T0
are the ones in which either P or ~P is provable in T0*, in which case
one can assert that P (~P) is "true" with respect to T0. That is,
truth in NAFL is with respect to its theories and is not a Platonic
truth. Note though that NAFL does not object to the classical assertion
that either P or ~P must be Platonically true, say in the real world
(e.g. the Schrodingder cat must be either alive or dead) even when P is
undecidable in T0. It is just that this classical picture must be
outside of NAFL semantics, which instead asserts P&~P as above, and
which is equally true.
The requirement of the existence of the non-classical model comes from
the NAFL truth definition (the Main Postulate of NAFL), which is a
metamathematical principle. The referee does not accept that truth
definitions should influence the logic (i.e., theoremhood, etc.) and so
felt that I was "mixing the object and the meta-language". This is not
quite true. Even classical logic is formulated with certain
metamathematical principles in mind. Thus the law of the excluded
middle comes from the metamathematical principle that every proposition
P must turn out either "true" or "false" regardless of our knowledge of
the truth value and regardless of the provability of P in axiomatic
theories. Can I accuse classical logicians of "mixing object language
and meta-language"? Certainly not. NAFL just does not accept that this
classical principle should be allowed to determine theoremhood of Pv~P
in T, in the case when P is undecidable in T, although it can still be
"metalogicially" "true" in the real world (i.e. outside of both syntax
and semantics of the NAFL theory T). E.g. the Schrodinger cat can still
be considered as "really" either alive or dead (Pv~P) at any moment of
time, but the NAFL theory T will admit non-classical models in which
P&~P is the case, which can be interpreted as "We cannot prove in the
interpretation T* of T that the cat is alive and we cannot prove in T*
that the cat is dead". In this sense P&~P is just as "true" as the
classical Pv~P in the real world.
Finally, you might ask "what is the rationale of leaving Pv~P as a
legitimate deduction in the proof syntax of T, when P is undecidable in
T? From the NAFL point of view Pv~P (or P -> P) asserts that "If P(~P)
is the case, then ~P(P) is not the case". I assert that this determines
theoremhood of Pv~P in the NAFL theories T+P (T+~P) in the case when P
is undecidable in T, and *not* theoremhood in T. So the presence of
Pv~P in the proof syntax will prevent us from constructing NAFL
theories in which both P and ~P are provable (although *models* of T
can exist in which P&~P is the case).
Regards, RS
Charlie-Boo
The trivial fallacy that Sagan refutes is not part of the psychological
and economic principles to which these various authors refer in their
explanations of resistance to change.
I would hope that we can all distinguish between Astronomers Halton C.
Arp and Thomas Gold, biologist Lynn Margulis, and comedian Bozo the
Clown.
C-B
Charlie-Boo
> What do YOU think of NAFL? Something to be laughed at? If so, please
> feel free to explain your rationale in straghtforward technical terms.
> In particular, tell me what YOU think of
> http://arxiv.org/abs/math.LO/0506475
First you need to give here the simplest possible completely
self-contained example of something that your system does that existing
systems don't and that has demonstrable value e.g. solving a problem
than has been addressed by others and never solved.
C-B
> Regards, RS
Charlie-Boo
> >What do YOU think of NAFL? Something to be laughed at?
> Chill, dude.
Chill? You introduced emotions (ridicule), not him.
> I just provided a quote, hopefully to balance the
> self-righteous tone of the other list of quotes.
Which quote had a self-righteous tone - John Locke, Niccolo
Machiavelli, Einstein?
> If in fact you are a revolutionary then
> congratulations, but /a priori/ the odds are slim.
But there is actually the odds times the potential reward = the
expected benefit, as well as principles of fairness and scientific
pursuit.
> Where in my reply did I engage in laughing or making vague derogatory
> comments? But if the shoe fits....
(1) The reference to Bozo (2) The 2nd sentence above (an obvious
reference to Bozo's big red shoes.*)
C-B
* Or is that Ronald McDonald? Same comedian?
Charlie-Boo
> Barb Knox wrote:
> > "sradhakr" <srad...@in.ibm.com> wrote:
> > >Charlie-Boo wrote:
>
> > >This is certainly an inspirational post, one that deserves to be saved
> > >to my computer. You can be sure that I will be getting back to it from
> > >time to time, whenever I need a morale-booster.
>
> > And here's another quote, for when you need a reality-check:
> >
> > "But the fact that some geniuses were laughed at does not imply that all
> > who are laughed at are geniuses. They laughed at Columbus, they laughed
> > at Fulton, they laughed at the Wright brothers. But they also laughed
> > at Bozo the Clown." -- Carl Sagan
Actually, the guy who played Bozo was a comedic genius and also a
Biologist. He dabbled in both science and art, and sometimes himself
complained that people can't tell the difference between science and
comedy - too often mistaking the writings of college professors as
having any honest significance and especially when compared to the
creations of industry where BS in product claims means unhappy
customers and a failed business venture.
sradhakr
Charlie-Boo wrote:
>
> First you need to give here the simplest possible completely
> self-contained example of something that your system does that existing
> systems don't and that has demonstrable value e.g. solving a problem
> than has been addressed by others and never solved.
>
> C-B
>
<http://arxiv.org/abs/math.LO/0506475>,
which outlines the strictly finitary formulation of real
analysis/computabilty theory that results from NAFL.
(1) Open/semi-open intervals of real numbers do not exist
(2) dy/dx is 0/0 -- Weierstrass epsilon-delta method fails because of
(1)
(3) Zeno's paradoxes and other paradoxes of classical real analysis
(Banach-Tarski, etc.) result from (1) and are successfully resolved in
the NAFL version of real analysis. In particular, the resolution of
Zeno that is given here is the only satisfactory one and cannot be
given in any other version of real analysis.
(4) I think the correct way to interpret the NAFL version of real
analysis is that it provides the correct criteria for the limits of
strictly finitary, "safe" reasoning. Which is the question that Hilbert
asked (his second problem, I think).
(5) The most important application that I foresee is in quantum
mechanics and fundamental theoretical physics. I firmly believe that
NAFL will provide the correct logical reasons for quantization (of
space, time, matter, energy, etc.).
Regards, RS
Charlie-Boo
Carg Sagan may be a good astronomer (or at least a famous one), but his
logic is woefully flawed (as is yours if you believe his quote.) This
is a false analogy. Bozo the Clown's comedy routine is not an instance
of the principle of people not taking seriously one's accomplishments
or goals that were later reached. Bozo presented his comedy routine
and it was accepted as such. All of these famous scientists (including
Cantor, BTW) presented their theories and were not accepted.
Perhaps Carl Sagan simply could not think of an actual example of a
famous person being laughed at and the public was right. Can you?
C-B
"I am very suspicious about anything that most people believe, because
most people are fucking idiots." - Bill Maher, upon receiving the
annual First Amendment Award.
Charlie-Boo
Are these examples complete and self-contained? "I am not convinced."
(You need to convince people to take the effort to look further, my
friend. Furthermore, there's not enough time to check into
everything!)
C-B
> Regards, RS
sradhakr
(quant-ph/0504115) is still under review. If that goes well, I plan to
write up an overview of my work that will highlight the main results
and maybe try to put it into Wikipedia. Maybe that will convince a few
to look into my work.
Regards, RS