Chinese Room
Daryl McCullough
>>> The Godel Incompleteness Theorem doesn't say any such thing.
>>
>>Yes it does, the Godel Proof of incompleteness asserts that there are
>>assertions of algebria which are true but can not be proven by formal
>>logic.
>
>No, it says that there is no consistent mathematical theory of
>arithmetics that does not contain undecidable propositions.
Yes, but the particular undecidable proposition that Godel
constructs is *true* (if the theory is consistent). The statement
G constructed via Godel's proof is provably equivalent to the
consistency of the theory. That is,
G_T <-> Con(T)
So, if we assume that T is consistent, then, under that assumption,
G is *true*. Of course, truth is relevant to an interpretation, but
to the extent that we've already decided to interpret Con(T) as
"T is consistent", then we've already chosen an interpretation
of the relevant parts of the theory T.
For the record, I don't think Godel's theorem tells us anything
about the possibility or impossibility of AI.
--
Daryl McCullough
CoGenTex, Inc.
Ithaca, NY
Janosch Zwerensky
>>No, it says that there is no consistent mathematical theory of
>>arithmetics that does not contain undecidable propositions.
>
>Yes, but the particular undecidable proposition that Godel
>constructs is *true* (if the theory is consistent). (...)
In some models of T, yes. If there were no models of T were G(T) is wrong, by
Godels completeness theorem there would be a proof of non G(T) in T and G(T)
would not be undecidable.
Of course, whenever we are working in a model of T where every x that fulfills
the "equation" dem(x,n) can be decoded to a proof of the proposition with the
Godel number n, then G(T) is true in that model (since T has a model, T must be
consistent).
Regards,
Janosch.
Daryl McCullough
>>>No, it says that there is no consistent mathematical theory of
>>>arithmetics that does not contain undecidable propositions.
>>
>>Yes, but the particular undecidable proposition that Godel
>>constructs is *true* (if the theory is consistent). (...)
>
>In some models of T, yes.
Not just *some* models of T, it is true in the model
that we are interested in. If T = Peano arithmetic,
then G is actually true in the standard model of
arithmetic.
Since G is just an ordinary statement of arithmetic,
we no more need to say G is true in some models than
we need to say "2+2 = 4" in some models. G is true
by the usual meanings of the arithmetic terms out
of which G is constructed.
>If there were no models of T were G(T) is wrong,
Yes, G(T) is false in *nonstandard* models. But when
we are talking about statements of arithmetic then
we just say "S is true" when we mean "S is true of
the standard model of arithmetic".
If you want to explicitly talk about truth within
a model, then the claim is this:
If T is a recursively enumerable theory, all
of whose axioms are true in the standard model
of arithmetic, then Godel's proof allows us to
construct a new sentence G such that G is true
in the standard model, but G is not provable by T.
So Godel's proof really does give us new *true* statements,
(true in the standard model) not just undecidable statements.
Janosch Zwerensky
>>In some models of T, yes.
>
>Not just *some* models of T, it is true in the model
>that we are interested in. If T = Peano arithmetic,
>then G is actually true in the standard model of
Never would deny that ;).
However, for some theories to which the GIT can be applied, the standard model
of arithmetic does not actually seem to be a model. For example, the natural
numbers are no model of ZFC.
Regards,
Janosch.
Daryl McCullough
But omega (the set of natural numbers) is definable in ZFC. So,
even if there isn't as clear an idea of the "standard model" for
ZFC as for Peano arithmetic, we would call any model of ZFC nonstandard
if its definition of omega is nonstandard. As with Peano arithmetic,
it is also true for ZFC---G is true in any model of ZFC that has a
standard model for PA.
aatu
Daryl McCullough <da...@cogentex.com> wrote in message
news:95ulu...@edrn.newsguy.com...
[about Goedel's theorem]
Goedel's theorem tirvially implies there are *true* undecidable statements
in any
sufficiently rich system.
For example, consider an undecidable statement G and a model M. By
definition,
either M |= G or M |= ~G. Since G is undecidable, so is ~G, and therefore
there is
a true statement (either G or ~G) which is true (in M).
--
Aatu Koskensilta (aa...@mediaclick.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
John Ahlstrom
Daryl McCullough wrote:
-- snip snip
> Yes, but the particular undecidable proposition that Godel
> constructs is *true* (if the theory is consistent). The statement
> G constructed via Godel's proof is provably equivalent to the
> consistency of the theory. That is,
--snip snip
> Daryl McCullough
> CoGenTex, Inc.
> Ithaca, NY
What does "*true*" in this statement mean?
I know of a couple of meanings:
a. Epistemology: corresponding to reality
b. Logic: Assumption (as of an axiom)
Derived from a collection of axioms
by an accepted collection of rules of inference
Clearly the statement cannot mean that
"the Godel construct is "derived ..." and that it is not
derivable from the collection of axioms.
Is there any logical meaning of "true" or are we reduced
to something like:
"the Godel construct is epistemologically true but logically
undecidable"?
JKA
--
Any system of neural organization sufficiently complex to
generate the axioms of arithmetic is too complex to be
understood by itself.
Kaekel's Conjecture
Daryl McCullough
>
>Also sent directly to Daryl
>
>Daryl McCullough wrote:
>-- snip snip
>
>> Yes, but the particular undecidable proposition that Godel
>> constructs is *true* (if the theory is consistent). The statement
>> G constructed via Godel's proof is provably equivalent to the
>> consistency of the theory. That is,
>
>I know of a couple of meanings:
> a. Epistemology: corresponding to reality
Yes, that's what "true" means, except that I would replace "reality" by
"the intended domain of discourse" which might be the physical world, if
we are talking about physical objects, or it might be some mathematical
structure, if we are talking about mathematics. In Godel's theorem, the
domain of discourse is the natural numbers.
Actually, I wouldn't use the word "epistemology" here. Doesn't
epistemology have to do with how we come to know things? In that
case, truth doesn't have much to do with epistemology---whether
something is true is independent of whether we know that it is
true.
> b. Logic: Assumption (as of an axiom)
> Derived from a collection of axioms
> by an accepted collection of rules of inference
No, that's definitely *not* what is meant by truth. "True" never means
"derived from a collection of axioms by accepted rules of inference". If the
axioms and rules of inference were chosen carefully, then we can hopefully
arrange things so that it is impossible to derive a statement that is not
true, but if our axioms are false (in the intended interpretation) then
some of the derived conclusions will be false, as well.
>Clearly the statement cannot mean that
> "the Godel construct is "derived ..." and that it is not
> derivable from the collection of axioms.
>
>Is there any logical meaning of "true"
No, there is no logical definition of truth. However, we can
convince ourselves that our logic is *sound*, which means that
it is impossible to derive a false conclusion from true axioms.
Truth has to do with the *meaning* of sentences, not with how
the sentence is proved.
>or are we reduced
>to something like:
> "the Godel construct is epistemologically true but logically
>undecidable"?
I would say that "The Godel sentence is true of the natural numbers,
but it is not provable using the axioms of Peano arithmetic". It
is provable using more powerful theories, such as 2nd order arithmetic
or ZFC.
Many people think that if a mathematical statement is neither
provable nor refutable, then it should be considered neither true
nor false. But that really isn't the way that we normally use
the word "true". Suppose that I have a steel safe that nobody
knows the combination to. If I tell you that the safe contains
100 dollars and it really does contain 100 dollars, then I'm
telling the truth, whether or not anyone ever can prove it.
If it doesn't contain 100 dollars, then I'm telling a falsehood,
whether or not anyone can ever prove it.
The same holds for mathematical statements. If I say
"There is no positive integers x,y,z, and n, such that
n > 2 and x^n + y^n = z^n"
then you understand what I'm saying, even if you don't
know how to prove or disprove it. If there happen to be
x,y,z,n satisfying n > 2 and x^n + y^n = z^n, then I'm
lying. If not, I'm telling the truth. It doesn't matter
whether anyone knows how to prove it. (Actually, we now
know that this true, but it was certainly true before
anyone knew how to prove it).
--
Greg Lee
...
> Many people think that if a mathematical statement is neither
> provable nor refutable, then it should be considered neither true
> nor false. But that really isn't the way that we normally use
> the word "true". Suppose that I have a steel safe that nobody
> knows the combination to. If I tell you that the safe contains
> 100 dollars and it really does contain 100 dollars, then I'm
> telling the truth, whether or not anyone ever can prove it.
> If it doesn't contain 100 dollars, then I'm telling a falsehood,
> whether or not anyone can ever prove it.
This little story isn't very persuasive. Do _you_ know what's
in the safe? If so, somebody can prove it. If not, and it
really does contain 100 dollars, we'd call your statement
lucky guess, not the truth, even though what you said turned
out to be true.
> The same holds for mathematical statements. If I say
> "There is no positive integers x,y,z, and n, such that
> n > 2 and x^n + y^n = z^n"
> then you understand what I'm saying, even if you don't
> know how to prove or disprove it. If there happen to be
> x,y,z,n satisfying n > 2 and x^n + y^n = z^n, then I'm
> lying.
No, then you would be mistaken, unless you knew there were.
The way we customarily use the words "true", "lie" can't be
divorced from epistemology. If you'd like to divorce them in
your own usage, of course you can do that.
> If not, I'm telling the truth. It doesn't matter
> whether anyone knows how to prove it. (Actually, we now
> know that this true, but it was certainly true before
> anyone knew how to prove it).
That's as may be, but it doesn't mean we would say that those who
said so were speaking the truth.
--
Greg Lee <l...@hawaii.edu>
Torkel Franzen
> No, there is no logical definition of truth.
That's an odd statement. If we're speaking of sentences of limited
logical complexity, we can define truth in arithmetic itself, in
particular "true P1-sentence". It is simply a theorem of arithmetic
that if T is consistent, the Godel sentence for T is true but
unprovable in T.
James Dolan
|The way we customarily use the words "true", "lie" can't be
|divorced from epistemology.
true no, lie yes. (alternatively, true yes, lie no.)
|If you'd like to divorce them in your own usage, of course you can do
|that.
daryl's usage here is the standard one; yours is bizarrely
idiosyncratic as usual.
you seem to have taken the customary epistemological baggage of "know"
and bestowed it upon "true". normal people don't do that. if someone
makes a true statement as the result of a wild lucky guess, we say "it
was a lucky guess" and "it's a true statement". we also say "you
didn't actually know that, it was just a lucky guess".
|That's as may be, but it doesn't mean we would say that those who
|said so were speaking the truth.
now you're trying to rescue your hopelessly lost argument by confusing
the issue with irrelevant idioms like "speaking the truth" whose
idiomatic meaning goes far beyond the ordinary meaning of the word
"true".
Jim Balter
>
> John says...
> >
> >Also sent directly to Daryl
> >
> >Daryl McCullough wrote:
> >-- snip snip
> >
> >> Yes, but the particular undecidable proposition that Godel
> >> constructs is *true* (if the theory is consistent). The statement
> >> G constructed via Godel's proof is provably equivalent to the
> >> consistency of the theory. That is,
> >
> >What does "*true*" in this statement mean?
It means that "the particular undecidable proposition" follows from the
consistency of the theory -- Godel proved that. Of course, it
isn't decidable in the theory because the theory's consistency
isn't decidable in the theory -- Godel proved that.
> >I know of a couple of meanings:
> > a. Epistemology: corresponding to reality
>
> Yes, that's what "true" means,
This is strikingly odd coming from you, Daryl, especially after
having recently stated that "truth is relative to an interpretation"
and "The simplest, and I think, most consistent, way to interpret
claims about truth is the eliminationist interpretation."
Have you suddenly become a Platonist?
> except that I would replace "reality" by
> "the intended domain of discourse" which might be the physical world, if
> we are talking about physical objects, or it might be some mathematical
> structure, if we are talking about mathematics. In Godel's theorem, the
> domain of discourse is the natural numbers.
>
> Actually, I wouldn't use the word "epistemology" here. Doesn't
> epistemology have to do with how we come to know things? In that
> case, truth doesn't have much to do with epistemology---whether
> something is true is independent of whether we know that it is
> true.
Whether something is true is dependent on whether it follows
from our axioms of discourse, whether we know that it follows
or not.
> > b. Logic: Assumption (as of an axiom)
> > Derived from a collection of axioms
> > by an accepted collection of rules of inference
>
> No, that's definitely *not* what is meant by truth.
Of course it is -- denying it is inconsistent with eliminativism.
> "True" never means
> "derived from a collection of axioms by accepted rules of inference".
I mean that; Godel meant that.
> If the
> axioms and rules of inference were chosen carefully, then we can hopefully
> arrange things so that it is impossible to derive a statement that is not
> true, but if our axioms are false (in the intended interpretation)
Axioms can't be false (unless you are an essentialist/Platonist
rather than an eliminativist), they can only be inconsistent. Of course,
if they are inconsistent with our axioms of discourse ("the intended
interpretation") then we say they are false, but this just means
that we take some axioms as more primitive -- and thus undeniable --
than others.
> then
> some of the derived conclusions will be false, as well.
I.e., they will be inconsistent with the axioms of discourse.
Are you an eliminativist, or not?
> >Clearly the statement cannot mean that
> > "the Godel construct is "derived ..." and that it is not
> > derivable from the collection of axioms.
The Godel construct is derivable from the collection of axioms
plus the assumption that they are consistent. Thus the truth
of the Godel construct *must* *always* be conditioned on
the consistency of the theory. The mistakes of Lucas, Penrose,
and the like are based on ignoring the fact that (the truth of)
the Godel construct is a *consequent* whose *antecedent* is
the consistency of the theory -- it does not and cannot stand alone.
> >Is there any logical meaning of "true"
>
> No, there is no logical definition of truth.
Whatever happened to your eliminativism?
> However, we can
> convince ourselves that our logic is *sound*, which means that
> it is impossible to derive a false conclusion from true axioms.
If our logic is sound, then it's impossible to derive conclusions
that contradict the axioms. Are you an eliminativist or an
essentialist/Platonist?
> Truth has to do with the *meaning* of sentences, not with how
> the sentence is proved.
"how the sentence is proved" is a red herring. What is true
must be derivable, whether we know how to do so or not.
The truth of the Godel construction is derivable from the axioms
of the theory plus the consistency of the theory -- that's how
we know it's true! It wasn't via divine revelation, it was via
a theorem of arithmetic.
> >or are we reduced
> >to something like:
> > "the Godel construct is epistemologically true but logically
> >undecidable"?
>
> I would say that "The Godel sentence is true of the natural numbers,
> but it is not provable using the axioms of Peano arithmetic".
You can only say it if you accept that PA is consistent, which is
of course not provable using the axioms of PA -- Godel proved all this.
What happened to your eliminativism?
> It
> is provable using more powerful theories, such as 2nd order arithmetic
> or ZFC.
"more powerful" meaning that the consistency of PA is derivable from
the axioms of such theories. It certainly doesn't mean "more
able to divine truth" or some such, although you seem to
be implying that.
> Many people think that if a mathematical statement is neither
> provable nor refutable,
In what theory?
> then it should be considered neither true
> nor false.
In what theory?
> But that really isn't the way that we normally use
> the word "true".
"normally" we don't pay attention to implicit assumptions.
That's not a good approach in this discourse.
> Suppose that I have a steel safe that nobody
> knows the combination to. If I tell you that the safe contains
> 100 dollars and it really does contain 100 dollars,
What does "really" mean here? This is a case where you *are*
using a God's eye view. If you never ever have the means to
establish whether the safe contains 100 dollars, then you have
no right to assert it, unless it is a stipulation/assumption/axiom.
> then I'm
> telling the truth, whether or not anyone ever can prove it.
This is God's truth -- it is not truth as we "normally use" the
word. It is downright circular -- you have *stipulated* that
the safe "really does contain 100 dollars" -- of course it is
true that the safe contains 100 dollars, since that is
*derivable* from the stipulation. Nothing else can justify
the claim (of course opening the safe and finding 100 dollars
justifies it, but then it follows from the entirety of our
axioms concerning observation and the use of evidence).
> If it doesn't contain 100 dollars, then I'm telling a falsehood,
> whether or not anyone can ever prove it.
it doesn't contain 100 dollars -> "it contains 100 dollars" is false.
It's just an inference -- a derivation. You seem to have quite
forgotten your eliminativism.
> The same holds for mathematical statements. If I say
>
> "There is no positive integers x,y,z, and n, such that
> n > 2 and x^n + y^n = z^n"
>
> then you understand what I'm saying, even if you don't
> know how to prove or disprove it.
You are saying that, for any set of positive integers x,y,z,n,
n > 2, the statement x^n + y^n = z^n contradicts the axioms
of arithmetic. But you seem not to recognize what you are saying,
having fallen down some Platonistic rathole.
> If there happen to be
> x,y,z,n satisfying n > 2 and x^n + y^n = z^n, then I'm
> lying.
Side issue: only if you believe otherwise, according to standard
usage of "lie".
> If not, I'm telling the truth. It doesn't matter
> whether anyone knows how to prove it.
Knowing how isn't the issue; contradiction is. If the statement
doesn't contradict the axioms, then there are no grounds for
claiming it is false. Mathematical statements aren't "just plain
false" as inscribed in Platonic heaven -- they are false in so far
as they contradict the axioms of a particular theory.
> (Actually, we now
> know that this true, but it was certainly true before
> anyone knew how to prove it).
This is a strawman. Suppose that FLT had been undecidable
in PA. Then it would have been neither true nor false *in PA*.
If there were some stronger theory in which FLT was decidable,
then it would be true or false in that theory, just as G(PA)
is undecidable in PA but is decidable in PA+con(PA).
There is no absolute sense in which mathematical statements are
true or false -- you already knew that, prior to this article,
and both your eliminativism and your recognition that
"truth is relative to an interpretation" are inconsistent
with the notion of the absolute truth of mathematical statements
that you expound here.
--
<J Q B>
Torkel Franzen
> > "True" never means
> > "derived from a collection of axioms by accepted rules of inference".
>
> I mean that; Godel meant that.
Godel certainly didn't mean that.
> Axioms can't be false (unless you are an essentialist/Platonist
> rather than an eliminativist), they can only be inconsistent.
You seem to have some weird notion of "eliminativism".
Jim Balter
>
> Jim Balter <j...@digisle.net> writes:
>
> > > "True" never means
> > > "derived from a collection of axioms by accepted rules of inference".
> >
> > I mean that; Godel meant that.
>
> Godel certainly didn't mean that.
Prove it.
> > Axioms can't be false (unless you are an essentialist/Platonist
> > rather than an eliminativist), they can only be inconsistent.
>
> You seem to have some weird notion of "eliminativism".
You seem to have some weird inability to express yourself
beyond authoritative exclamations.
I am using "eliminativism" to refer to what Daryl called
"eliminationism". Whether or not you find the notion "weird"
is irrelevant.
--
<J Q B>
Torkel Franzen
>Prove it.
Godel observed that the Godel sentence for T is true if T is
consistent. What gives you the idea that by this he meant that
the Godel sentence for T is "derived from a collection of axioms by
accepted rules of inference"?
Jim Balter
He showed that G(T) follows from T+con(T), presumably via accepted
rules of inference. What do *you* suppose "T is true" means?
--
<J Q B>
Torkel Franzen
>He showed that G(T) follows from T+con(T), presumably via accepted
>rules of inference.
Any sentence A, then, is true on your view, since it follows from
T+A?
Jim Balter
It's true in T+A -- if you are concerned with my view, it might
help if you read what I already wrote, which was a critique of
the notion of absolute truth. "is true on my view" without
an implicit or explicit agreement as to the underlying assumptions
is incoherent on my view.
You still haven't said what *you* think "G(T) is true" means.
But, you wrote
Godel observed that the Godel sentence for T is true if T is
consistent.
You also wrote
It is simply a theorem of arithmetic
that if T is consistent, the Godel sentence for T is true but
unprovable in T.
As it is a theorem of arithmetic, Godel didn't merely "observe"
something by looking around and saying "hey, see that?" --
He *showed* it. He showed that G(T) if con(T). And the way
things are shown in theorems is by deriving them from a
collection of axioms by accepted rules of inference.
If "G(T) if con(T)" can be derived from the axioms of arithmetic,
then G(T) can be derived from the axioms of arithmetic + con(T).
And by "eliminationism", "G(T) can be derived ..." is the same
as "``G(T) is true'' can be derived ...".
Ok, it's late here, I'm going to bed. If you have more you
think worth saying, I'll read it in the morning.
--
<J Q B>
Torkel Franzen
> It's true in T+A -- if you are concerned with my view, it might
> help if you read what I already wrote, which was a critique of
> the notion of absolute truth.
What I read was a claim that this is what Godel meant by "true". It
is not. You may of course correctly observe that G(T) is provable in
T+ConT (whether or not T is consistent) and that ~G(T) is provable in
T+~ConT (whether or not T is consistent). And you may of course choose
to eccentrically formulate these statements as "G(T) is true in
T+ConT" and "~G(T) is true in T+~ConT". Godel's observation, however,
was simply that G(T) is true if T is consistent. In this observation,
that G(T) is true means that every natural number satisfies the
predicate P(x), given that G(T) is formalized as (x)P(x).
Daryl McCullough
>
>Daryl McCullough wrote:
>Whether something is true is dependent on whether it follows
>from our axioms of discourse, whether we know that it follows
>or not.
Truth is relative to an interpretation, not relative to a
particular set of axioms. Of course, we try to choose axioms
so that they agree with our intended interpretation, but
we can make a mistake.
>> > b. Logic: Assumption (as of an axiom)
>> > Derived from a collection of axioms
>> > by an accepted collection of rules of inference
>>
>> No, that's definitely *not* what is meant by truth.
>
>Of course it is -- denying it is inconsistent with eliminativism.
>
>> "True" never means
>> "derived from a collection of axioms by accepted rules of inference".
>
>I mean that; Godel meant that.
No, I don't think he did. We choose our axioms and rules of
inference so as to be true (in the intended interpretation,
if there is one).
>> >Clearly the statement cannot mean that
>> > "the Godel construct is "derived ..." and that it is not
>> > derivable from the collection of axioms.
>
>The Godel construct is derivable from the collection of axioms
>plus the assumption that they are consistent. Thus the truth
>of the Godel construct *must* *always* be conditioned on
>the consistency of the theory. The mistakes of Lucas, Penrose,
>and the like are based on ignoring the fact that (the truth of)
>the Godel construct is a *consequent* whose *antecedent* is
>the consistency of the theory -- it does not and cannot stand alone.
The reason we *believe* that the Godel sentence is true is
because it is provably equivalent to the assumption that the
axioms are consistent.
>> >Is there any logical meaning of "true"
>>
>> No, there is no logical definition of truth.
>
>Whatever happened to your eliminativism?
I haven't abandoned it. To say that truth is relative
to an interpretation is not to say that it is relative
to a set of axioms.
>> Truth has to do with the *meaning* of sentences, not with how
>> the sentence is proved.
>
>"how the sentence is proved" is a red herring. What is true
>must be derivable, whether we know how to do so or not.
I don't agree with that.
>The truth of the Godel construction is derivable from the axioms
>of the theory plus the consistency of the theory -- that's how
>we know it's true!
Yes, that's how we *know* it's true. I'm distinguishing between
knowledge and truth.
>> I would say that "The Godel sentence is true of the natural numbers,
>> but it is not provable using the axioms of Peano arithmetic".
>
>You can only say it if you accept that PA is consistent, which is
>of course not provable using the axioms of PA -- Godel proved all this.
Once again, you're talking about what is *known* to be true.
The set of truths of arithmetic contains sentences that we will
never know to be true.
>What happened to your eliminativism?
>> It is provable using more powerful theories, such as 2nd order
>> arithmetic or ZFC.
>
>"more powerful" meaning that the consistency of PA is derivable from
>the axioms of such theories. It certainly doesn't mean "more
>able to divine truth" or some such, although you seem to
>be implying that.
I am implying that. ZFC is better at coming up with arithmetic
truths than PA is.
>> Many people think that if a mathematical statement is neither
>> provable nor refutable,
>
>In what theory?
Pick one.
>> then it should be considered neither true
>> nor false.
>
>In what theory?
Truth and falsity are not relative to a theory, they
are relative to a model, or interpretation.
>> Suppose that I have a steel safe that nobody
>> knows the combination to. If I tell you that the safe contains
>> 100 dollars and it really does contain 100 dollars,
>
>What does "really" mean here? This is a case where you *are*
>using a God's eye view.
Only if you equate truth with knowledge.
>If you never ever have the means to establish whether the safe
>contains 100 dollars, then you have no right to assert it,
>unless it is a stipulation/assumption/axiom.
>> then I'm
>> telling the truth, whether or not anyone ever can prove it.
>
>This is God's truth -- it is not truth as we "normally use" the
>word.
Only if you equate truth with knowledge. It's God's knowledge,
I would say.
>It is downright circular -- you have *stipulated* that
>the safe "really does contain 100 dollars" -- of course it is
>true that the safe contains 100 dollars, since that is
>*derivable* from the stipulation. Nothing else can justify
>the claim (of course opening the safe and finding 100 dollars
>justifies it, but then it follows from the entirety of our
>axioms concerning observation and the use of evidence).
I'm not talking about *justifying* the claim.
>> If it doesn't contain 100 dollars, then I'm telling a falsehood,
>> whether or not anyone can ever prove it.
>
>it doesn't contain 100 dollars -> "it contains 100 dollars" is false.
>It's just an inference -- a derivation. You seem to have quite
>forgotten your eliminativism.
No, I'm saying exactly
If it doesn't contain 100 dollars, then
"It contains 100 dollars" is false.
>> The same holds for mathematical statements. If I say
>>
>> "There is no positive integers x,y,z, and n, such that
>> n > 2 and x^n + y^n = z^n"
>>
>> then you understand what I'm saying, even if you don't
>> know how to prove or disprove it.
>
>You are saying that, for any set of positive integers x,y,z,n,
>n > 2, the statement x^n + y^n = z^n contradicts the axioms
>of arithmetic.
No, I'm not.
>But you seem not to recognize what you are saying,
>having fallen down some Platonistic rathole.
No. Intended interpretations can be in our heads, rather
than in Plato's heaven.
>> If there happen to be
>> x,y,z,n satisfying n > 2 and x^n + y^n = z^n, then I'm
>> lying.
>
>Side issue: only if you believe otherwise, according to standard
>usage of "lie".
>
>> If not, I'm telling the truth. It doesn't matter
>> whether anyone knows how to prove it.
>
>Knowing how isn't the issue; contradiction is. If the statement
>doesn't contradict the axioms, then there are no grounds for
>claiming it is false.
Right. There is difference between
X is false.
and
I know that X is false.
My eliminationism doesn't imply the equivalence of those two.
>Mathematical statements aren't "just plain
>false" as inscribed in Platonic heaven -- they are false in so far
>as they contradict the axioms of a particular theory.
In ZFC there is a definition of what it means for a sentence
of arithmetic to be true. That definition is *not* "provable
according to certain axioms", because most true sentences are
not provable in ZFC or PA.
The way in ZFC that we talk about truth of sentences is via
a *model*, not via a set of axioms.
>> (Actually, we now
>> know that this true, but it was certainly true before
>> anyone knew how to prove it).
>
>This is a strawman. Suppose that FLT had been undecidable
>in PA. Then it would have been neither true nor false *in PA*.
It would have been neither provable nor disprovable in PA.
It still would have been true in the standard model. It's an
easy inference:
FLT undecidable in PA --> FLT
>If there were some stronger theory in which FLT was decidable,
>then it would be true or false in that theory,
You are using "true" and "false" to mean "provable" and
"disprovable". That is *not* what I mean by the terms.
Truth and falsity is not relative to a theory, they are
relative to an interpretation, or a model.
>There is no absolute sense in which mathematical statements are
>true or false
Right, it's relative to an interpretation. But statements
in the language of arithmetic have an intended interpretation.
-- you already knew that, prior to this article,
>and both your eliminativism and your recognition that
>"truth is relative to an interpretation" are inconsistent
>with the notion of the absolute truth of mathematical statements
>that you expound here.
No, it isn't. I'm still talking about truth relative to an
interpretation, but an interpretation is *not* a set of axioms.
John Ahlstrom
Back to my first question. What is the meaning of
"true" in "T is true if T is consistent".
It cannot mean the T is derivable from the axioms
by an agreed set of rules of inference, because he just
showed that it is undecidable from those axioms and rules.
If Goedel did not mean "derivable..." what did he mean by
"true"?
JKA
--
The best way to get information on Usenet
is not to ask a question,
but to post the wrong information.
--Aahz' Law
Daryl McCullough
>
>Torkel Franzen wrote:
>> You seem to have some weird notion of "eliminativism".
>
>You seem to have some weird inability to express yourself
>beyond authoritative exclamations.
>
>I am using "eliminativism" to refer to what Daryl called
>"eliminationism". Whether or not you find the notion "weird"
>is irrelevant.
No, you are not using the term the way I was. I'm not sure
I was using the word correctly, but all I meant was a program
for eliminating sentences involving the word "true" to equivalent
sentences that don't involve the word true. So, we can convert
"2+2=4" is true
==>
2+2=4
That reduction doesn't require that I provide a set of axioms
for arithmetic. All it means is that, however I interpret
"2+2=4", I'm interpreting "'2+2=4' is true" the same way.
So, my eliminationist approach to truth doesn't replace claims
involving truth with any procedure for deciding such claims. It
only replaces claims involving truth with other claims that *don't*
involve truth.
George Greene
: Godel observed that the Godel sentence for T is true if T is
: consistent.
He did NOT merely OBSERVE it.
It is called Godel's 1st Incompleteness THEOREM, which
implies that he PROVED it.
: What gives you the idea that by this he meant that
: the Godel sentence for T is "derived from a collection of axioms by
: accepted rules of inference"?
I could answer this, but if I did, would I be interrupting your
pedagogy of someone else?
--
---
"It's difficult ... you need to be united to have any
strength, but internal issues have to be addressed."
--- E. Ray Lewis, on liberalism in America
George Greene
: > consistent. What gives you the idea that by this he meant that
: > the Godel sentence for T is "derived from a collection of axioms by
: > accepted rules of inference"?
Jim Balter <j...@digisle.net> writes:
: He showed that G(T) follows from T+con(T), presumably via accepted
: rules of inference. What do *you* suppose "T is true" means?
It certainly doesn't mean that.
And the issue was *not* whether T is true (T is a theory),
but rather whether G(T) is true. Since T is a 1st-order
theory, one of the first questions you have to ask
yourself about the meaning of "G(T) is true" is,
what kind of statement is G(T)? Is it, for example,
universally quantified?
George Greene
: >
: >
: > >He showed that G(T) follows from T+con(T), presumably via accepted
: > >rules of inference.
: Torkel Franzen wrote:
: >
: > Any sentence A, then, is true on your view, since it follows from
: > T+A?
:
: It's true in T+A --
No, it isn't.
T+A is a THEORY.
"In T+A" means "in" a world were what matters is what is PROVABLE.
Things are provable (or not) in THEORIES.
NOTHING is "true in" ANY *theory*. That is a category mistake.
Things are "true in" MODELS or INTERPRETATIONS of theories,
if the specifications of those interpretations or models are
succiently complete.
GaryForbis
news:vcb4rxl...@beta13.sm.luth.se...
Is that how G(T) is formalized? I could have swore that the Godel showed
there were some statements that were true but not provable. Maybe I'm
being confused because T is commonly used as a symbol for True rather than
Theory and I've seen Godel's proof using P as the specific predicate
"Provable"
where for G(T) to be formalized as (x)P(x) P must be some amalgam of
"is true" and "is provable".
Daryl McCullough
I guess what I meant by no logical definition of truth is that
(for a subject as complicated as arithmetic, anyway) there is
no set of inference rules such that a sentence is true
if and only if it is derived by those inference rules.
Jim Balter
>
> Torkel Franzen wrote:
>
> > Jim Balter <j...@digisle.net> writes:
> >
> > >Prove it.
> >
> > Godel observed that the Godel sentence for T is true if T is
> > consistent. What gives you the idea that by this he meant that
> > the Godel sentence for T is "derived from a collection of axioms by
> > accepted rules of inference"?
>
> Back to my first question. What is the meaning of
> "true" in "T is true if T is consistent".
First, it's "G(T) is true if T is consistent".
> It cannot mean the T is derivable from the axioms
> by an agreed set of rules of inference,
*If* T is consistent, then the truth of G(T) is derivable
from the axioms of arithmetic.
> because he just
> showed that it is undecidable from those axioms and rules.
No, he showed that G(T) is not derivable from T, absent
an assumption that T is consistent.
> If Goedel did not mean "derivable..." what did he mean by
> "true"?
Franzen writes that
G(T) is true means that every natural number satisfies the
predicate P(x), given that G(T) is formalized as (x)P(x).
but this just passes the buck. How do we know that
"every natural number satisfies the predicate P(x)"
is true? Not by divine inspiration. Godel showed it, by
deriving this truth from the axioms of arithmetic.
There is no *warrant* to make any claim of truth unless
the claim is derivable from something that we *stipulate* to
be true.
--
<J Q B>
Jim Balter
>
> : > Godel observed that the Godel sentence for T is true if T is
> : > consistent. What gives you the idea that by this he meant that
> : > the Godel sentence for T is "derived from a collection of axioms by
> : > accepted rules of inference"?
> :
>
> Jim Balter <j...@digisle.net> writes:
> : He showed that G(T) follows from T+con(T), presumably via accepted
> : rules of inference. What do *you* suppose "T is true" means?
>
> It certainly doesn't mean that.
Then say what it *does* mean.
> And the issue was *not* whether T is true (T is a theory),
That was a typo, which I later corrected.
> but rather whether G(T) is true. Since T is a 1st-order
> theory, one of the first questions you have to ask
> yourself about the meaning of "G(T) is true" is,
> what kind of statement is G(T)? Is it, for example,
> universally quantified?
Godel made clear what quantification applies.
--
<J Q B>
Torkel Franzen
> Back to my first question. What is the meaning of
> "true" in "T is true if T is consistent".
I don't think you mean "T is true", but rather "G
is true". This is an arithmetically definable property
of statements G of the form (x)P(x). A statement of this
form is true if every natural number satisfies P.
More generally, "A is a true arithmetical statement"
is definable in second order number theory.
Torkel Franzen
> Franzen writes that
>
> G(T) is true means that every natural number satisfies the
> predicate P(x), given that G(T) is formalized as (x)P(x).
>
> but this just passes the buck. How do we know that
> "every natural number satisfies the predicate P(x)"
> is true?
Of course in general we have no idea whether or not the
Godel sentence of T is true! This is completely irrelevant
to the question what is meant by saying that the Godel
sentence of T is true.
Torkel Franzen
> Is that how G(T) is formalized? I could have swore that the Godel showed
> there were some statements that were true but not provable.
Provability is relative to a theory.
Jim Balter
Fine, then A is true in the standard model of T+A.
But I think it best to avoid talk of "truth"
and just stick to what can or cannot be shown. If you can't
show it, then any claim as to something being true is just
hot air.
--
<J Q B>
Jim Balter
>
> Jim Balter <j...@digisle.net> writes:
>
> > Franzen writes that
> >
> > G(T) is true means that every natural number satisfies the
> > predicate P(x), given that G(T) is formalized as (x)P(x).
> >
> > but this just passes the buck. How do we know that
> > "every natural number satisfies the predicate P(x)"
> > is true?
>
> Of course in general we have no idea whether or not the
> Godel sentence of T is true!
Of course, but this is irrelevant; we're only talking about
the inference conT -> G(T). Again, how do we know that
"every natural number satisfies the predicate P(x)"
is true when conT? Replacing "G(T) is true" with
"every natural number satisfies the predicate P(x), given
that G(T) is formalized as (x)P(x)" tells us nothing
about *truth*, it's just a syntactic trick -- we could as
well say
G(T) means that every natural number satisfies the
predicate P(x), given that G(T) is formalized as (x)P(x).
or
G(T) means that "every natural number satisfies the
predicate P(x), given that G(T) is formalized as (x)P(x)"
is true.
or
G(T) is true means that "every natural number satisfies the
predicate P(x), given that G(T) is formalized as (x)P(x)"
is true.
> This is completely irrelevant
> to the question what is meant by saying that the Godel
> sentence of T is true.
Godel didn't just *say* it, he *showed* that conT -> G(T).
So he showed that conT -> every natural number satisfies
the predicate P(x). Since it's a proof, it follows
from axioms via inference rules.
--
<J Q B>
Jim Balter
And, Godel showed that some statements were true *if* the theory
is consistent. He did *not* show that G(PA) is true!
The truth of G(PA) is also relative to the theory -- G(PA)
is *not* true given PA+~conPA.
--
<J Q B>
Jim Balter
If not, then what warrants a claim that the sentence is true?
Why can't I simply deny your claim that something is true
if you can't show it to be true? You wrote
I'm not talking about *justifying* the claim.
Yeah, you and Jerry Falwell. As soon as you *make* a claim,
the rest of the world has the right to talk about justifying it.
You wrote
I am implying that. ZFC is better at coming up with arithmetic
truths than PA is.
All this says is that ZFC better matches our pretheoretical
intuitions -- we want conPA to be true, so we pick a theory
with axioms from which we can derive conPA. This doesn't
make conPA "true" in any non-circular sense. I think it is
a serious mistake to equate our intuitions with "truth".
We should have learned that from Euclid's 5th postulate.
For those who would like the continuum hypothesis or the axiom
of choice to be true, there are theories that are better at coming
up with such "truths" than other theories, but what of it?
You wrote
It's God's knowledge, I would say.
Sorry, there is no God, and thus no God's knowledge.
There is only our knowledge -- that which we know how to
derive from our stipulations -- which is a subset of our truths,
that which is derivable from our stipulations. The more we
stipulate, the more truths we have -- we ironically call
theories with more stipulations "stronger", but that strength
is obtained at quite a cost. TANSTAAFL -- there ain't no such
thing as a free lunch, or an absolute truth. Gregory Chaitin's
work is relevant here -- at some point, you only get out what
you put in. conPA -> G(PA); ~conPA -> ~G(PA) -- take your pick;
*God* doesn't care, only we do.
--
<J Q B>
Torkel Franzen
> Of course, but this is irrelevant; we're only talking about
> the inference conT -> G(T). Again, how do we know that
> "every natural number satisfies the predicate P(x)"
> is true when conT?
By the proof, of course. Why do you imagine that this has
a bearing on the meaning of "true"?
Let's make a resolute effort to take your statements seriously. Do
you take "A is true" to mean "A is provable in T, for some T" (which
makes "A is true" vacuous), or do you take it to mean "A is provable
in T", for some particular T? If so, which T? Or do you renounce
"true" as a predicate and use only the relation "A is true in T",
meaning "A is provable in T"? Whatever your answer, how do you
interpret the observation (=simple mathematical theorem) that for any
consistent recursively axiomatizable T there are infinitely many true
statements of the form (x)P(x) with primitive recursive P that are not
provable in T?
Torkel Franzen
> The truth of G(PA) is also relative to the theory -- G(PA)
> is *not* true given PA+~conPA.
To say that "the truth of G(PA) is relative to a theory" makes
no sense at all on the ordinary meaning of "true".
Daryl McCullough
>> Godel's observation, however,
>> was simply that G(T) is true if T is consistent. In this observation,
>> that G(T) is true means that every natural number satisfies the
>> predicate P(x), given that G(T) is formalized as (x)P(x).
>
>Is that how G(T) is formalized? I could have swore that the Godel showed
>there were some statements that were true but not provable.
That's right. Godel proved
G(T) <-> not Prove(G(T),T) <-> con(T)
In other words, G(T) is true if and only if it
is not provable in T if and only if T is consistent.
George Greene
: > ... cannot mean that G(T) is derivable from the axioms
: > by an agreed set of rules of inference,
Jim Balter <j...@digisle.net> writes:
: *If* T is consistent, then the truth of G(T) is derivable
: from the axioms of arithmetic.
That is almost the DENIAL of Godel's theorem, Jim.
: > because he just
: > showed that it is undecidable from those axioms and rules.
:
: No, he showed that G(T) is not derivable from T, absent
: an assumption that T is consistent.
No, Jim, he showed that G(T) is not derivable from T,
PRESENT the assumption that T is consistent.
To see why it has to be this way, you have to learn
that there is more than one definition of "inconsistent".
The "first" or usual definition of "T is inconsistent" is that there
exists some statement s such that both s and ~s are provable from T.
BUT
most of the logics we are considering have the property
that you can derive ANYthing from a contradiction.
Symbolically (p&~p)->q is tautologous for ALL p & q
-- including, for our purposes, p=s and q=G(T).
Thus we learn ANOTHER, second definition of "inconsistent"
(equivalent, under this false-implies-anything rule, to the first
-- which is WHY you didn't know there were 2 definitions,
since they are sort of only 1) which is that
T is inconsistent iff EVERY statement s
is provable/derivable from T.
In other words, if T were INconsistent,
it would follow trivially and definitionally that EVERY
statement -- THEREFORE OBVIOUSLY INCLUDING G(T) -- was
provable; that wouldn't even be interesting. But if
T *is* consistent -- i.e., if there are ANY unprovable statements at all,
of ANY kind, from T -- then Godel's theorem says
that G(T) must be one of them
(in addition to the denials of all the proved theorems).
: > If Goedel did not mean "derivable..." what did he mean by
: > "true"?
That depends on what kind of statement it is.
If it is a disjunction, it is true iff either of its disjuncts is
(yes, that passes the buck, recursively, but to SMALLER statements,
so the recursion WILL terminate). If it is a conjunction, it
is true iff each of its conjuncts is. If it is existentially quantified,
it is true iff it has a true instance (generated by replacing the
outer existentially quantified variable by some value from the domain).
FINALLY (this is the part that MATTERS for your question),
If (as G(T) is) it is universally quantified, it is true iff *all* of
its instances are true. When this all becomes infinitely many, we get
the best easy example of a gap bewteen true and provable, because
PROOF IS FINITARY. To prove this universal statement might in
some sense require proving its instance for 0, proving its instance
for 1, proving its instance for 2, and proving its instance for 3, ... and
for 999999999999999999999999999999999999998, and proving its instance
for 999999999999999999999999999999999999999, ..., etc.
Each of those might be doable. But even if they were, all
that together would not COUNT as a proof because *PROOFS* HAVE TO BE *FINITE*!
: Franzen writes that
:
: G(T) is true means that every natural number satisfies the
: predicate P(x), given that G(T) is formalized as (x)P(x).
: but this just passes the buck.
Of course!
But what you need to appreciate is that that is simply COMPLETELY
(pun intended or not??) unavoidable. I mean, suppose you were
basing your confidence in the truth of statement s on the fact
that you had PROVED it from theory T: how would you know if T
were consistent? How would you know that your rules of inference
in your proof from T were reliable? The only way you could
justify your confidence would be to PROVE the consistency of T
and the validity of your scheme-of-application of inference rules!
HOW would you prove that?? Godel shows that you are CERTAINLY NOT
going to be able to prove it INSIDE T ITSELF. You will HAVE to
conduct such a proof at a higher level, in a meta-language ABOUT
T and its logic. Of course, after you prove this meta-proof, how
do you know you proved IT reliably?
It's a deep result, Jim.
: How do we know that
: "every natural number satisfies the predicate P(x)"
: is true?
We don't, if all we have is Peano Arithmetic.
: Not by divine inspiration. Godel showed it,
No, he didn't. He showed that it had to be true
IF ARITHMETIC IS CONSISTENT. But the point is, you
don't know THAT, either. At least, not if all you have to
justify it is Peano Arithmetic.
: by deriving this truth from the axioms of arithmetic.
No, actually, he proved that that this truth CANNOT BE derived
from the axioms of arithmetic, IF arithmetic is consistent (if
it were not, EVERYthing could be so derived).
: There is no *warrant* to make any claim of truth unless
: the claim is derivable from something that we *stipulate* to
: be true.
No, really, that's not true.
That's the whole point, except that there are LEVELS of stipulation.
Some truths have the property that their derivability has to be
stipulated on a higher level than the one on which they
are normally asserted and evaluated.
Daryl McCullough
>But I think it best to avoid talk of "truth"
>and just stick to what can or cannot be shown.
Why?
>If you can't show it, then any claim as to something
>being true is just hot air.
What does truth have to do with it? Why not say "If you
can't show it, then any claim is just hot air"?
You seem to be saying that "X is true" is only meaningful
if you have some way of proving X. But I would say that
"X is true" is meaningful whenever X is meaningful. That's
an immediate consequence of my "eliminationist" approach
to truth---"X is true" means exactly the same thing as X.
I understand what the claim "No rabbit was ever born
with blue fur" means, even though I have no way to prove
it. To me, it seems like an unacceptable straight-jacket
to say that sentences are meaningless until they are proved
or disproved. The reason people *tried* to prove Fermat's
last theorem is because it is a meaningful claim. It was
perfectly meaningful before it was ever proved. Even if
it had *never* been proved, it would still have been a
meaningful claim.
I'm sorry if you mistakenly thought that I was on the
side of those who equated truth with provability. I've
never advocated such a position, and have argued against
such a position every time it has come up.
Jim Balter
The *ordinary* meaning of "true", as used by the population at
large, allows for such statements as "That's true, if you assume ...".
If you entertain a notion of "true" that allows for absolute truths
independent of assumptions, then your notion is incoherent.
Whether or not you think my notion "makes no sense at all",
people use it and talk about it. See, e.g.,
http://www.eeng.dcu.ie/~tkpw/mail_archives/critical-cafe/Y1996-M12/msg00085.html
--
<J Q B>
George Greene
: Fine, then A is true in the standard model of T+A.
A is true in *all* models of T+A.
And G(T) is true in all models of T+G(T).
Assuming hereafter that T is Peano Arithmetic,
G(T) is also true in the standard model of T.
But there are non-standard models of T in which G(T) is NOT true;
that is *why* G(T) is not provable from T.
Another way of saying this is that our intended standard
true arithmetic is not 1st-order "categorical"; 1st-order
axiom-sets complicated enough to do arithmetic are unfortunately
going to have, in addition to the standard model, OTHER models
that differ from it in surprising ways.
: But I think it best to avoid talk of "truth"
: and just stick to what can or cannot be shown. If you can't
: show it, then any claim as to something being true is just
: hot air.
Not if it's infinitary.
By *definition*, Ax[P(x)] is TRUE iff all its instances are TRUE,
EVEN when there is no finitary "showing" of Ax[P(x)].
There might be a finitary showing of every instance.
That's enough, for truth, even if not for showability.
It's more than enough, actually.
And there are plenty of other questions, like,
what if you CAN show it, but just not *IN* T?
Or what if you use it every day, LIKE ARITHMETIC?
How do you justify your belief that PA is consistent,
given that Godel proved you can't show that in PA?
Daryl McCullough
>Again, how do we know that
>"every natural number satisfies the predicate P(x)"
>is true when conT?
By proving the implication
Con(T) -> for all x, P(x)
>Replacing "G(T) is true" with
>"every natural number satisfies the predicate P(x),
>given that G(T) is formalized as (x)P(x)" tells us nothing
>about *truth*, it's just a syntactic trick
I don't know what you want it tell you about truth. G(T)
is a name for a statement of the form "for all x, P(x)".
So, saying "G(T) is true" is the same thing as saying
"for all x, P(x)".
>> This is completely irrelevant
>> to the question what is meant by saying that the Godel
>> sentence of T is true.
>
>Godel didn't just *say* it, he *showed* that conT -> G(T).
>So he showed that conT -> every natural number satisfies
>the predicate P(x). Since it's a proof, it follows
>from axioms via inference rules.
Right. The *implication* Con(T) -> G(T) is provable. But
G(T) by itself is not. So when Godel said "If T is consistent,
then G(T) is true" he was not saying "It T is consistent,
then G(T) is provable".
Daryl McCullough
>
>Torkel Franzen wrote:
>>
>> "GaryForbis" <GaryF...@email.msn.com> writes:
>>
>> > Is that how G(T) is formalized? I could have swore that the Godel showed
>> > there were some statements that were true but not provable.
>>
>> Provability is relative to a theory.
>
>And, Godel showed that some statements were true *if* the theory
>is consistent. He did *not* show that G(PA) is true!
Nobody here claimed otherwise. (Although Penrose and Lucas
may have.)
>The truth of G(PA) is also relative to the theory'
No, it's relative to a model. It's true in the standard
model, but not true in nonstandard models.
>G(PA) is *not* true given PA+~conPA.
You mean it is not true in a model of PA + ~con(PA)
(which is necessarily a nonstandard model of PA).
Daryl McCullough
>
>George Greene wrote:
>> Jim Balter <j...@digisle.net> writes:
>> : He showed that G(T) follows from T+con(T), presumably via accepted
>> : rules of inference. What do *you* suppose "T is true" means?
>>
>> It certainly doesn't mean that.
>
>Then say what it *does* mean.
"G(T) is true"
means
"There does not exist a natural number coding a
proof of G(T) from the axioms of T"
(or something provably equivalent to that)
Jim Balter
>
> Jim Balter <j...@digisle.net> writes:
>
> > Of course, but this is irrelevant; we're only talking about
> > the inference conT -> G(T). Again, how do we know that
> > "every natural number satisfies the predicate P(x)"
> > is true when conT?
>
> By the proof, of course. Why do you imagine that this has
> a bearing on the meaning of "true"?
Uh, that was *my* point. *You* were purporting to answer the
question of what "G(T) is true" *means*, but in fact what you
said had no bearing on the question.
We can say conT -> G(T) or we can say conT -> "G(T) is true";
they are the same thing. "is true" is just syntactic sugar;
there is no predicate "is true"; there are not statements
that are or are not true all by themselves; there are only
*consequents* of *antecedents*. G(T) is a consequent
of conT.
> Let's make a resolute effort to take your statements seriously. Do
> you take "A is true" to mean "A is provable in T, for some T" (which
> makes "A is true" vacuous), or do you take it to mean "A is provable
> in T", for some particular T? If so, which T? Or do you renounce
> "true" as a predicate and use only the relation "A is true in T",
> meaning "A is provable in T"? Whatever your answer, how do you
> interpret the observation (=simple mathematical theorem) that for any
> consistent recursively axiomatizable T there are infinitely many true
> statements of the form (x)P(x) with primitive recursive P that are not
> provable in T?
I do not believe you are taking me seriously. If you want to say that
your phrasing is a theorem, then what matters is *your* interpretation
of "true". I could easily pick an interpretation by which it isn't
a theorem at all. But surely the theorem can be stated in such a
way that it doesn't use the word "true" at all, and then we can see what
you must mean by the word in this particular context.
--
<J Q B>
Daryl McCullough
>
>Daryl McCullough wrote:
>> I guess what I meant by no logical definition of truth is that
>> (for a subject as complicated as arithmetic, anyway) there is
>> no set of inference rules such that a sentence is true
>> if and only if it is derived by those inference rules.
>
>If not, then what warrants a claim that the sentence is true?
I'm not talking about what *warrants* a claim that a sentence
is true, I'm talking about what it *means* for a sentence to
be true.
>Why can't I simply deny your claim that something is true
>if you can't show it to be true?
You can certainly deny anything you like.
>You wrote
>
> I'm not talking about *justifying* the claim.
>
>Yeah, you and Jerry Falwell. Yeah, you and Jerry Falwell.
>As soon as you *make* a claim, the rest of the world has
>the right to talk about justifying it.
Of course it does. I never claimed otherwise. If I
claim that there is life on Mars, you can certainly
ask me to justify why I would say such a thing. But
the *meaning* of my claim is not the same as its
justification.
>You wrote
>
> I am implying that. ZFC is better at coming up with arithmetic
> truths than PA is.
>
>All this says is that ZFC better matches our pretheoretical
>intuitions -- we want conPA to be true,
We only want Con(PA) to be true if it PA is consistent.
If PA is inconsistent, then we don't want Con(PA) to be
true.
>so we pick a theory with axioms from which we can derive conPA.
That's not the whole story. ZFC allows us to say what we *mean*
by the natural numbers in a much more direct way than PA does.
A nice benefit of this expressiveness is that it allows us to
prove a few more things, such as Con(PA).
>This doesn't make conPA "true" in any non-circular sense.
*If* PA is consistent, then con(PA) is true. That's not circular,
because it is relating a proof-theoretic notion, the consistent
of a particular theory, to a number-theoretic notion.
In the theory T' = PA + not Con(PA), then Con(PA) will be disprovable,
and so will Con(T'). That *doesn't* mean that T' is inconsistent. What
it means is that in any model of T', Con is not interpreted to mean
"consistent".
The thing that is nice about the standard model is that it is nice
(via Godel's encoding) to interpret proof-theoretic claims as
number-theoretic claims.
>I think it is a serious mistake to equate our intuitions with "truth".
As long as we are clear that it is truth relative to an interpretation,
then what is the problem?
To me, it is a much bigger mistake to equate truth with provability.
>We should have learned that from Euclid's 5th postulate.
I don't see how Euclid's 5th postulate supports your case.
It's not provable (from the other axioms) but it is true in
certain geometries, and false in other geometries. There is
more than one model of Euclid's axioms. That's why truth is
always relative to a model, as I've always said.
>For those who would like the continuum hypothesis or the axiom
>of choice to be true, there are theories that are better at coming
>up with such "truths" than other theories, but what of it?
I don't know---what of it?
Daryl McCullough
>You wrote
>
> It's God's knowledge, I would say.
>
>Sorry, there is no God, and thus no God's knowledge.
I didn't bring up God, you did. I said that if you
equate truth with knowledge (which I *don't*) then
my notion of truth would be God's knowledge. Since
I don't equate truth and knowledge, I don't need a
God.
>There is only our knowledge -- that which we know how to
>derive from our stipulations -- which is a subset of our truths,
>that which is derivable from our stipulations. The more we
>stipulate, the more truths we have
I don't agree. I can stipulate that there are only 10,000,000
pairs of twin primes, but that doesn't make it true.
>TANSTAAFL -- there ain't no such
>thing as a free lunch, or an absolute truth.
I've never advocated absolute truth. I'm only talking
about truth relative to an interpretation.
>Gregory Chaitin's work is relevant here -- at some point,
>you only get out what you put in. conPA -> G(PA);
>~conPA -> ~G(PA) -- take your pick;
You can't just "take your pick". We can study models of
PA, and we can show that any model satisfying ~con(PA)
must be nonstandard. So, the two choices are not equally
good, if we are interested in the standard model.
Daryl McCullough
>The *ordinary* meaning of "true", as used by the population at
>large, allows for such statements as "That's true, if you assume ...".
I don't interpret such statements as claims that something is
*derivable* under those assumptions. Saying "A is true is true,
if you assume B" just means "B implies A". That doesn't mean
that A is true "relative to a theory in which B is true", it
means that *either* B and A are both true, or B is false.
>If you entertain a notion of "true" that allows for absolute truths
>independent of assumptions, then your notion is incoherent.
I don't think so.
>Whether or not you think my notion "makes no sense at all",
>people use it and talk about it. See, e.g.,
>
>http://www.eeng.dcu.ie/~tkpw/mail_archives/critical-cafe/Y1996-M12/msg00085.html
I don't think that facts being theory-laden means that
facts must be *derivable* from a set of axioms.
Jim Balter
>
> Jim says...
>
> >But I think it best to avoid talk of "truth"
> >and just stick to what can or cannot be shown.
>
> Why?
To avoid solipsism. You can make all sort of
claims, but if there is no possible way to demonstrate
them, then they aren't worth discussing. David Chalmers
talks about a world just like this one except that
horses have completely undetectable horns. To me, this is
just a jumble of words -- it doesn't *say anything* at all;
I have no idea what it means for a horse to "have" such
a horn, or even to talk about whether such a horn "exists"
or not.
> >If you can't show it, then any claim as to something
> >being true is just hot air.
>
> What does truth have to do with it? Why not say "If you
> can't show it, then any claim is just hot air"?
First, by "can't show it", I don't mean that you are
currently unable to show it, I mean that there is no
possible means to show it. Secondly, "truth" is implicit
in "claim", so it has a lot to do with it. Or, if there
are other sorts of claims, then that is the basis for the
distinction.
> You seem to be saying that "X is true" is only meaningful
> if you have some way of proving X. But I would say that
> "X is true" is meaningful whenever X is meaningful.
First, I'm not so interested in what is meaningful as in what
is *justified*. It's one thing to say that "Euclid's 5th postulate
is true" is meaningful -- after all, it *might* have been derivable
from the other 4, but it's another thing to *claim* that it is
or isn't true. Just because something is meaningful, that doesn't
warrant uttering it.
Second, is ``"chocolate ice cream tastes better than vanilla ice cream"
is true'' meaningful whenever "chocolate ice cream tastes
better than vanilla ice cream" is meaningful? I don't
think so -- when I say that c tastes better than v, I am
talking about my preferences, but I can't talk about the
truth of the statement when it is divorced from my uttering it.
The problem here is that, in natural language, we elide various
sorts of context that affect the meaning of a statement.
If we include the context, e.g., "I prefer c to v" and
``"I prefer c to v" is true'', then the elimination will more
often apply.
But there is an important category of statements where this
doesn't apply, and that is our *stipulations*. Certainly
"There is one and only one line passing through a given
point that is parallel to a given line" is *meaningful*,
but is it *true*? Is it meaningful to talk about it being
true? Only after we *stipulate* it, in which case of course
it follows from itself.
> That's
> an immediate consequence of my "eliminationist" approach
> to truth---"X is true" means exactly the same thing as X.
>
> I understand what the claim "No rabbit was ever born
> with blue fur" means, even though I have no way to prove
> it.
But there is a *possible* means of showing it. "No rabbit
was ever born with blue health", OTOH, is meaningless.
And again, I'm interested in the warrant for claims, more
than I am in whether people can agree as to how to interpret
a claim.
> To me, it seems like an unacceptable straight-jacket
> to say that sentences are meaningless until they are proved
> or disproved.
Of course, but that's radically different from anything I've said.
> The reason people *tried* to prove Fermat's
> last theorem is because it is a meaningful claim. It was
> perfectly meaningful before it was ever proved. Even if
> it had *never* been proved, it would still have been a
> meaningful claim.
Strawman.
> I'm sorry if you mistakenly thought that I was on the
> side of those who equated truth with provability. I've
> never advocated such a position, and have argued against
> such a position every time it has come up.
I thought you agreed that truth doesn't float free
in Plato's heaven or God's mind, such that a safe does
or does not contain 100 dollars independently of whether
it is possible to establish which. For all we know,
the safe is topologically equivalent to a Moebius strip
and the notion of "contains" is ill-defined for it.
Or the safe is in a superposition of quantum states.
Or it isn't really a safe at all, but merely a facade,
like a movie set or like Quine's cow billboards
(maybe there's a real safe behind the facade and *it*
contains 100 dollars, but does "the safe" refer to *that*
safe?). Or possibly other, unimagined, conditions apply for
which "the safe contains 100 dollars" is ill-defined.
Just as with Euclid's 5th postulate, we can stipulate
that the safe does or does not contain 100 dollars and
consider what follows from that, but imagining that there is a
"truth" to the statement when such truth is not accessible
is just that -- imagining.
--
<J Q B>
Jim Balter
[stuff]
Thanks your insightful comments, George. I'll have to
take time to absorb them. I've certainly bitten off
more than I can chew!
Thanks also to Daryl and Torkel to take me seriously
enough to attempt (and probably succeed) to rebut me.
--
<J Q B>
Daryl McCullough
>
>Daryl McCullough wrote:
>>
>> Jim says...
>>
>> >But I think it best to avoid talk of "truth"
>> >and just stick to what can or cannot be shown.
>>
>> Why?
>
>To avoid solipsism. You can make all sort of
>claims, but if there is no possible way to demonstrate
>them, then they aren't worth discussing.
The problem is that claims don't come equipped with
the procedures for investigating their truth. We have
to work to discover such procedures. I think it is
important that we can discuss the consequences of
various possible claims, to decide whether they are
worth investigating.
>David Chalmers talks about a world just like this one except that
>horses have completely undetectable horns. To me, this is
>just a jumble of words -- it doesn't *say anything* at all;
>I have no idea what it means for a horse to "have" such
>a horn, or even to talk about whether such a horn "exists"
>or not.
I don't consider that a matter of whether it is *verifiable*.
It's a matter of whether the claim has *consequences*. Those
aren't the same things. Theories of physics typically aren't
verifiable, in the sense that there is no way to prove them,
but they have deducible consequences.
>> You seem to be saying that "X is true" is only meaningful
>> if you have some way of proving X. But I would say that
>> "X is true" is meaningful whenever X is meaningful.
>
>First, I'm not so interested in what is meaningful as in what
>is *justified*.
Well, that isn't what I'm talking about in this thread. I'm
talking about the *meaning* of sentences such as "G is true".
>It's one thing to say that "Euclid's 5th postulate
>is true" is meaningful -- after all, it *might* have been derivable
>from the other 4, but it's another thing to *claim* that it is
>or isn't true.
Of course.
>Just because something is meaningful, that doesn't
>warrant uttering it.
I don't think that anyone has said otherwise.
>Second, is ``"chocolate ice cream tastes better than vanilla ice cream"
>is true'' meaningful whenever "chocolate ice cream tastes
>better than vanilla ice cream" is meaningful? I don't
>think so -- when I say that c tastes better than v, I am
>talking about my preferences, but I can't talk about the
>truth of the statement when it is divorced from my uttering it.
You're right there.
>But there is an important category of statements where this
>doesn't apply, and that is our *stipulations*. Certainly
>"There is one and only one line passing through a given
>point that is parallel to a given line" is *meaningful*,
>but is it *true*?
It depends on how you are interpreting the terms "line"
and "point". Of course, the axioms constrain the interpretation,
but they don't determine it.
>Is it meaningful to talk about it being
>true? Only after we *stipulate* it, in which case of course
>it follows from itself.
I would say only if we are limiting our discussion to geometries
for which it is true.
>> I understand what the claim "No rabbit was ever born
>> with blue fur" means, even though I have no way to prove
>> it.
>
>But there is a *possible* means of showing it.
Okay, that's a different point. Yes, I certainly
can make a meaningful distinction between some claims, which
would be testable if only I lived long enough, and was in the
right place at the right time, etc. and claims that I can't
even *imagine* how I would test them/
>"No rabbit was ever born with blue health", OTOH, is meaningless.
That's because we don't have an agreed-upon interpretation
of "blue health", not because I have no way of testing the
blue health of rabbits.
>And again, I'm interested in the warrant for claims, more
>than I am in whether people can agree as to how to interpret
>a claim.
But it seems to me that meaning must come *prior* to
establishing the criterion for how we determine the
truth of something. It's often quite a bit of work
to make a claim testable. Claims don't (usually) come
with test kits.
>> To me, it seems like an unacceptable straight-jacket
>> to say that sentences are meaningless until they are proved
>> or disproved.
>
>Of course, but that's radically different from anything I've said.
Maybe so, but it seemed to me that you were saying that.
>> The reason people *tried* to prove Fermat's
>> last theorem is because it is a meaningful claim. It was
>> perfectly meaningful before it was ever proved. Even if
>> it had *never* been proved, it would still have been a
>> meaningful claim.
>
>Strawman.
I don't see how it is a strawman. It seemed to me that you
were claiming that the truth of statements is only meaningful
if there is a way to establish their truth.
>> I'm sorry if you mistakenly thought that I was on the
>> side of those who equated truth with provability. I've
>> never advocated such a position, and have argued against
>> such a position every time it has come up.
>
>I thought you agreed that truth doesn't float free
>in Plato's heaven or God's mind,
No, I haven't expressed an opinion about Platonism.
>such that a safe does or does not contain 100 dollars
>independently of whether it is possible to establish which.
>For all we know, the safe is topologically equivalent to a
>Moebius strip and the notion of "contains" is ill-defined for it.
Sure, that's a possibility. Some claims may be neither true nor
false.
>Or the safe is in a superposition of quantum states.
>Or it isn't really a safe at all, but merely a facade,
>like a movie set or like Quine's cow billboards
>(maybe there's a real safe behind the facade and *it*
>contains 100 dollars, but does "the safe" refer to *that*
>safe?). Or possibly other, unimagined, conditions apply for
>which "the safe contains 100 dollars" is ill-defined.
>Just as with Euclid's 5th postulate, we can stipulate
>that the safe does or does not contain 100 dollars and
>consider what follows from that, but imagining that there is a
>"truth" to the statement when such truth is not accessible
>is just that -- imagining.
Sure.
Alan Smaill
> Torkel Franzen wrote:
> >
> > Jim Balter <j...@digisle.net> writes:
> >
> > > The truth of G(PA) is also relative to the theory -- G(PA)
> > > is *not* true given PA+~conPA.
> >
> > To say that "the truth of G(PA) is relative to a theory" makes
> > no sense at all on the ordinary meaning of "true".
>
> The *ordinary* meaning of "true", as used by the population at
> large, allows for such statements as "That's true, if you assume ...".
> If you entertain a notion of "true" that allows for absolute truths
> independent of assumptions, then your notion is incoherent.
You seem to have departed from your original claim, which
was about Goedel's understanding of his theorems.
Are you saying that Goedel thought that "the truth of G(PA)
is relative to a theory", as you seem to say?
> --
> <J Q B>
--
Alan Smaill email: A.Sm...@ed.ac.uk
Division of Informatics tel: 44-131-650-2710
Edinburgh University
Frank Rudzicz
main point was that there are true 'statements' (specifically, about natural
numbers) that cannot be proven, therefore Math is a trivial pursuit
(personally, I think that Godel was probably just bitter that no
mathematician ever took him seriously). In other words, if I were to say
"this is not an implementation of Godel's Theorem", then it would have to be
a true statement (if it were false, it WOULD be an implementation of the
theorem, but since it's false it can't be a theorem, contradiction, kaput).
Now I just read the theorem 5 minutes ago but the way I understand it is
that Godel is saying that the fact that the above statement is true cannot
be determined from our corpora which consists, unfortunately, of that single
sentence.
What I don't understand is why we can't add new axioms, or corpora to our
knowledge base? When I was really young I had no idea that, say, P=I^2R
(electricity). That was a true statement that I just couldn't prove given my
knowledge base. Eventually, I learnt the rules V=IR and P=IV and used these
new axioms to come to that original true conclusion. What real world
application can Godel's Theorem have except to validate the efforts of
researchers in probability-based AI, whose main selling-point is
uncertainty? All I'm getting out of this no-name is that to find hidden
truths I must keep gathering new knowledge, which seems intuitive anyway.
Hmm...maybe Godel was saying that nobody can know absolutely everything?
Genius!
BTW, whatever happened to John Searle in his little padded room?
Franko
George Greene <gre...@eagle.cs.unc.edu> wrote in message
news:xessnl5...@eagle.cs.unc.edu...
George Greene
: > I guess what I meant by no logical definition of truth is that
: > (for a subject as complicated as arithmetic, anyway) there is
: > no set of inference rules such that a sentence is true
: > if and only if it is derived by those inference rules.
:
: If not, then what warrants a claim that the sentence is true?
There is a definition, although it is recursive and model-
relative, not absolute.
: Why can't I simply deny your claim that something is true
: if you can't show it to be true? You wrote
:
: I'm not talking about *justifying* the claim.
:
: Yeah, you and Jerry Falwell. As soon as you *make* a claim,
: the rest of the world has the right to talk about justifying it.
Not exactly.
The law of the excluded middle really does ensure that
there are claims that have to be true despite the
fact that they can't be justified. Suppose there are
7 people: I, who will be rolling a standard 6-sided die,
and 6 others, each of whom has a distinct favorite number,
from 1-6. Suppose these people are my friends and agree
with me to play and joke on you. I am going to roll the
die, where you can see it but they can't, and each of them
promises to claim "He rolled my favorite number: _ " (fill
in the blank, 1 different number, for each person).
One of the claimants' claims MUST be TRUE, despite
the fact that all of them have zero justification.
Knowing that the odds are 5-1 against something
happening means that claiming it DIDN'T happen might
have a justification, but claiming that it did can't.
Fundamentally, what forces 1 of the claims to be true,
despite the fact that none of them is justifiable, is
that the die has to come down/up on some face or another.
In classical logic, statements are like coins: because
P v ~P is tautologous for ALL P, EVERY statement HAS
to be EITHER true or false in EVERY fully-specified model.
The standard model in particular is fully specified (despite
the fact that Peano Arithmetic doesn't determine it).
: You wrote
:
: I am implying that. ZFC is better at coming up with arithmetic
: truths than PA is.
This is irrelevant, period.
: All this says is that ZFC better matches our pretheoretical
: intuitions -- we want conPA to be true, so we pick a theory
: with axioms from which we can derive conPA.
ZFC is not such a theory, and even if it were, it wouldn't
matter, because we still wouldn't know that ZFC was consistent,
so we would have to doubt our proof-of-Con(PA)-in-ZFC.
It *wouldn't*help*.
: This doesn't make conPA "true" in any non-circular sense.
No, THAT doesn't, but it is true in the standard model of PA.
It is true in any model of ANY axioms that looks enough like
standard arithmetic that we would want to use it. The existence
of the intended/standard model makes it true.
: I think it is a serious mistake to equate our intuitions with "truth".
It is an even bigger mistake to equate axiomatic derivability
with truth, but you continue to insist on making it. Godel
would be frustrated.
: We should have learned that from Euclid's 5th postulate.
No, you are drawing entirely the wrong lesson.
Euclid's 5th postulate WAS true.
It still is, in the relevant models.
The reason it was not provable was that in addition
to the standard model, there also existed other models.
The fact that the consistency of the alternative axioms
was discovered BEFORE the other models was really a big
VICTORY for the difference between truth and derivability.
Euclid's 5th postulate (and its alternatives) are all
TRUE in SOME models of the smaller postulate-set, but
not provable from it.
: For those who would like the continuum hypothesis or the axiom
: of choice to be true, there are theories that are better at coming
: up with such "truths" than other theories, but what of it?
The axiom of choice is obviously true.
The fact that there are consistent theories of
first-order axiom-sets in which it is false is first-
order-logic's problem, or the problem of the non-standard
kinds of sets that result from those axioms, NOT the
axiom of choice's (or set theory's) problem.
: there is no God, and thus no God's knowledge.
That's ridiculous.
Arithmetic is a classic example of "God's knowledge".
: There is only our knowledge -- that which we know how to
: derive from our stipulations -- which is a subset of our truths,
: that which is derivable from our stipulations.
Truth is NOT derivable from stipulations, at least not
completely. The point is, EVERY time you add a NEW stipulation,
you create NEW truths, which are NOT derivable in the framework
to which you just added the stipulation. THat is provable,
but only in a higher framework with other stipulations.
: The more we stipulate, the more truths we have -- we ironically call
: theories with more stipulations "stronger", but that strength
: is obtained at quite a cost. TANSTAAFL -- there ain't no such
: thing as a free lunch, or an absolute truth.
Yes, there is, but the problem is, all the absolute truth
is about either abstractions or history. There is absolute
truth about yesterday's weather and yesterday's elections,
despite the fact that we don't absolutely know it completely.
: Gregory Chaitin's
: work is relevant here -- at some point, you only get out what
: you put in.
In terms of what's justifiable or provable, yes.
In terms of what's true, no; there is always MORE truth
than provability.
There is a difference between recursive, recursively enumerable,
and not-even-recursively-enumerable (there are degrees of the
latter).
: conPA -> G(PA);
This is true; you can also reverse the arrow.
But you need to clarify the axiomatic context
in which you are asserting it.
: ~conPA -> ~G(PA)
This is vacuous.
If PA is not consistent then it can prove everything,
including ~G(PA). If PA is consistent then the
hypothesis is false so the statement is vacuously true.
: -- take your pick;
: *God* doesn't care, only we do.
No, God cares that we act halfway consistent
about it all. God cares that equality remain
reflexive, symmetric, and transitive, for example.
James Dolan
<some stuff, deleted>
hi, jim. i have a question for you. actually it's more of an
interactive sequence of questions, but since i don't find this medium
sufficiently interactive for my tastes, i'll try to get all of the
preliminary interactive part out of the way all at once by stating a
few assumptions and/or conventions and/or stipulations which i hope
you and i can agree to stick to for the moment, before asking you my
main questions.
stipulation #1:
there's a recursively axiomatizable first-order logical theory called
"zfc" (short for "zermelo-frankel set theory plus the axiom of
choice") that's reasonably unambiguously defined in the mathematical
logic literature accessible to you and me. thus in principle you and
i could write a computer program to print out the infinite sequence of
axioms of zfc. furthermore, on good days we both feel capable of
grappling with this r.a.f.o.l.t. to the extent of trying to prove some
very simple theorems in it or engaging in similar exercises, possibly
at an informal level but formalizable in principle.
(the specific choice of zfc rather than some other reasonable formal
system is not important here; indeed eventually it's important that
what i'm talking about here not be tied down to just a single system.
i chose zfc just because of its fame.)
stipulation #2:
there's a specific predicate formula p in the language of zfc with one
free variable x such that p(x) expresses the idea that "x is a
statement in the language of first-order peano arithmetic" via some
specific system of goedel-numbering. p is sufficiently unambiguously
specified (up to insignificant variations) that in principle you and i
could actually write it down on paper or in a text-file.
stipulation #3:
there's a specific predicate formula q with one free variable x such
that the statement "for any x, q(x) implies p(x)" is provable in zfc,
and such that q(x) expresses the idea that "x is a provable statement
of first-order peano arithmetic". q is sufficiently unambiguously
specified (up to insignificant variations) that in principle you and i
could actually write it down on paper or in a text-file.
(q(x) is a rigorously formalized version of some informal predicate
formula that says something like "there exists a finite list of
statements with x last in the list, with each entry in the last either
an axiom or derivable by a rule of inference from...blah blah
blah...".)
a reasonable name for q is "the provability predicate formula for
first-order peano arithmetic", or if you want to be more specific "the
provability predicate formula in zfc for first-order peano
arithmetic".
stipulation #4:
there's a specific predicate formula r with one free variable x such
that the statement "for any x, r(x) implies p(x), and q(x) implies
r(x)" is provable in zfc, and such that r(x) is a rigorously
formalized version of some informal predicate formula that says "x
belongs to every subset s of the set of statements of first-order
peano arithmetic such that [s contains all valid arithmetic equations,
and if y is a conjunction of elements of s then y is also in s, and if
y is a disjunction of statements at least one of which is in s then y
is also in s, and if [y starts with a universal quantifier symbol and
for any natural number n the result of stripping off the quantifier
symbol and the variable following it and replacing all other
occurrences of that variable by n belongs to s] then y is also in s,
and if...blah blah blah...]". r is sufficiently unambiguously
specified (up to insignificant variations) that in principle you and i
could actually write it down on paper or in a text-file.
now for the main questions:
1. what would you consider to be a reasonable name for the predicate
formula r, if not "the truth predicate formula for first-order
arithmetic" or "the truth predicate formula in zfc for first-order
arithmetic", or something very similar?
2. do you understand that r, or perhaps more accurately the entire
complex of ideas resulting from formally (or not-so-formally)
translating r into other formal (or not-so-formal) systems besides
zfc, is an inescapably important object in mathematics, and that when
people like daryl mccullough and torkel franzen talk about "truth"
what they're talking about is, roughly, this entire complex of ideas?
of course, another way of describing what this concept of "truth" is
is simply to say that it's the same old ordinary concept of "truth"
that intelligent third-graders generally seem to have little trouble
understanding, but the point of introducing r is to try to demonstrate
that certain attempts to escape the concept of "truth" by taking
refuge in "formalism" (the mindless it's-only-a-meaningless-game
variety that usually only turns up when opponents of "formalism" use
it as a straw-man) don't succeed.
i can't guarantee that mccullough or franzen would think that i've
described their ideas accurately here. i might be misunderstanding
the mathematical situation, for one thing. for another, i couldn't be
sure that my description was inaccurate even if one or both of them
thought that it was.
Torkel Franzen
> But surely the theorem can be stated in such a
> way that it doesn't use the word "true" at all, and then we can see what
> you must mean by the word in this particular context.
That a statement of the form (x)P(x), with P primitive recursive
and the quantifier ranging over the natural numbers, is true means
that every natural number satisfies the predicate P. This
definition can be given in arithmetic.
It is a consequence of this definition that "(x)P(x) is true"
is equivalent to (x)P(x), for every such sentence. That is, this
equivalence is also provable in arithmetic, for every particular (x)P(x).
Now consider the theorem: "if T is a recursively axiomatizable
consistent theory, there are infinitely many true statments of the
form (x)P(x) not provable in T", which is also provable in arithmetic.
"True" can of course be eliminated, using the definition. But
there is no elimination consisting in replacing "A is true" with A,
for any A.
Torkel Franzen
> I'd like to admit ignorance of Godel's Theorem.
This would have been a good place to stop.
GaryForbis
news:97602...@drn.newsguy.com...
> No, I don't think he did. We choose our axioms and rules of
> inference so as to be true (in the intended interpretation,
> if there is one).
> The reason we *believe* that the Godel sentence is true is
> because it is provably equivalent to the assumption that the
> axioms are consistent.
I think these are very important points.
What a strange world we'd be living in if we designed our theories
so that their intended interpretation showed them to be inconsistent.
GaryForbis
Thanks, I got it, again. I lost it for awhile.
(x)P(x) is the set of all provable theorems in T and con(T) is the assertion
that all provable theorems are true. (Why else have a theory?) PA can't
prove all provable theorems of PA are true.
Torkel Franzen
> (x)P(x) is the set of all provable theorems in T and con(T) is the assertion
> that all provable theorems are true. (Why else have a theory?) PA can't
> prove all provable theorems of PA are true.
Con(T) is equivalent (for the theories at issue) to "all Pi-0
theorems of T are true". It does not imply that all theorems of
T are true. For example, PA+~Con(PA) is consistent, but proves a
false theorem.
Pierre-Normand Houle
> No, you are drawing entirely the wrong lesson.
> Euclid's 5th postulate WAS true.
> It still is, in the relevant models.
> The reason it was not provable was that in addition
> to the standard model, there also existed other models.
> The fact that the consistency of the alternative axioms
> was discovered BEFORE the other models was really a big
> VICTORY for the difference between truth and derivability.
> Euclid's 5th postulate (and its alternatives) are all
> TRUE in SOME models of the smaller postulate-set, but
> not provable from it.
Aren't there models of the smaller postulate-set in which
Euclid's 5th postulate is indeterminate? (And not only
undecidable)
Neil W Rickert
>I'd like to admit ignorance of Godel's Theorem.
Your ignorance shows.
> If I'm not mistaken, Godel's
>main point was that there are true 'statements' (specifically, about natural
>numbers) that cannot be proven, therefore Math is a trivial pursuit
You are mistaken.
Jim Balter
> : there is no God, and thus no God's knowledge.
>
> That's ridiculous.
> Arithmetic is a classic example of "God's knowledge".
Well, here a crucial matter. Many people disagree
with you, so it isn't "ridiculous". In a radically
different universe with creatures that aren't
"object-oriented", something quite different might be
considered "God's knowledge".
--
<J Q B>
Jim Balter
Arrogance can turn even the most knowledgeable people
into fools.
--
<J Q B>
Jim Balter
> : Yeah, you and Jerry Falwell. As soon as you *make* a claim,
> : the rest of the world has the right to talk about justifying it.
>
> Not exactly.
> The law of the excluded middle really does ensure that
> there are claims that have to be true despite the
> fact that they can't be justified.
No, it ensures that the *conjunction* of some claims
has to be true, not that any one of the claims has to
be true.
> Suppose there are
> 7 people: I, who will be rolling a standard 6-sided die,
> and 6 others, each of whom has a distinct favorite number,
> from 1-6. Suppose these people are my friends and agree
> with me to play and joke on you. I am going to roll the
> die, where you can see it but they can't, and each of them
> promises to claim "He rolled my favorite number: _ " (fill
> in the blank, 1 different number, for each person).
People can promise to say whatever they like, but so what?
These claims are still unjustified, and these folks
would do better to remain silent than to utter what
is most likely a lie.
> One of the claimants' claims MUST be TRUE, despite
> the fact that all of them have zero justification.
So what? The claim that one of these claims must be true
is quite a different claim from any one of the claims.
> Knowing that the odds are 5-1 against something
> happening means that claiming it DIDN'T happen might
> have a justification, but claiming that it did can't.
So don't claim it. Sheesh. Like I said,
"As soon as you *make* a claim, the rest of the world
has the right to talk about justifying it." And in the
case of you and your arrogant friends, I challenge
their justification for making unsupported claims.
> Fundamentally, what forces 1 of the claims to be true,
Nothing forces any of the claims to be true.
Only the conjunction is true. This is just the standard
confusion about quantifiers when translated into English.
> despite the fact that none of them is justifiable, is
> that the die has to come down/up on some face or another.
> In classical logic, statements are like coins: because
> P v ~P is tautologous for ALL P, EVERY statement HAS
> to be EITHER true or false in EVERY fully-specified model.
> The standard model in particular is fully specified (despite
> the fact that Peano Arithmetic doesn't determine it).
This suggests that classic logic isn't adequate for our
purposes. The kinds of statements we make are more
like "P" than "P v ~P". Would you say that P has
to be either true or false? You don't even know what
statement it is. In fact, all statements made by humans
are like that -- their meaning is contextual, and the
context is never *fully* articulated. Classic logic
has no room for that sort of linguistic indeterminism.
> Euclid's 5th postulate WAS true.
> It still is, in the relevant models.
It's quite false on this here globe, so I have to
wonder about your concept of "relevant".
> The reason it was not provable was that in addition
> to the standard model, there also existed other models.
> The fact that the consistency of the alternative axioms
> was discovered BEFORE the other models was really a big
> VICTORY for the difference between truth and derivability.
> Euclid's 5th postulate (and its alternatives) are all
> TRUE in SOME models of the smaller postulate-set, but
> not provable from it.
This is a rather useless sort of "TRUE". Perhaps you also
consider it true that nothing can be simultaneously
red-all-over and green-all-over? For decades or even centuries
philosophers debated whether this was an analytic truth or
a synthetic truth, but in fact it's true only due to the nature
of our perceptual system, and can in fact be falsified with the
right lab setup.
> : For those who would like the continuum hypothesis or the axiom
> : of choice to be true, there are theories that are better at coming
> : up with such "truths" than other theories, but what of it?
>
> The axiom of choice is obviously true.
This is a religious claim.
> Truth is NOT derivable from stipulations, at least not
> completely. The point is, EVERY time you add a NEW stipulation,
> you create NEW truths, which are NOT derivable in the framework
> to which you just added the stipulation.
> : The more we stipulate, the more truths we have -- we ironically call
> : theories with more stipulations "stronger", but that strength
> : is obtained at quite a cost. TANSTAAFL -- there ain't no such
> : thing as a free lunch, or an absolute truth.
>
> Yes, there is,
No there isn't.
> but the problem is, all the absolute truth
> is about either abstractions or history. There is absolute
> truth about yesterday's weather and yesterday's elections,
> despite the fact that we don't absolutely know it completely.
Sigh. Are you totally unaware of the arguments against your
naive realism, or are you just too arrogant to pay any attention
to anything that contradicts what you find "obviously true"?
There is no absolute truth about these things, as historians
know very well. In order to even make a statement about the
weather, you need a language in which to express it, and
such a language carries numerous assumptions -- what I called
"axioms of discourse". In philosophy of science, reference
is made to "theory-laden observation". Platonic mathematicians
way consider such things to be "ridiculous", but their arrogant
insistence as to what is true, and their appeal to God,
is just a lot of hot air.
> : *God* doesn't care, only we do.
>
> No, God cares that we act halfway consistent
> about it all.
That's nonsense. Just what does this "caring" consist of?
You just have delusions of grandeur, raising what *you* care
about to the divine. Many people who aren't mathematicians
go about there way in quite an inconsistent fashion, and are
often happier and live longer lives than some "halfway
consistent" mathematician.
> God cares that equality remain
> reflexive, symmetric, and transitive, for example.
That's deep linguistic nonsense.
--
<J Q B>
Jim Balter
> of course, another way of describing what this concept of "truth" is
> is simply to say that it's the same old ordinary concept of "truth"
> that intelligent third-graders generally seem to have little trouble
> understanding,
There are so many erroneous Platonic assumptions packed in there that
I really don't know where to begin. But do consider reading a wide
variety of writings on epistemology from outside your community of
mathematics.
--
<J Q B>
JRStern
wrote:
I dunno, third graders? Pretty much an operant conditioning or
associative model at that age, are those Platonic? Academic, maybe.
<g>
J.
Jim Balter
I wouldn't be surprised to find that academics tend to have had an
unusually strong attachment to their third grade teachers relative to
the population at large. *My* experience with my 3rd grade teacher
(dear Miss Gleason, a sweet young thing) was that she insisted that
"gymnasium" had three syllables because it was written in the school
dictionary as "gym.na'si.um" and the school dictionary said that
syllables were separated by dots; apparently the authors just assumed
that it was understood that accent marks accent syllables -- what
else? But the fertile Miss Gleason wasn't swayed by arguments of such
limited imagination. We had a class debate, and eventually everyone
went to her side (my best friend at the time, the last to leave me,
said "I know you're right, Jim, but ..." -- not surprisingly, he later
he became a member of Barry Goldwater's jingo squad, Young Americans
for Freedom) -- the pressure of the academy won out. When I asked
whether this proved I was wrong, she was wise enough to reject
consensus gentium, at least. The next day, through the greatest of
fortune, my mother took me with her to the market where there were
paperback dictionaries in a display at the checkout stand (ah, the
good old days -- circa 1958), and when I opened it up, I found the
marvelously redundant "gym.na.'si.um". "Mommy, mommy! I need to get
this!" -- ah, the simple joys of 8 year olds. Miss Gleason's
concession before the class the next day that I was right determined
my relationship with academia ever after.
But just what sort of "truth" is it that "gymnasium" contains has four
syllables, and what insight do the conceptualizations of 3rd graders
or 3rd grade teachers offer? I myself would be embarrassed to justify
my position on complex subjects by appeal to the conceptual levels of
such folks. For an alternative view to that of 3rd graders, consider
a few bits gathered from the web:
http://www.susx.ac.uk/Units/philosophy/ugpage/Wittlec7.html
http://www.math.psu.edu/simpson/fom/postings/9801/msg00398.html
http://www.uncletaz.com/library/philclass/Steiner/truthknowledge.html
http://www.bu.edu/wcp/Papers/Math/MathLand.htm
--
<J Q B>
Neil W Rickert
I didn't take Jim (Balter) to be accusing third graders of platonic
assumptions. I took that comment to refer to James Dolan.
If a particularly precoceous third grader were to discuss the concept
of "truth" as it is needed for AI, in the same simplistic terms, then
the criticism might reasonably fit there too.
James Dolan
could you please show me where you think the first significant
erroneous platonic assumption was in what i wrote? i don't think
there are a bunch of platonic assumptions in there; all i wrote about
was some text-strings that i want you to contemplate and answer a few
questions about. is it the platonic assumption of the existence of
those text-strings that you're objecting to? i hope you're not
objecting to the harmless easily removable "platonic assumptions" i
used merely to help evoke awareness of particular text-strings; that
would be sillier than, for example, me objecting to your apparent
"erroneous platonic assumption" of the concrete existence of certain
objects called "erroneous platonic assumptions" evidently in some sort
of "erroneous-platonic-assumption-land-in-the-sky".
or when you spoke of "erroneous platonic assumptions packed in there",
did you mean specificly in the one sentence of mine that you quoted
rather than in the rest of my post? if so, then just ignore that one
sentence; i assure you it was not directed to you but rather to any
intelligent third-graders who might happen to be listening in.
Jim Balter
>
> jim balter wrote:
>
> |James Dolan wrote:
> |
> |> of course, another way of describing what this concept of "truth" is
> |> is simply to say that it's the same old ordinary concept of "truth"
> |> that intelligent third-graders generally seem to have little trouble
> |> understanding,
> |
> |There are so many erroneous Platonic assumptions packed in there that
> |I really don't know where to begin. But do consider reading a wide
> |variety of writings on epistemology from outside your community of
> |mathematics.
>
> could you please show me where you think the first significant
> erroneous platonic assumption was in what i wrote?
The first: you assume that there is such a thing as "a concept".
The understandings of third graders can be determined operationally,
but it is seriously mistaken to suppose that there is "a concept"
to which they correspond.
> i assure you it was not directed to you but rather to any
> intelligent third-graders who might happen to be listening in.
This assurance assures me of your bad faith.
--
<J Q B>
Frank Rudzicz
Thanks for the insight! Since the only exposure I've had to this funny
little theorem has previously been through the inconsistencies of this
newsgroup, I luckily can feel no shame in any misinterpretations I might
have had. Since you were gracious enough to provide the details of this
principle of which you have complete understanding, I had no reason to look
elsewhere for further information. I did anyhow, and this is what I turned
up:
Godel's 'proof' states that there are logical statements within a corpus
that cannot be proven because of the possibility of constructing an axiom
that cannot be proven within the same corpus. Therefore, to prove the
self-consistency of the system, methods of proof from outside the system are
required. Hmm...that sounds like my original interpretation.
Again, I wonder what practical application this masterful accomplishment of
logic could serve. Likely, there is no more value in a real-world
application of this theorem than there is in the utterance of the phrase "I
am lying" ( If he's telling the truth then he's lying but if he's lying then
he's not telling the truth! Oh my!). BTW, the next time somebody requests
information or at least gives you the opportunity to display what knowledge
you have, perhaps stamping your foot and calling names would be best left to
high-schoolers.
Frank
Neil W Rickert <ricke...@cs.niu.edu> wrote in message
news:978lfd$9...@euclid.cs.niu.edu...
Jim Balter
>
> jim balter wrote:
>
> |James Dolan wrote:
> |
> |> of course, another way of describing what this concept of "truth" is
> |> is simply to say that it's the same old ordinary concept of "truth"
> |> that intelligent third-graders generally seem to have little trouble
> |> understanding,
> |
> |There are so many erroneous Platonic assumptions packed in there that
> |I really don't know where to begin. But do consider reading a wide
> |variety of writings on epistemology from outside your community of
> |mathematics.
> all i wrote about
> was some text-strings that i want you to contemplate and answer a few
> questions about.
There is the question of whether these text strings have anything to do
with the use of the word "truth" in common discourse. We have your
bald assertion about third graders, which I find not merely erroneous
but incoherent. As a sort of response, I offer
http://wings.buffalo.edu/philosophy/faculty/smith/articles/truthmakers/tm.html
Something you might want to contemplate if you can get off your
pedagogical pedestal for a while.
--
<J Q B>
Neil W Rickert
>Thanks for the insight! Since the only exposure I've had to this funny
>little theorem has previously been through the inconsistencies of this
>newsgroup, I luckily can feel no shame in any misinterpretations I might
>have had. Since you were gracious enough to provide the details of this
>principle of which you have complete understanding, I had no reason to look
>elsewhere for further information. I did anyhow, and this is what I turned
>up:
>Godel's 'proof' states that there are logical statements within a corpus
>that cannot be proven because of the possibility of constructing an axiom
>that cannot be proven within the same corpus. Therefore, to prove the
>self-consistency of the system, methods of proof from outside the system are
>required. Hmm...that sounds like my original interpretation.
Your original assertion:
If I'm not mistaken, Godel's main point was that there are
true 'statements' (specifically, about natural numbers) that
cannot be proven, therefore Math is a trivial pursuit
That assertion claimed that the statements are "true". Goedel's
theorem concludes only that they can neither be proved nor
disproved within the system.
More egregious is your conclusion "therefore Math is a trivial
pursuit". If anything, Goedel's theorem provides evidence for the
contrary.
Your parenthetic remark "(personally, I think that Godel was probably
just bitter that no mathematician ever took him seriously)" was
particularly stupid.
James Dolan
|James Dolan wrote:
|>
|> jim balter wrote:
|>
|> |James Dolan wrote:
|> |
|> |> of course, another way of describing what this concept of "truth" is
|> |> is simply to say that it's the same old ordinary concept of "truth"
|> |> that intelligent third-graders generally seem to have little trouble
|> |> understanding,
|> |
|> |There are so many erroneous Platonic assumptions packed in there that
|> |I really don't know where to begin. But do consider reading a wide
|> |variety of writings on epistemology from outside your community of
|> |mathematics.
|>
|> could you please show me where you think the first significant
|> erroneous platonic assumption was in what i wrote?
|
|The first: you assume that there is such a thing as "a concept".
|The understandings of third graders can be determined operationally,
|but it is seriously mistaken to suppose that there is "a concept"
|to which they correspond.
that example comes from the second-to-last sentence in my post, the
one i explicitly told you was not directed to you. (neither was the
last sentence directed to you, by the way.) are you saying that you
don't find any serious erroneous platonic assumptions in the prior
part of the post, which is where the substance was?
|> i assure you it was not directed to you but rather to any
|> intelligent third-graders who might happen to be listening in.
|
|This assurance assures me of your bad faith.
i'm not sure why you're reacting so strongly to my harmless joke
alluding to torkel franzen as an "intelligent third grader". in
writing my post, i was cringing as i imagined franzen looking over my
shoulder asking "why are you belaboring the obvious by translating it
into a formal system?" (or something like that; i'm sure i can't
imitate franzen's style).
intelligent third graders often have a naive but practical
understanding of significant parts of their world, taking certain
platonic assumptions for granted. then as they get older they
sometimes develop doubts about those naive platonic assumptions, but
they realize that to get on in the world it's expedient to "pretend"
that they still believe in those assumptions. sometimes they
eventually realize that this sort of "pretending" is so pervasive and
inescapable that it's expedient to define "belief" as this sort of
"pretense of belief" except perhaps in some very unusual special
cases, and then they give the outward appearance of agreeing with the
intelligent third graders. inward appearance too, for the most part.
i see that your posts are littered with indications of harmlessly
tentative platonic assumptions of existence of abstractions ("this
assurance", "your bad faith", "your bald assertion", etc.) just like
everyone else's are, so why bother pointing to my equally harmless
usages? why not just try to give an honest answer to my honest
questions from my original post?
Pierre-Normand Houle
> "Frank Rudzicz" <f_ru...@alcor.concordia.ca> writes:
>
> >Thanks for the insight! Since the only exposure I've had to this funny
> >little theorem has previously been through the inconsistencies of this
> >newsgroup, I luckily can feel no shame in any misinterpretations I might
> >have had. Since you were gracious enough to provide the details of this
> >principle of which you have complete understanding, I had no reason to
look
> >elsewhere for further information. I did anyhow, and this is what I
turned
> >up:
>
> >Godel's 'proof' states that there are logical statements within a corpus
> >that cannot be proven because of the possibility of constructing an axiom
> >that cannot be proven within the same corpus. Therefore, to prove the
> >self-consistency of the system, methods of proof from outside the system
are
> >required. Hmm...that sounds like my original interpretation.
>
> Your original assertion:
>
> If I'm not mistaken, Godel's main point was that there are
> true 'statements' (specifically, about natural numbers) that
> cannot be proven, therefore Math is a trivial pursuit
>
> That assertion claimed that the statements are "true". Goedel's
> theorem concludes only that they can neither be proved nor
> disproved within the system.
Goedel also exhibits a statement that is true if the system is
consistent. Under one interpretation this statement G means
"Neither G nor not G can be proven". The truth of this statement
follows from the consistency of the system.
James Dolan
|that example comes from the second-to-last sentence in my post, the
|one i explicitly told you was not directed to you. (neither was the
|last sentence directed to you, by the way.)
sorry, that was bad counting on my part. to correct the mistake,
replace both occurrences of "sentence" in the above paragraph by
"paragraph".
Jim Balter
>
> In article <3A96EA15...@digisle.net>, Jim says...
> >such that a safe does or does not contain 100 dollars
> >independently of whether it is possible to establish which.
> >For all we know, the safe is topologically equivalent to a
> >Moebius strip and the notion of "contains" is ill-defined for it.
>
> Sure, that's a possibility. Some claims may be neither true nor
> false.
Well, I'm glad to see you deny semantic realism, contrary to
George Greene's call upon classic logic and the excluded middle,
since the reason I entered this thread was to respond to your
Many people think that if a mathematical statement is neither
provable nor refutable, then it should be considered neither true
nor false. But that really isn't the way that we normally use
the word "true". Suppose that I have a steel safe that nobody
knows the combination to. If I tell you that the safe contains
100 dollars and it really does contain 100 dollars, then I'm
telling the truth, whether or not anyone ever can prove it.
If it doesn't contain 100 dollars, then I'm telling a falsehood,
I don't believe this *is* ``the way we normally use the word
"true"''. "The way" can only be determined by looking at
the entirety of how we use the word, especially how we
respond to problem cases such as safes without insides.
It's my contention that those statements that are either
true or false are precisely those statements that are
fully semantically determined, which are precisely those
statements for which there are unambiguous agreed upon
criteria as to their truth or falsity. As you say,
statements don't come with test kits. But it isn't merely
a matter of taking a statement as having a pre-determined
(by God or Plato) semantics and designing a test kit; the
indeterminate semantics of a statement is made determinate
by defining/refining its terms into testable form. Of course,
this isn't absolute -- the semantic net is never fully formed,
and there are always cases where the truth is indeterminate,
because the semantics are underdetermined.
This has great importance in the realm of c.a.p (vs. sci.logic),
where you and I hang out. Consider the statement "Machines can
think." Turing famously noted its semantic indeterminacy,
and suggested an alternative approach (although his Test
is still seriously underdetermined). Or consider "Zombies
are conceivable". Certainly they are conceivable as long
as we aren't too clear on just what it is we are conceiving --
Chalmers says zombies are just like us but "dark inside" --
quite a test, that. But sharpen up the criteria for determining
whether something is a zombie or not and the distinction
between zombies and us disappears -- either zombies really
are conscious and thus aren't zombies, or we are like
Dennett's "zimboes" -- we aren't "really" conscious, but we
put on a very good act, fooling even ourselves.
Consider especially this from Chalmers' _The Conscious Mind_:
Does a mouse have conscious experience? Does a virus? These
are not matters for stipulation. Either there is something
that it is like to be a mouse or there is not, and it is not
up to us to define the mouse's experience into or out of
existence.
I believe that Chalmers is deeply wrong here, and this flies
in the face of everything we have learned about linguistics
and philosophy of science. These questions are underdetermined
-- there is no fact of the matter of whether something has
conscious experience absent a model of consciousness that
provides some means of settling the question -- and as we
gather more neurophysiological data and develop conceptual
frameworks such as computationalism then we will become clearer
on what precisely we *want* consciousness to mean, and will
settle on criteria for deciding what is or is not conscious;
the facts of the matter, as always, will follow the formulation
of definitions. And I believe that philosophy is riddled with
this sort of thing -- great debates about questions that are not
settleable because the questions themselves are semantically
indeterminate.
--
<J Q B>
Neil W Rickert
>> Your original assertion:
Since we cannot prove the consistency of the system, this does not
actually add anything.
Jim Balter
>
> jim balter wrote:
>
> |James Dolan wrote:
> |>
> |> jim balter wrote:
> |>
> |> |James Dolan wrote:
> |> |
> |> |> of course, another way of describing what this concept of "truth" is
> |> |> is simply to say that it's the same old ordinary concept of "truth"
> |> |> that intelligent third-graders generally seem to have little trouble
> |> |> understanding,
> |> |
> |> |There are so many erroneous Platonic assumptions packed in there that
> |> |I really don't know where to begin. But do consider reading a wide
> |> |variety of writings on epistemology from outside your community of
> |> |mathematics.
> |>
> |> could you please show me where you think the first significant
> |> erroneous platonic assumption was in what i wrote?
> |
> |The first: you assume that there is such a thing as "a concept".
> |The understandings of third graders can be determined operationally,
> |but it is seriously mistaken to suppose that there is "a concept"
> |to which they correspond.
>
> that example comes from the second-to-last sentence in my post, the
> one i explicitly told you was not directed to you. (neither was the
> last sentence directed to you, by the way.) are you saying that you
> don't find any serious erroneous platonic assumptions in the prior
> part of the post, which is where the substance was?
I disagree that that is where the substance is. Or at least,
I have a different view as to what is important. Your claim
is that the zfc notion of truth is "the" one we use use every
day, and I disagree.
As for erroneous platonic assumptions, I already stated where
I found them -- namely, in what to me was your important substantive
claim. I'm not interested in scrutinizing your entire
body of work to see where else they may have manifested.
> |> i assure you it was not directed to you but rather to any
> |> intelligent third-graders who might happen to be listening in.
> |
> |This assurance assures me of your bad faith.
>
> i'm not sure why you're reacting so strongly to my harmless joke
> alluding to torkel franzen as an "intelligent third grader".
Ok, I'll accept that that is how you intended the latter,
although I see no internal evidence of it. But I do not
accept that that was the intent of the original comment
about third graders and their concepts -- perhaps you've simply
forgotten your intent. That comment was in a note addressed to me
and seems integral to the intent of the note, and was intended to
bolster your claim that this is what McCullough and Franzen are talking
about when they talk about "truth". Well, it may well be what
they *claim* to be talking about, but I'm not so sure it's what
they *are* talking about, especially Daryl. And if it is what they
are talking about, then they are likely to be miscommunicating
with third graders and others. In order to establish that,
we would need to look harder at how third graders and others use the
term "truth", which is why your claim about third graders is crucial.
> i see that your posts are littered with indications of harmlessly
> tentative platonic assumptions of existence of abstractions ("this
> assurance", "your bad faith", "your bald assertion", etc.) just like
> everyone else's are, so why bother pointing to my equally harmless
> usages?
This sort of syntactic concreteness is not of the same sort as your
equation of a formal definition of truth with "the same ordinary concept".
Your platonic assumptions are crucial to the claim -- it isn't
"harmless" at all.
> why not just try to give an honest answer to my honest
> questions from my original post?
Because I have finite time and don't wish to dwell on what I
don't consider important. I said something about Godel, Franzen
said "certainly not", and rather than defend my position I said
"prove it" -- at that point I abdicated any positive claim.
I later wrote "I've certainly bitten off
more than I can chew! Thanks also to Daryl and Torkel to take me
seriously enough to attempt (and probably succeed) to rebut me" --
I was referring here largely to my comment about Godel, and I haven't
addressed Godel since -- it was a side issue in my original note
to Daryl. You can lecture me on honesty and demand that I play the
subject to your pedagogy, but you would be better off putting your
energies elsewhere.
--
<J Q B>
TOCoons
writes:
>Many people think that if a mathematical statement is neither
>provable nor refutable, then it should be considered neither true
>nor false. But that really isn't the way that we normally use
>the word "true". Suppose that I have a steel safe that nobody
>knows the combination to. If I tell you that the safe contains
>100 dollars and it really does contain 100 dollars, then I'm
>telling the truth, whether or not anyone ever can prove it.
>If it doesn't contain 100 dollars, then I'm telling a falsehood,
>whether or not anyone can ever prove it.
>
The idea that if a statement (mathematical or otherwise) is neither
verifiable nor refutable, then it should be considered neither true nor false
has been one of the motivations for attempts to develop nonclassical logic by
using a third logical value to describe such statements.
Lukasiewicz tried this and made some progress, but this interpretation of
his logic runs into serious problems when the "law" of the excluded middle
[~ (P & ~P)]. can be applied, so this approach appears to be unproductive.
In working with multi-valued logics I have found it possible and useful to
describe and work with statements that are either definitely true or definitely
false, but whose actual truth value may or may not be known (or even knowable).
It is also possible to describe statements that are neither definitely
true nor definitely false, but somewhere in between. However, this is a
distinct case. These two kinds of statements do have remarkably similar logical
properties, so it is understandable that they would be easily confused, but
confusing them appears to be a serious impediment to the success of
multi-valued logic.
Frank Rudzicz
validity of a statement that could not have been determined given a limited
amount of knowledge (a conundrum you are likely familiar with), can be
resolved ("...methods of proof from outside the system are required") given
new knowledge. If a statement cannot be proven to be true or false within
the system, that does not negate the fact that the statment is either true
or false nor the fact that you require more information to disambiguate it.
The fact that I originally chose to use the term 'true' is obviously
arbitrary as if I had chosen 'false' and 'disproven', my original statement
would have lost absolutely no generality. (I'd like to point out that if we
were discussing probability-based rather than boolean-based logic, this
conversation would happily not be taking place).
I personally do not believe that Math is a trivial pursuit and that
impression of Godel's work may have been an unfair criticism of what I
believe to be a weak mathematical 'idea'. I have yet to hear of a practical
purpose that this theorem can serve and I believe that it offers no logical
implication (you can't build on this theorem as you can build on, say
transitivity or syllogism). I don't doubt that there is a reason that Godel
is mentioned in almost no discreet mathematics textbook. This sort of
logical concept is interesting in a Star Trek kind of way but in general I
doubt that it contributes in any way to the advancement of Artificial
Intelligence, which is why we are here in the first place.
Your annecdotal remark was typically useless.
Frank
Neil W Rickert <ricke...@cs.niu.edu> wrote in message
news:979e77$b...@euclid.cs.niu.edu...
JRStern
wrote:
>I didn't take Jim (Balter) to be accusing third graders of platonic
>assumptions. I took that comment to refer to James Dolan.
James Dolan was assuming the Platonic existence of third graders?
>If a particularly precoceous third grader were to discuss the concept
>of "truth" as it is needed for AI, in the same simplistic terms, then
>the criticism might reasonably fit there too.
Better get them reading Wittgenstein in second grade, I guess.
J.
Pierre-Normand Houle
> "Pierre-Normand Houle" <hou...@attglobal.net> writes:
> >Goedel also exhibits a statement that is true if the system is
> >consistent. Under one interpretation this statement G means
> >"Neither G nor not G can be proven". The truth of this statement
> >follows from the consistency of the system.
>
> Since we cannot prove the consistency of the system, this does not
> actually add anything.
How do you know that such a proof can not be achieved? Goedel's
theorem does not exclude the possibility of our producing a metamathematical
proof of the consistency of PA or ZFT.
(I skipped "of the consistency" in a previous reply that does not
appear on my server yet)
Neil W Rickert
>"Neil W Rickert" <ricke...@cs.niu.edu> wrote in message
>news:979iro$b...@euclid.cs.niu.edu...
>> "Pierre-Normand Houle" <hou...@attglobal.net> writes:
>> >Goedel also exhibits a statement that is true if the system is
>> >consistent. Under one interpretation this statement G means
>> >"Neither G nor not G can be proven". The truth of this statement
>> >follows from the consistency of the system.
>>
>> Since we cannot prove the consistency of the system, this does not
>> actually add anything.
>How do you know that such a proof can not be achieved?
It probably can. But that proof will likely be given in another
system whose consistency is unproven. Thus it will not really
provide any additional evidence of the consistency of the original
system.
Neil W Rickert
>My original assertion was that with the addition of new knowledge, the
>validity of a statement that could not have been determined given a limited
>amount of knowledge (a conundrum you are likely familiar with), can be
>resolved ("...methods of proof from outside the system are required") given
>new knowledge.
No, your original assertion was a false claim about what Goedel had
shown.
> If a statement cannot be proven to be true or false within
>the system, that does not negate the fact that the statment is either true
>or false nor the fact that you require more information to disambiguate it.
Whether that is a "fact" is disputed.
>I personally do not believe that Math is a trivial pursuit and that
>impression of Godel's work may have been an unfair criticism of what I
>believe to be a weak mathematical 'idea'.
Goedel's theorem settled one of Hilbert's problems (or perhaps I
should say that it unsettled the problem). Hilbert is usually not
accused of being a weak mathematician.
> I have yet to hear of a practical
>purpose that this theorem can serve and I believe that it offers no logical
>implication (you can't build on this theorem as you can build on, say
>transitivity or syllogism).
If it has stopped people from wasting their time trying to prove what
could not be proven, surely it is not wasted effort.
James Dolan
|But I do not
|accept that that was the intent of the original comment
|about third graders and their concepts -- perhaps you've simply
|forgotten your intent. That comment was in a note addressed to me
|and seems integral to the intent of the note,
if you read it again you'll see that the internal evidence shows that
the comment you're interpreting as my main substantive remark to you
was merely an afterword, and not even especially addressed to you. i
announced the form of my post at the beginning: some stipulations i
think we can agree to (there's where you should look for erroneous
assumptions i might be making), plus my _main questions_ for you, as i
twice referred to them. you have decided to completely ignore the
main part of my post. that's unfortunate, because if you tried to
understand it you might learn something useful.
the structure of my argument is to show you that even if i
scrupulously adhere to your own standards of abstraction-skepticism i
can still show you (up to the point where we have to decide whether to
interpret a certain formal system as having "real meaning") what is
this concept of "truth" that people like franzen and mccullough talk
about but which you refuse to believe even exists. the after-comments
that you use as an excuse to avoid thinking about my argument occur
explicitly outside of the "stipulation" section where i scrupulously
adhere to your restrictions.
even a selectively puritanical abstraction-skeptic such as yourself
must acknowledge the existence of "truth" if "truth" is defined as
merely a formal concept in a formal system (in principle just marks on
paper, etc.). treating the formal system formally, not trying to
ascribe any "meaning" to it, you can see how within the system "truth"
presents an interesting contrast to other parallel formal concepts
such as "provability". then you can begin to understand the thought
processes of people who tentatively ascribe some level of "real
meaning" to the formal system, and you can decide whether and how
tentatively you want to adopt such a stance yourself. if you think
about it much at all you tend to realize that tentatively ascribing
enough real meaning to the system to be able to talk meaningfully
about the "truth" of (for example) statements of first-order
arithmetic as something quite apart from their provability involves
engaging in only about the same amount of abstraction-making as is
indispensable in daily life (which is quite a lot).
when you feel comfortable ascribing real meaning to a formal system in
this way it means that to you it's more than just a formal system; you
recognize it as more or less isomorphic to a part of the way you
think. in this case the effect is to recognize that the naive concept
of "truth" you had in third grade is more or less isomorphic to the
formal concept of "truth" appearing in certain formal systems.
sometimes progress comes in doubting the obvious, and sometimes it
comes in realizing that the obvious idea was essentially correct after
all. if you doubt the obvious and turn out to be wrong about it, then
you end up looking foolish in the eyes of people who didn't even have
the imagination to doubt the obvious; you can resent that if you like.
even if you're not ready to ascribe meaning to the formal systems in
question, it should be educational for you to understand what are the
formal concepts in those formal systems that correspond to what people
are talking about. it's clear that you haven't previously understood
it.
Pierre-Normand Houle
On this view no evidence could be offered for any statement
about anything ever.
Can there not be evidences that are not to be identified with
derivability from a set of consistent axioms?
Neil W Rickert
>"Neil W Rickert" <ricke...@cs.niu.edu> wrote in message
>news:979qma$b...@euclid.cs.niu.edu...
>> "Pierre-Normand Houle" <hou...@attglobal.net> writes:
>> >"Neil W Rickert" <ricke...@cs.niu.edu> wrote in message
>> >news:979iro$b...@euclid.cs.niu.edu...
>> >> "Pierre-Normand Houle" <hou...@attglobal.net> writes:
>> >> >Goedel also exhibits a statement that is true if the system is
>> >> >consistent. Under one interpretation this statement G means
>> >> >"Neither G nor not G can be proven". The truth of this statement
>> >> >follows from the consistency of the system.
>> >> Since we cannot prove the consistency of the system, this does not
>> >> actually add anything.
>> >How do you know that such a proof can not be achieved?
>> It probably can. But that proof will likely be given in another
>> system whose consistency is unproven. Thus it will not really
>> provide any additional evidence of the consistency of the original
>> system.
>On this view no evidence could be offered for any statement
>about anything ever.
That does not follow from anything I wrote.
>Can there not be evidences that are not to be identified with
>derivability from a set of consistent axioms?
There is plenty of evidence for the consistence of PA. However, it
falls short of being mathematical proof.
Pierre-Normand Houle
> "Pierre-Normand Houle" <hou...@attglobal.net> writes:
> >> >> >Goedel also exhibits a statement that is true if the system is
> >> >> >consistent. Under one interpretation this statement G means
> >> >> >"Neither G nor not G can be proven". The truth of this statement
> >> >> >follows from the consistency of the system.
>
> >> >> Since we cannot prove the consistency of the system, this does not
> >> >> actually add anything.
>
> >> >How do you know that such a proof can not be achieved?
>
> >> It probably can. But that proof will likely be given in another
> >> system whose consistency is unproven. Thus it will not really
> >> provide any additional evidence of the consistency of the original
> >> system.
>
> >On this view no evidence could be offered for any statement
> >about anything ever.
>
> That does not follow from anything I wrote.
You seem to have said that you can not accept a proof of the consistency
of (a model of) arithmetic that is not performed in a system shown to be
consistent. And you said that this later condition is unlikely
to be satisfied. So, I ask you, do you know anything that is evidence
(or proof) for something else? And if so, what consistent set of axioms do
you use to derive this evidence (or proof)?
> >Can there not be evidences that are not to be identified with
> >derivability from a set of consistent axioms?
>
> There is plenty of evidence for the consistence of PA. However, it
> falls short of being mathematical proof.
Is your standard of "mathematical proof" derivability from a set of
axioms that has been shown to be consistent? If it is, then you ought
to say that the demonstration all theorems of arithmetic fall short of
being mathematical proofs.
Jim Balter
> that's unfortunate, because if you tried to
> understand it you might learn something useful.
> it's clear that you haven't previously understood it.
You seem to find some pleasure in finding something that I
haven't understood. There is no other apparent reason
for your post.
--
<J Q B>
Neil W Rickert
>> >> >> Since we cannot prove the consistency of the system, this does not
>> >> >> actually add anything.
>> >> >How do you know that such a proof can not be achieved?
>> >> It probably can. But that proof will likely be given in another
>> >> system whose consistency is unproven. Thus it will not really
>> >> provide any additional evidence of the consistency of the original
>> >> system.
>> >On this view no evidence could be offered for any statement
>> >about anything ever.
>> That does not follow from anything I wrote.
>You seem to have said that you can not accept a proof of the consistency
>of (a model of) arithmetic that is not performed in a system shown to be
>consistent.
I haven't said that either. I only said that such a proof does not
actually demonstrate the consistency of the original axiom system.
However, I do not deny that it might be a valid mathematical proof
within the meta-system where the proof was given.
> So, I ask you, do you know anything that is evidence
>(or proof) for something else? And if so, what consistent set of axioms do
>you use to derive this evidence (or proof)?
I am not claiming that formal proof within a system is the only
evidence acceptable. It is, however, the situation that Goedel's
theorem deals with.
>> >Can there not be evidences that are not to be identified with
>> >derivability from a set of consistent axioms?
>> There is plenty of evidence for the consistence of PA. However, it
>> falls short of being mathematical proof.
What I was actually referring to here, is that PA has been
extensively exercised, and used for proving many interesting
results. No inconsistency has yet turned up. That is good informal
evidence for the consistency of PA, but it is not a mathematical
proof of the consistency of PA.
>Is your standard of "mathematical proof" derivability from a set of
>axioms that has been shown to be consistent?
I was not requiring such a strict standard in my comment above.
> If it is, then you ought
>to say that the demonstration all theorems of arithmetic fall short of
>being mathematical proofs.
This is surely wrong. Many theorems of arithmetic are proved from
the axioms of a field, rather than from PA.
Greg Lee
...
> (dear Miss Gleason, a sweet young thing) was that she insisted that
> "gymnasium" had three syllables because it was written in the school
> dictionary as "gym.na'si.um" and the school dictionary said that
> syllables were separated by dots; apparently the authors just assumed
> that it was understood that accent marks accent syllables -- what
> else?
It's quite possible to pronounce this with three syllables (and a "zh"),
as in "conclusion", "Cartesian", and perhaps "Malthusian", "magnesium".
But like the question of how "true" is used, on its face it seems
to be a question of fact that should be settled by observation. Instead,
you tell us about dictionary conventions. A very revealing anecdote.
...
--
Greg Lee <l...@hawaii.edu>
Peter Percival
>
> Torkel Franzen wrote:
>
> > Jim Balter <j...@digisle.net> writes:
> >
> > >Prove it.
> >
> > Godel observed that the Godel sentence for T is true if T is
> > consistent. What gives you the idea that by this he meant that
> > the Godel sentence for T is "derived from a collection of axioms by
> > accepted rules of inference"?
>
> Back to my first question. What is the meaning of
> "true" in "T is true if T is consistent".
See Tarski's "The Concept of Truth in Formalized Languages", accessible
as Chapter VIII in "Logic Semantics Metamathematics". See also McKinsey
"A New Definition of Truth" in Synthese, vii, 1948-9, pp428-33, which
avoids the notion of satisfaction.
>
> It cannot mean the T is derivable from the axioms
> by an agreed set of rules of inference, because he just
> showed that it is undecidable from those axioms and rules.
>
> If Goedel did not mean "derivable..." what did he mean by
> "true"?
For Godel, derivable meant formally derivable in a formal system. True
meant informally provable in the meta-system. If you wish you can study
PA (the formal system) within a formalized meta-system, say set theory
as used by Tarski. "6 = 2x3 (say) is a truth of PA" will then be a
theorem of that set theory. When Godel wrote his incompleteness paper
Tarski's work was not available to him.
>
> JKA
>
> --
> The best way to get information on Usenet
> is not to ask a question,
> but to post the wrong information.
> --Aahz' Law
--
The from address is fictional
peter dot percival at cwcom dot net
Torkel Franzen
> For Godel, derivable meant formally derivable in a formal system. True
> meant informally provable in the meta-system.
What gives you this idea?
Torkel Franzen
> I haven't said that either. I only said that such a proof does not
> actually demonstrate the consistency of the original axiom system.
Why not? Does any mathematical proof ever demonstrate anything?
I see no justification for being skeptical about consistency proofs
that doesn't equally justify skepticism with respect to mathematical
proofs in general.
Anders Goeransson
<not...@nowhere.invalid> wrote:
> When Godel wrote his incompleteness paper
>Tarski's work was not available to him.
In a certain sense this is of course true, Tarski's paper coming after
Goedel's own. But as for the ideas and results given in Tarski's paper
these were certainly known to Goedel. Why, the very starting point for
Goedel's undecidability result was just the fact that 'truth' couldn't
be generally defined in arithmetic while 'provability' could. Goedel
tried to give a relative consistence proof for analysis while he
stumbled over this fact.
As you know, he then went on to elaborate on this result, constructing
an explicit undecidable formula etc.
For philosophical purposes it would however, in many cases, be
sufficient to consider the abovementioned difference between 'truth'
and 'provability'.
regards Anders
Pierre-Normand Houle
"Neil W Rickert" <ricke...@cs.niu.edu> wrote in message
news:97b875$e...@euclid.cs.niu.edu...
> "Pierre-Normand Houle" <hou...@attglobal.net> writes:
> >"Neil W Rickert" <ricke...@cs.niu.edu> wrote
> >> "Pierre-Normand Houle" <hou...@attglobal.net> writes:
>
> >> >> >> Since we cannot prove the consistency of the system, this does
not
> >> >> >> actually add anything.
>
> >> >> >How do you know that such a proof can not be achieved?
>
> >> >> It probably can. But that proof will likely be given in another
> >> >> system whose consistency is unproven. Thus it will not really
> >> >> provide any additional evidence of the consistency of the original
> >> >> system.
>
> >> >On this view no evidence could be offered for any statement
> >> >about anything ever.
>
> >> That does not follow from anything I wrote.
>
> >You seem to have said that you can not accept a proof of the consistency
> >of (a model of) arithmetic that is not performed in a system shown to be
> >consistent.
>
> I haven't said that either. I only said that such a proof does not
> actually demonstrate the consistency of the original axiom system.
So you accept that it would be a proof but you do not accept that this
proof would demonstrate what we intend it to demonstrate on the
ground this it would not consist in a derivation from the axioms of
the original system. What would this proof demonstrate then?
And if it does not demonstrates anything why accept it as a proof?
> However, I do not deny that it might be a valid mathematical proof
> within the meta-system where the proof was given.
But statements of the meta-system refer to the original system
just as theorems of arithmetic refer to natural numbers. So,
if there can be valid proofs of statements about natural numbers
in arithmetic why couldn't there be valid proofs of statements
about arithmetic in the meta-theory?
> > So, I ask you, do you know anything that is evidence
> >(or proof) for something else? And if so, what consistent set of axioms
do
> >you use to derive this evidence (or proof)?
>
> I am not claiming that formal proof within a system is the only
> evidence acceptable. It is, however, the situation that Goedel's
> theorem deals with.
What is the system within which Goedel's theorem is formulated?
This system is not arithmetic is it? Yet the theorem is about
the statements of arithmetic. Does not Goedel's theorem tell
us anything about these? And is it not a valid proof?
> >> >Can there not be evidences that are not to be identified with
> >> >derivability from a set of consistent axioms?
>
> >> There is plenty of evidence for the consistence of PA. However, it
> >> falls short of being mathematical proof.
>
> What I was actually referring to here, is that PA has been
> extensively exercised, and used for proving many interesting
> results. No inconsistency has yet turned up. That is good informal
> evidence for the consistency of PA, but it is not a mathematical
> proof of the consistency of PA.
Again, I have never claimed that the consistency of PA was
proven. I was inquiring as to what would count as a proof
that PA or any other axiomatization of arithmetic and what
the existence of such a proof would entail for the truth of
Goedel's statement (that "neither G nor not G can be derived...).
I still don't know what would count for you as "a mathematical
proof of the consistency of PA.)
> >Is your standard of "mathematical proof" derivability from a set of
> >axioms that has been shown to be consistent?
>
> I was not requiring such a strict standard in my comment above.
What is your standard then?
> > If it is, then you ought
> >to say that the demonstration all theorems of arithmetic fall short of
> >being mathematical proofs.
>
> This is surely wrong. Many theorems of arithmetic are proved from
> the axioms of a field, rather than from PA.
I don't see how this is relevant. On what ground would a derivation
from Peano axioms be any worse than the derivation of a corresponding
statement from the axioms of a field?