lifting a infinitely heavy rock question
Kevin Flowers
professionals think about this theory.
Thanks
>Regarding the Rocks of Unusual Size, it suffices to imagine the
>collection of all possible rock sizes; it's an infinite
>collection, corresponding roughly to the positive integer
>numbers. To say that God is omnipotent is to say that God can
>lift any rock whose size occurs in the collection, and can
>create a rock of any size in the collection. No circular
>definitions (I've tried to eliminate the need for God to create
>every rock first, by specifying "size;" I think this works, and
>the only objection is that a creature may be forced to imagine
>only those things which the creator can create). No mysterious
>RUSes that God cannot lift, because every thing that is a rock
>is specified in our collection. There is no limitation to God's
>power to create, because anything that is a rock can be created.
>To say that God cannot create a Rock of Unusual Size is to
>assert that there is a rock whose size is outside of the set of
>all rock sizes, ie, true is false. No deal.
****
This is probably the best response I've seen on this topic so
far. If I may paraphrase, it says that God can lift an
infinitely heavy rock. That means that a rock too heavy for him
to lift would have to be more than infinitely heavy, which is
impossible. Since God can't be expected to do the logically or
mathematically impossible, his omnipotence is preserved.
The problem is that God can't lift an infinitely heavy rock.
The reason he can't do that is bound up in the definition of
infinity. Infinity is not a number, it's a concept. Queer
things happen when you try to do arithmatic with infinity as
if it were a number.
For instance, you cannot add any number to infinity or subtract
any number from it. That is,
infinity + 1 = infinity
infinity + any_number = infinity
infinity - 1 = infinity
infinity - any_number = infinity
Although this seems counter-intuitive at first, a moments
thought will show that it has to be this way. If you can add
one to a number, the result will be a bigger number and
therefore your original number could not have been infinity.
Similarly, if you subtract one from infinity, all you have to do
is add two to the answer and you have a new number that is
greater than the number you started with and therefore it could
not have been infinity, either.
Since multiplication is repeated addition and division is
repeated subtraction, you can't multiply or divide infinity by
any number either.
infinity * 2 = infinity
infinity / 2 = infinity
Mathematical operations involving only infinity also have some
surprises:
infinity + infinity = infinity
infinity - infinity = zero
Bearing these quirky rules in mind, consider this scenario:
God Lifts a Ten Kilo Rock
You have a 10 kilogram rock sitting on a table. It's a cube the
size of a bowling ball and it has a little handle on top. You
pray for God to lift the ten kilogram rock. God hears your
prayer and decides to do so. He grasps the rock, lifts upwards
with exactly ten kilograms of force and the rock ... just sits
there. The ten kilograms of lift is exactly enough to
counteract the pull of gravity, but there's no extra lift to
accelerate the rock upwards, so it remains at rest. God
realizes his error, blushes and increases his lift to eleven
kilograms. Now the rock rises smoothly into the air and all the
earth rejoices.
God Attempts to Lift an Infinitely Heavy Rock
You have an infinitely heavy rock sitting on a (very strong)
table. It's also a cube the size of a bowling ball because it's
made of a material that's infinitely dense. Like the ten kilogram
rock, it has a little handle on top. You pray for God to lift
the infinitely heavy rock. God hears your prayer and decides to
do so. He grasps the rock, lifts upwards with an infinite
amount of force and the rock ... just sits there. The infinite
lift is exactly enough to counteract the pull of gravity, but
there's no extra lift to actually move the rock, so it remains
at rest.
God realizes his error, blushes and increases his lift to
infinity plus one kilogram. The rock continues to sit there
because infinity plus one equals infinity and there is still no
extra lift to accelerate the rock, so it remains at rest. You
can repeat this an infinite number of times because infinity
plus infinity equals infinity.
You can't lend God a hand and help him lift the rock, either.
If you grab the handle and lift upwards with, say, one kilogram
of extra lift, your one kilogram plus God's infinite kilograms
of lift will still "only" equal an infinite amount of lift,
exactly enough to counterbalance the infinite pull of gravity,
with nothing left over to accelerate the rock upwards.
Now God could probably suspend an infinitely heavy rock. For
instance, if God applied an infinite amount of upwards lift on
the infinitely heavy rock and someone slid the table out from
under it, God should be able to support the rock because his
infinite lift is equal and opposite to the infinite pull of
gravity. The two pulls should cancel out and the rock will
remain motionless.
Some quibblers might say that this was "lifting" the rock, but
as far as I'm concerned "lifting" something implies imparting an
upwards motion to it. People who wish to assert differently
should picture the infinitely heavy rock laying on the ground,
with God pulling furiously on it with all his infinite strength
and the rock not budging off the ground and then try calling
that "lifting" with a straight face. God's not lifting a rock
that's staying on the ground. If anyone insists that this IS
lifting anyway, then the question becomes why God can't _raise_
the rock.
Curiously, once God was supporting that infinite rock, I don't
think he could put it down gently. To do that, he'd have to
reduce his upwards lift to a little less than the downwards pull
of gravity and because of the peculiar way arithmatic works with
infinity, he can't do that. If he tried to reduce his infinite
lift by one kilogram, he would be quickly be reminded that
infinity minus one equals infinity because his lift would not be
diminished and therefore the rock would not accellerate
downwards. Luckily, you need not fear God getting stuck holding
the rock forever, unable to put it down. Infinity minus
infinity equals zero, so if he just lets go of the rock, it will
fall (hard).
I think the reason there's so much difficulty here is because
theologians and philosophers don't write most sacred literature.
They come along afterwards and try to repair the damage.
Dave Seaman
Kevin Flowers <Kevin_...@ColPal.Com> wrote:
>I read this on another news group. I am interested in what math
>professionals think about this theory.
>
>Thanks
>
>
>>Regarding the Rocks of Unusual Size, it suffices to imagine the
>>collection of all possible rock sizes; it's an infinite
>>collection, corresponding roughly to the positive integer
>>numbers. To say that God is omnipotent is to say that God can
>>lift any rock whose size occurs in the collection, and can
>>create a rock of any size in the collection. No circular
>>every thing that is a rock
>>is specified in our collection. There is no limitation to God's
>>power to create, because anything that is a rock can be created.
>>To say that God cannot create a Rock of Unusual Size is to
>>assert that there is a rock whose size is outside of the set of
>>all rock sizes, ie, true is false. No deal.
>
>****
>
>This is probably the best response I've seen on this topic so
>far. If I may paraphrase, it says that God can lift an
>infinitely heavy rock.
That's not what it says. It says that anything infinitely heavy is not
a rock. If you accept the offered definition of a rock, then this
logically follows, but I think it's just playing word games.
>That means that a rock too heavy for him
>to lift would have to be more than infinitely heavy, which is
>impossible. Since God can't be expected to do the logically or
>mathematically impossible, his omnipotence is preserved.
There are different orders of infinity -- infinitely many of them,
in fact. There is no largest infinity. Therefore, the argument fails.
>The problem is that God can't lift an infinitely heavy rock.
>The reason he can't do that is bound up in the definition of
>infinity. Infinity is not a number, it's a concept. Queer
>things happen when you try to do arithmatic with infinity as
>if it were a number.
When you are doing cardinal arithmetic, infinity is a number. More
precisely, each transfinite cardinal is a number.
>For instance, you cannot add any number to infinity or subtract
>any number from it. That is,
>
> infinity + 1 = infinity
> infinity + any_number = infinity
>
> infinity - 1 = infinity
> infinity - any_number = infinity
You just contradicted yourself by doing what you claimed can't be done.
The only quibble I have with this is that the last line should read:
infinity - any_FINITE_number = infinity
or, more generally,
infinity - any_SMALLER_number = infinity
where any_SMALLER_number may be infinite, as long as it's a smaller
infinity.
For example, letting aleph_null be the cardinality of the integers,
and letting c = the cardinality of the continuum (= cardinality of
the real numbers), we have
c - aleph_null = c
but
aleph_null - aleph_null is undefined
c - c is undefined.
>Although this seems counter-intuitive at first, a moments
>thought will show that it has to be this way. If you can add
>one to a number, the result will be a bigger number and
>therefore your original number could not have been infinity.
First, adding one to infinity (any infinity) does not give you a bigger
infinity. It gives you the same one, as you pointed out yourself.
Second, the existence of a bigger number does not imply that the original
number is finite. Aleph_null is infinite, even though c is bigger.
I'll skip the rest of your comments about arithmetic, which are partly
correct and partly nonsense, and cut to the chase:
2 ^ infinity = a_bigger_infinity.
For example, 2 to the power aleph_null is equal to c, a bigger infinity.
>Bearing these quirky rules in mind, consider this scenario:
[ ... ]
> God Attempts to Lift an Infinitely Heavy Rock
> [...] He grasps the rock, lifts upwards with an infinite
>amount of force and the rock ... just sits there. The infinite
>lift is exactly enough to counteract the pull of gravity, but
>there's no extra lift to actually move the rock, so it remains
>at rest.
>
>God realizes his error, blushes and increases his lift to
>infinity plus one kilogram.
Ignoring the fact that the kilogram is a unit of mass, not force,
suppose God increases his lift to 2^infinity of the appropriate units
(newstons, perhaps)? It doesn't really matter, because a force of
2^infinity newtons is the same as a force of 2^infinity pounds-force,
or any other unit of force you care to use.
You are still stuck with the same dilemma. Assuming that God is
omnipotent leads to a contradiction. This saves me from having to
think about what effect the existence of an infinitely massive rock
would have on our universe.
Dave Seaman
David Ullrich
>I read this on another news group. I am interested in what math
>professionals think about this theory.
>
existence of an omnipotent being without totally senseless gibbering
you're ultimately reduced to saying that God is omnipotent because he
can do anything except the things he can't do. I can do everything that
I can do - this is a very puny sort of "omnipotence".
Yeah I know, the actual assertion is that God can do anything
except the "logically or mathematically impossible". I don't think this
is very well-defined.
Let's say hypothetically that I dropped a rock yesterday. Is it
a "logical impossibility" for God to rearrange things so that in fact
I did not drop a rock yesterday?
--
David Ullrich
Don't you guys find it tedious typing the same thing
after your signature each time you post something?
I know I do, but when in Rome...
Toby Bartels
>Kevin Flowers <Kevin_...@ColPal.Com> wrote:
>> God Attempts to Lift an Infinitely Heavy Rock
>> [...] He grasps the rock, lifts upwards with an infinite
>>amount of force and the rock ... just sits there. The infinite
>>lift is exactly enough to counteract the pull of gravity, but
>>there's no extra lift to actually move the rock, so it remains
>>at rest.
>>God realizes his error, blushes and increases his lift to
>>infinity plus one kilogram. [But oo + 1 = oo, so he still can't lift it.]
>Ignoring the fact that the kilogram is a unit of mass, not force,
>suppose God increases his lift to 2^infinity of the appropriate units
>(newstons, perhaps)? It doesn't really matter, because a force of
>2^infinity newtons is the same as a force of 2^infinity pounds-force,
>or any other unit of force you care to use.
Now God can lift the rock.
>You are still stuck with the same dilemma. Assuming that God is
>omnipotent leads to a contradiction. This saves me from having to
>think about what effect the existence of an infinitely massive rock
>would have on our universe.
After all this good explanation about infinity, Dave,
why do you stop immediately before the conclusion
that God can lift the infinitely heavy rock?
Instead, you say Kevin is still stuck with a dilemma.
What dilemma? God can lift every rock there is - even the infinite ones.
There is no contradiction.
(I think we are all quite aware that,
although there are such things as infinitely heavy rocks,
the size of no rock is a proper class;
this is by definition of the word `rock',
which can be defined in ZF set theory alone.)
-- Toby
to...@ugcs.caltech.edu
Dave Seaman
Toby Bartels <to...@mince.ugcs.caltech.edu> wrote:
>After all this good explanation about infinity, Dave,
>why do you stop immediately before the conclusion
>that God can lift the infinitely heavy rock?
>Instead, you say Kevin is still stuck with a dilemma.
>What dilemma? God can lift every rock there is - even the infinite ones.
>There is no contradiction.
The dilemma is that if God is omnipotent, then God should be able to
make a rock so big that it cannot be lifted.
More formally, let P(x) be the statement, "x can make a rock so big
that x can't lift it." Now suppose God is omnipotent. Is P(God) true,
or is it false?
This is the dilemma the original poster was trying to avoid, but I was
pointing out that the attempt was unsuccessful, at least according to
logic. There are some who claim that since God is omnipotent, God can
also violate the rules of logic.
My reply is that before God can nullify logic, He has to exist first.
That's why I was careful to state the dilemma in the form (omnipotent
God exists) ==> contradiction, rather than (God is omnipotent) ==> (God
can't do X). It's a preemptive strike to keep God from existing
before He gets a chance to nullify our logic. :-)
Dave Seaman
Toby Bartels
>Toby Bartels (to...@ugcs.caltech.edu) wrote:
>>After all this good explanation about infinity, Dave,
>>why do you stop immediately before the conclusion
>>that God can lift the infinitely heavy rock?
>>Instead, you say Kevin is still stuck with a dilemma.
>>What dilemma? God can lift every rock there is - even the infinite ones.
>>There is no contradiction.
>The dilemma is that if God is omnipotent, then God should be able to
>make a rock so big that it cannot be lifted.
That is not the dilemma in question here. Please reread Kevin's post.
>More formally, let P(x) be the statement, "x can make a rock so big
>that x can't lift it." Now suppose God is omnipotent. Is P(God) true,
>or is it false?
P (God) is a logical contradiction,
so its falsehood doesn't deny God's omnipotence.
God's omnipotence doesn't require It to do the logically impossible.
>This is the dilemma the original poster was trying to avoid, but I was
>pointing out that the attempt was unsuccessful, at least according to
>logic. There are some who claim that since God is omnipotent, God can
>also violate the rules of logic.
Did Kevin claim this?
On the contrary, I think he (or those he quoted) denied this.
>My reply is that before God can nullify logic, He has to exist first.
>That's why I was careful to state the dilemma in the form (omnipotent
>God exists) ==> contradiction, rather than (God is omnipotent) ==> (God
>can't do X). It's a preemptive strike to keep God from existing
>before He gets a chance to nullify our logic. :-)
I applaud this approach but remind you that you have succeed at it
only for omnipotence that includes the ability to do logical impossibilities.
-- Toby
to...@ugcs.caltech.edu
Dave Seaman
Toby Bartels <to...@chop.ugcs.caltech.edu> wrote:
>Dave Seaman (a...@seaman.cc.purdue.edu) wrote:
>>The dilemma is that if God is omnipotent, then God should be able to
>>make a rock so big that it cannot be lifted.
>
>That is not the dilemma in question here. Please reread Kevin's post.
I stated one aspect of the dilemma, an aspect that was not emphasized
in Kevin's post because he was concerned with the other aspect:
whether God can lift an infinitely heavy rock. However, it takes both
aspects in order for the discussion to have any substance at all.
>>More formally, let P(x) be the statement, "x can make a rock so big
>>that x can't lift it." Now suppose God is omnipotent. Is P(God) true,
>>or is it false?
>
>P (God) is a logical contradiction,
>so its falsehood doesn't deny God's omnipotence.
>God's omnipotence doesn't require It to do the logically impossible.
The last time I got into a discussion on this topic, I was accused of
"underestimating omnipotence," which I consider to be a logical
impossibility. The guy thought I was arguing "God is omnipotent" ==>
"God can't do X", and he argued (with some justification) that this
is "underestimating omnipotence." However, he misunderstood my argument.
I was arguing "omnipotent God exists" ==> contradiction, which does not
place any limits on omnipotence.
However, I think you are at least attempting to "underestimate
omnipotence" here. Perhaps you are talking about some limited form of
omnipotence?
>>[...] There are some who claim that since God is omnipotent, God can
>>also violate the rules of logic.
>
>Did Kevin claim this?
Did I say Kevin claimed it?
>On the contrary, I think he (or those he quoted) denied this.
Did I deny that he denied it?
May I remind you that this discussion is taking place in a public
forum? I was trying to anticipate objections that I have encountered
before with this type of argument -- not necessarily from you or Kevin,
but from others who may be reading this thread. I happen to know from
experience that there are some who will argue very vehemently that:
(1) Yes, God can make a rock so big that he can't lift it,
(2) Yes, God can then turn around and lift that very rock,
(3) Yes, it's a logical contradiction. God is omnipotent.
He can defy logic.
[Re: omnipotent God exists ==> contradiction]
>I applaud this approach but remind you that you have succeed at it
>only for omnipotence that includes the ability to do logical impossibilities.
Omnipotence is omnipotence.
You can believe in an omnipotent God, or you can believe in logic. I choose
logic.
Dave Seaman
William Schneeberger
One theory I have heard on this matter is that an omnipotent God could
make such a rock but chooses not to.
In other words, an omnipotent being would have the capability of
destroying omnipotence, but obviously wouldn't be doing so.
Seems at least logically sound. Of course, if you stretch the
definition of omnipotence far enough, there will be other
contradictions.
--
Will Schneeberger Hi There !!
wil...@math.Princeton.EDU
Dave Seaman
Timothy Chow <tc...@math.lsa.umich.edu> wrote:
>In article <44outt$g...@seaman.cc.purdue.edu>,
>Dave Seaman <a...@seaman.cc.purdue.edu> wrote:
>>Omnipotence is omnipotence.
>>You can believe in an omnipotent God, or you can believe in logic. I choose
>>logic.
>
>the naive notion of omnipotence is self-contradictory. But the naive notion
>of truth is self-contradictory (liar paradox) and the naive notion of "all
>things" (the "set of all sets") is self-contradictory (Cantor's paradox).
>Would you say, in response to the liar paradox, that "truth is truth; you
>can believe in truth, or you can believe in logic---I choose logic"? I
I would say that truth is not contradictory. What is contradictory is the
law of the excluded middle -- the naive notion that every proposition is
either true or false.
The notion of "all things" is not contradictory. What is contradictory
is the naive notion that the class of all things (class of all sets,
actually) is a set.
>would say instead that while the naive notion of truth is flawed, there are
>ways of defining truth that are sufficiently close to the naive notion of
>truth that they deserve the name of truth. I would argue similarly for
>omnipotence.
Fair enough. I am certainly no theologian, but I believe St. Thomas
Aquinas considered the question of whether God could create a triangle
having an angle sum different from two right angles, and concluded that
it is impossible. Today St. Thomas would no doubt amend the statement
to say that God cannot create a *Euclidean* triangle with the stated
property. [Is a spherical triangle on a Euclidean sphere a Euclidean
triangle?]
>To put it another way, I would reject the assumption, maintained implicitly
>by some theists (and apparently by you---perhaps for the sake of argument),
>that anything other than the naive notion of omnipotence is necessarily not
>"really" omnipotence but a "restricted" omnipotence.
I have no problem with that. As I said, my adoption of the literal
definition was primarily a defensive reaction. I get the feeling from
previous experience that some well-known theologian (possibly C. S.
Lewis?) has taken exactly that approach, and I was trying to insulate
my explanation from an attack from that direction.
Dave Seaman
David Ullrich
>In article <44outt$g...@seaman.cc.purdue.edu>,
>Dave Seaman <a...@seaman.cc.purdue.edu> wrote:
>>Omnipotence is omnipotence.
>>You can believe in an omnipotent God, or you can believe in logic. I choose
>>logic.
>
>The matter isn't quite as simple as this. What your argument shows is that
>the naive notion of omnipotence is self-contradictory. But the naive notion
>of truth is self-contradictory (liar paradox)
No, the liar paradox shows that certain constructions that appear a
priori to be sentences had better not be regarded as such.
[order slightly revised, in a manner I hope the author regards as faithful
to the meaning of the original]
>Would you say, in response to the liar paradox, that "truth is truth; you
>can believe in truth, or you can believe in logic---I choose logic"?
So I would in fact say that I could believe in logic or in the
idea that "This semtence is false" is a sentence, and of the two I'd choose
logic.
>and the naive notion of "all
>things" (the "set of all sets") is self-contradictory (Cantor's paradox).
The trouble there is the idea that the naive notion of "is an element
of" extends to "everything", ie that given any two "things" A and B the question
"Is A an element of B?" has a well-defined answer. I don't see any problem
with "the class of all things" (talking about "real" things, not mathematical
entities), and it's not clear to me that there's even any reason that the class
of all things cannot be regarded as a thing itself. The problem arises when we
consider the class of all things which are not members of themselves - luckily
(if we're talking about real things) this problem evaporates when we realize that
it doesn't make sense to ask whether a given thing is "a member of" another
thing we haven't said exactly what "is a member of" means, and coming up with
a precise definition (with literally _global_ scope) seems unlikely.
>To put it another way, I would reject the assumption, maintained implicitly
>by some theists (and apparently by you---perhaps for the sake of argument),
>that anything other than the naive notion of omnipotence is necessarily not
>"really" omnipotence but a "restricted" omnipotence. For example, I would
>reject the intuition that "all possible things" represents a *restriction*
>of "all things"; in fact, there aren't any "impossible things." Omnipotence,
>defined then as the ability to do all possible things, isn't a restricted
>omnipotence at all, and therefore still deserves the "omni" prefix.
Informally I'd agree that there are no impossible things. But to
take "can do anything which is possible" as the definition of "omnipotent"
requires that we define "possible" first. Good luck with that one.
>I don't claim that this solves all the problems with the concept of
>omnipotence, but I think it illustrates my point that questions about
>intuitive concepts (cf. the Church-Turing thesis and the intuitive notion
>of a "procedure") can never be completely "closed" by a logical proof.
I'm missing your point regarding Turing machines and Church's
thesis. Do you have an example of something that you intuitively regard as
an "algorithm" which cannot be realized by a Turing machine?
Dave Seaman
Toby Bartels <to...@puree.ugcs.caltech.edu> wrote:
>Dave Seaman (a...@seaman.cc.purdue.edu) wrote:
>>I stated one aspect of the dilemma, an aspect that was not emphasized
>>in Kevin's post because he was concerned with the other aspect:
>>whether God can lift an infinitely heavy rock. However, it takes both
>>aspects in order for the discussion to have any substance at all.
>
>(which is a reason why it is not later emphasized).
>Kevin was talking about that sort of omnipotence
>which allows God to do anything that is *not* logically contradictory.
Kevin's disposal of that aspect was based on the false premise that it
is logically impossible to lift a rock of infinite size, since that
would require more than an infinite lift. Therefore, Kevin concluded,
God *can* create a rock that is too big for Him to lift, and He is not
required to lift the rock in order to be omnipotent, because that would
be logically impossible.
What I did is show that there is no contradiction in lifting infinite
rocks, since there are different orders of infinity. Therefore, Kevin
is once again left with the original dilemma. Can God make a rock so
big that He can't lift it? It is logically conceivable that He can,
and it is logically conceivable that He can't. No contradiction either
way.
>>May I remind you that this discussion is taking place in a public
>>forum? I was trying to anticipate objections that I have encountered
>>before with this type of argument -- not necessarily from you or Kevin,
>>but from others who may be reading this thread.
>
>Then, for goodness's sake, tell us that that's what you're doing.
>But don't say Kevin is left with a dilemma - because *he* isn't.
Then, can God create a rock so big that He can't lift it, or not?
>>Omnipotence is omnipotence.
>
>But there are different kinds of omnipotence.
>You refer to those who define omnipotence to include
>the ability to perform logical contradictions.
>But this thread had been about a different kind of omnipotence
>- one that allows God to do only what is not logically contradictory.
Ok, let's use your definition of omnipotence. Are you saying,
(1) It is a logical impossibility that God can create a rock so
big that He can't lift it, or
(2) It is a logical impossibility that God can lift any rock He
can possibly create, or
(3) both, or
(4) neither?
Explain your answer.
>>You can believe in an omnipotent God, or you can believe in logic. I choose
>>logic.
>
>I also happen to be a fan of logic.
>That's why I react so badly to illogical statements.
Some of my statements were based on a different hypothesis. That does
not make them illogical. For the sake of argument, I am now willing to
adopt your hypothesis. Where is the illogic?
>For the record:
>by claiming omnipotence when God is shown to be capable
>of doing anything not logically contradictory,
>the authors quoted in Kevin's original post show themselves
>to be using a definition of omnipotence limited by the requirements of logic.
>Kevin extended the situation a bit and found a problem with infinity,
I didn't read it that way. I thought Kevin considered his conclusions
regarding infinity to be a way out of the dilemma. Since (he thought)
asking God to lift an infinitely heavy rock is asking Him to do the
impossible, we shouldn't expect Him to be able to do it, even though He
is omnipotent. Dilemma solved.
>which Dave neatly solved.
>So far, no logical contradiction has been found to result
>from the assumption that God can do anything that's not logically impossible.
>Obviously, no such a priori contradiction can ever be found.
Then can God make a rock so big that He can't lift it, or not? Explain
your answer.
>Equally obviously, omnipotence that is not limited by logic can violate logic.
>I, personally, believe neither kind of omnipotence exists in the world,
>in part for empirical reasons.
We do, at least, agree on something.
Dave Seaman
Timothy Chow
David Ullrich <ull...@math.okstate.edu> wrote:
>tc...@math.lsa.umich.edu (Timothy Chow) wrote:
>>The matter isn't quite as simple as this. What your argument shows is that
>>the naive notion of omnipotence is self-contradictory. But the naive notion
>>of truth is self-contradictory (liar paradox)
>
> No, the liar paradox shows that certain constructions that appear a
>priori to be sentences had better not be regarded as such.
Well, Dave Seaman seems to think that the problem lies with the law of the
excluded middle, so it seems your point of view is not uncontroversial.
In any case, let's say I grant you your resolution of the liar paradox.
Then my point is that one could respond to the omnipotence paradox by
saying, "I can either believe in logic or in the idea that 'Can God make
a rock too heavy for him to lift?' is a question, and of the two I'd choose
logic." Or, if I grant Dave Seaman's point of view, then I can respond to
the omnipotence paradox by rejecting the assumption that either "God can
make a rock too heavy for him to lift" or "God cannot make a rock too heavy
for him to lift" must be true, and not both.
Because the argument is an intuitive one, there are lots of different places
you can fiddle with the assumptions. Rejecting the concept of omnipotence
is not the only way out.
> I'm missing your point regarding Turing machines and Church's
>thesis. Do you have an example of something that you intuitively regard as
>an "algorithm" which cannot be realized by a Turing machine?
No. The Church-Turing thesis asserts that no intuitive notion of procedure
can be any more powerful than a Turing machine algorithm. My point is that
the Church-Turing thesis cannot be proved or disproved logically because it
makes reference to the vague concept of "an intuitive notion of a procedure."
Similarly, the intuitive notion of omnipotence cannot be *proved* to be
self-contradictory; one can show that *certain* intuitions about omnipotence
(say, the intuition that certain formal statements X, Y, and Z accurately
and consistently capture the notion of omnipotence) lead to contradiction,
but there is always enough vagueness in the "raw" intuition to escape total
disproof.
--
Tim Chow tyc...@math.mit.edu
Where a calculator on the ENIAC is equipped with 18,000 vacuum tubes and weighs
30 tons, computers in the future may have only 1,000 vacuum tubes and weigh
only 1 1/2 tons. ---Popular Mechanics, March 1949
David Ullrich
>David Ullrich <ull...@math.okstate.edu> writes:
>
>>>tc...@math.lsa.umich.edu (Timothy Chow) wrote:
>>>
>>>The matter isn't quite as simple as this. What your argument shows is that
>>>the naive notion of omnipotence is self-contradictory. But the naive notion
>>>of truth is self-contradictory (liar paradox)
>>
>> No, the liar paradox shows that certain constructions that appear a
>>priori to be sentences had better not be regarded as such.
>
>
>> So I would in fact say that I could believe in logic or in the
>>idea that "This sentence is false" is a sentence, and of the two I'd choose
>>logic.
>
>Oh, sure its a sentence---it's just maybe not a meaningful one. But
>the question is: why *isn't* it meaningful? Timothy Chow was pointing
>out (correctly, in my opinion) that it is the naive, unrestricted notion
>of "truth" that leads to difficulty. Some people think that it is the
>fact that the sentence "talks about itself" that somehow makes it
>a meaningless sentence. But that is dispelled by considering similar
>examples: "This sentence has five words" and "This sentence is written
>in English" which are self-referential but perfectly meaningful. For
>any well-defined property P of sentences, I believe we can make sense of
>"This sentence has property P". The failure of "This sentence is false"
>to make any sense shows that an unrestricted notion of "being true" or
>"being false" is *not* a well-defined property of sentences.
No, actually I have big troubles with "This sentence has five
words". Because "five" means the same as "four plus one", so that
"this sentence has five words" and "this sentence has four plus one words"
should be equivalent.
Sentences are not allowed to talk about themselves, except this one.
Timothy Chow
David Ullrich <ull...@math.okstate.edu> wrote:
> Sentences are not allowed to talk about themselves, except this one.
This is a pretty extreme view, at least if "talking about themselves"
includes talking about objects which happen to coincide with themselves.
The basic idea behind the Goedel trick is to construct a sentence which
talks about itself in this sense.
Tarski's theorem tells us (roughly) that a language can't contain its own
truth predicate, but it doesn't prohibit all sentence predicates. Rather,
it relies on *particular* properties of the truth predicate.
> No, actually I have big troubles with "This sentence has five
>words". Because "five" means the same as "four plus one", so that
>"this sentence has five words" and "this sentence has four plus one words"
>should be equivalent.
Why should they be equivalent? Saying that "five" means the same as "four
plus one" does not imply that every *sentence* containing the word "five"
will not change its meaning when "five" is replaced by "four plus one."
Let S be the sentence "This sentence has five words" and let S' be the
sentence "This sentence has four plus one words." Saying that "five"
means the same as "four plus one" just says that the proposition that S
has five words is the same as the proposition that S has four plus one
words. It does not imply that the syntactic transformation that changes
S to S' necessarily leaves the meaning of the sentence (i.e., the
proposition expressed by the sentence) invariant. The meaning of the
word "this" depends on context; if one changes the context (e.g., by
some kind of syntactic transformation) then naturally one expects the
meaning of the word "this" (and consequently the meaning of the sentence)
to change.
David Ullrich
>In article <44pfj7$10...@bubba.ucc.okstate.edu>,
>David Ullrich <ull...@math.okstate.edu> wrote:
>>tc...@math.lsa.umich.edu (Timothy Chow) wrote:
>>>The matter isn't quite as simple as this. What your argument shows is that
>>>the naive notion of omnipotence is self-contradictory. But the naive notion
>>>of truth is self-contradictory (liar paradox)
>>
>> No, the liar paradox shows that certain constructions that appear a
>>priori to be sentences had better not be regarded as such.
>
>excluded middle, so it seems your point of view is not uncontroversial.
>
Others in this thread have suggested that God has the power to abolish the
law of the excluded middle - I would agree that the most appropriate
response to anyone perhaps questioning the law of the excluded middle
is "ajhsgd ashsdn,kj wef dsf, so that UYGWd. And further, jgf sdf sd."
I was not claiming that the question of what's a sentence and what isn't
had anything to do with the "define omnipotence" problem, I was referring
specifically to the liar paradox - I didn't notice Seaman saying anything
about that. Anyway, your "the problem" seems a little ambiguous.
>In any case, let's say I grant you your resolution of the liar paradox.
>Then my point is that one could respond to the omnipotence paradox by
>saying, "I can either believe in logic or in the idea that 'Can God make
>a rock too heavy for him to lift?' is a question, and of the two I'd choose
>logic."
If you insist - assuming we know what "God" is I don't see any
problem with what the string "God can make a rock too heavy for him to
lift" _means_, there's no funny self-reference going on there. Seems like
it's either true or false to me.
>Or, if I grant Dave Seaman's point of view, then I can respond to
>the omnipotence paradox by rejecting the assumption that either "God can
>make a rock too heavy for him to lift" or "God cannot make a rock too heavy
>for him to lift" must be true, and not both.
>
Hmm. ajhsgd ashsdn,kj wef dsf, so that UYGWd. And further, jgf sdf sd.
>Because the argument is an intuitive one, there are lots of different places
>you can fiddle with the assumptions. Rejecting the concept of omnipotence
>is not the only way out.
>
>> I'm missing your point regarding Turing machines and Church's
>>thesis. Do you have an example of something that you intuitively regard as
>>an "algorithm" which cannot be realized by a Turing machine?
>
>No. The Church-Turing thesis asserts that no intuitive notion of procedure
>can be any more powerful than a Turing machine algorithm. My point is that
>the Church-Turing thesis cannot be proved or disproved logically because it
>makes reference to the vague concept of "an intuitive notion of a procedure."
>Similarly, the intuitive notion of omnipotence cannot be *proved* to be
>self-contradictory; one can show that *certain* intuitions about omnipotence
>(say, the intuition that certain formal statements X, Y, and Z accurately
>and consistently capture the notion of omnipotence) lead to contradiction,
>but there is always enough vagueness in the "raw" intuition to escape total
>disproof.
I haven't been saying that anything completely and accurately captures
the intuitive notion of anything. If anything just the opposite: The traditional
notion of "omnipotent" is in fact "can do anything", and yes there is some
ambiguity there. There is no point in discussing anything with someone who
believes in the possible existence of a being who can do things that are
impossible, while if we revise the definition of "omnipotent" to "can do
anything that's possible" we have serious difficulties defining the word
"possible".
David Ullrich
I have a big problem with this, namely the definition of
"logically impossible". I know what that means in certain formal
systems, but thay have nothing to do with the real world, so surely
something broader is intended.
The notion of "logically impossible" is not as well-defined as
you think it is. Somebody mentioned the triangle with angles adding up
to more than 180 degrees earlier - most of us would agree this is
impossible if we're careful to define everything first.
But what about time travel into the past? A lot of people would
say that's impossible, and prove it by saying "what if you shot your
grandfather?". But a lot of people are not convinced by this (I don't
see why not, but that's another question...)
What about simultaneity? Is it logically possible for me or God
or somebody clever to invent a gizmo that would turn on a light bulb on
Alpha Centauri precisely when I hit a certain button here on Earth? Most
people would see no logical contradiction there, but some people would
asert that there is simply no such thing as simultaneity: Event A there
and event B here may look simultaneous to one observer and not to another,
and so the word "simultaneous" is simply meaningless.
After something happens can God adjust the universe in such a way
that it did not in fact happen? I don't mean just revise the universe so
that it looks the same as it would if the something hadn't happened, I mean
revise the universe so that it did not happen. Is that a logical impossibility
or not? It certainly seems to me like it "should" count as logically
impossible, but I'm equally certain I'm not going to be able to prove it's
impossible.
Or in the other direction, what about turning matter into energy?
That's certainly impossible, except it isn't...
Dave Seaman
>>Well, Dave Seaman seems to think that the problem lies with the law of the
>>excluded middle, so it seems your point of view is not uncontroversial.
>[ ... ] I was referring
>specifically to the liar paradox - I didn't notice Seaman saying anything
>about that. Anyway, your "the problem" seems a little ambiguous.
Actually, I did mention the law of excluded middle in connection with
the liar paradox.
I think there are two basic ways to handle the liar paradox, both of
which have been discussed in this thread. Let L be the statement "This
statement is false."
1. L is not a statement, and therefore the law of excluded
middle is irrelevant, because the law of excluded middle
applies only to statements, and not gibberish.
2. L is a statement, but it doesn't have a truth value. The
law of excluded middle fails for L.
I originally had in mind the second approach, but either one works for
the liar paradox. Neither approach works for the heavy-rock paradox,
however. Let P(x) be the statement "x can make a rock so large that x
can't lift it." Is P(God) a statement? What about P(Fred)? Is either
of those collections of words self-contradictory? Does the law of
excluded middle fail for P(God)? For P(Fred)?
Whichever way you choose to look at the liar paradox, the heavy-rock
paradox has a fundamentally different nature. P(x) is not itself a
contradiction, and neither is "not P(x)". The contradiction appears
when you add O(x) = "x is omnipotent."
(1) P(x) ==> not O(x),
(2) not P(X) ==> not O(x),
(3) O(x).
From these we may derive not (P(x) or not P(x)). This does not mean
P(x) is itself a contradiction, but nevertheless a contradiction exists.
The contradiction is not resolved by saying that O(x) does not enable x
to do contradictory things, because that proviso does not apply to P.
Dave Seaman
Martin C. Glanvill
In article <44p5jo$5...@cnn.Princeton.EDU>, wil...@coffee.princeton.edu (William Schneeberger) writes:
->
->One theory I have heard on this matter is that an omnipotent God could
->make such a rock but chooses not to.
->
->In other words, an omnipotent being would have the capability of
->destroying omnipotence, but obviously wouldn't be doing so.
->
->Seems at least logically sound. Of course, if you stretch the
->definition of omnipotence far enough, there will be other
->contradictions.
The `fatal' assumption is that GOD is omnipotent in the first place...
there's always a minor catch to these sort of `proofs'.
--
______.+= From Martin Glanvill =+._____
/ _/ _/_/_/ _/ _/ _/ _/ _/ _/ /\
/ _/ _/ _/_/_/ _/ _/ _/ / /
/ _/_/_/ _/_/_/ _/ _/ _/_/ _/ _/ / /
/___/ i486-DX4-100, v1.3.30+kswap /____/ /
\___\/ Maths Department, G3.02 \____\/
Timothy Chow
Dave Seaman <a...@seaman.cc.purdue.edu> wrote:
>Let P(x) be the statement "x can make a rock so large that x
>can't lift it." Is P(God) a statement? What about P(Fred)? Is either
>of those collections of words self-contradictory? Does the law of
>excluded middle fail for P(God)? For P(Fred)?
How about P(a being that can lift anything and that can make anything)?
This has a stab at being something that is not a sentence, because the
part "rock so heavy that a being that can lift anything ... cannot lift
it" fails to refer.
>Whichever way you choose to look at the liar paradox, the heavy-rock
>paradox has a fundamentally different nature.
I'm not convinced. Consider this: number all "variable sentences"
(sentences with a variable in them, like your P(x)) and let S(x) be
the variable sentence "You get a false sentence if you set all the
variables in the xth variable sentence equal to x." There's no problem
with this variable sentence until you plug in its own index number.
The structure looks pretty similar to me.
Dave Seaman
Timothy Chow <tc...@math.lsa.umich.edu> wrote:
>In article <44s93m$k...@seaman.cc.purdue.edu>,
>Dave Seaman <a...@seaman.cc.purdue.edu> wrote:
>>Let P(x) be the statement "x can make a rock so large that x
>>can't lift it."
>How about P(a being that can lift anything and that can make anything)?
>This has a stab at being something that is not a sentence, because the
>part "rock so heavy that a being that can lift anything ... cannot lift
>it" fails to refer.
If you assume that God exists, then P(God) is a sentence and it is not
contradictory. The problem arises if you add the hypothesis that God
is omnipotent.
>>Whichever way you choose to look at the liar paradox, the heavy-rock
>>paradox has a fundamentally different nature.
>
>I'm not convinced. Consider this: number all "variable sentences"
>(sentences with a variable in them, like your P(x)) and let S(x) be
>the variable sentence "You get a false sentence if you set all the
>variables in the xth variable sentence equal to x." There's no problem
>with this variable sentence until you plug in its own index number.
>The structure looks pretty similar to me.
In other words, the sentence-with-variable scheme could be used to
construct a version of the liar paradox. True, but I wasn't using it
that way. The contradiction is not in P(x), but rather in O(x) = "x is
omnipotent." P(x) is merely being used as a tool to expose the problem
in O(x).
O(x) and P(x) ==> contradiction
O(x) and not P(x) ==> contradiction
Therefore, not O(x).
Notice, we didn't conclude that anything is not a sentence. We
concluded that O(x) is false for all x. "False" is a well-defined
truth value, unlike the truth value of the liar sentence.
Dave Seaman
Dave Seaman
Toby Bartels <to...@bradbert.ugcs.caltech.edu> wrote:
>Dave Seaman (a...@seaman.cc.purdue.edu) wrote:
>>Kevin's disposal of that aspect was based on the false premise that it
>>is logically impossible to lift a rock of infinite size, since that
>>would require more than an infinite lift. Therefore, Kevin concluded,
>>God *can* create a rock that is too big for Him to lift, and He is not
>>required to lift the rock in order to be omnipotent, because that would
>>be logically impossible.
>
>Some other fellows disposed of it, and Kevin merely concurred.
>By `the very beginning', I meant before Kevin even mentions infinity.
The "other fellows" said that rocks, by definition, have finite sizes,
and therefore it is sufficient that God can create any size rock and
also can lift any size rock. The only mention of infinity was to point
out that the collection of potential rock sizes is infinite.
Kevin immediately paraphrased this incorrectly as saying that God can
lift an infinitely heavy rock. As soon as Kevin allows infinitely
heavy objects to be considered rocks, the "other fellows'" disposal of
the problem goes out the window. Kevin's disposal of the problem was
as I said -- God can create infinitely heavy rocks, and he can't lift
them, but lifting them would be (he claimed) logically impossible.
It's true that when you introduce higher orders of infinity, then the
"other fellows'" disposal of the problem becomes feasible once again.
Unlike Kevin's explanation, we once again find that an omnipotent being
can create a rock of any size, and also can lift a rock of any size,
and there is no contradiction. I now think this is what you were
arguing, but it wasn't clear to me at the time.
I think the rest of the discussion was based on a misunderstanding, and
is best skipped.
Dave
David Ullrich
>In article <44s4jr$1o...@bubba.ucc.okstate.edu>,
>David Ullrich <ull...@math.okstate.edu> wrote:
>> Sentences are not allowed to talk about themselves, except this one.
>
>This is a pretty extreme view, at least if "talking about themselves"
>includes talking about objects which happen to coincide with themselves.
>The basic idea behind the Goedel trick is to construct a sentence which
>talks about itself in this sense.
>
anything, in fact it's not a sentence in the sense in which I was using the
word, it's just a string of symbols. If the formal sentences in the formal
language are given a certain interpretation then the sentence sort of appears
to talk about itself, in particular if you assume the sentence is provable
in the system it follows that it isn't, and conversely. But the Goedelian
sentence does _not_ talk about itself, it doesn't talk about anything.
And I'm not quibbling here, simply trying to clarify what I meant.
If my assertion that sentences must talk about themselves includes the
assertion that the moon is made of green cheese then it's an even more
extreme view, but luckily it does not include any such assertion. The
notion that sentences must not talk about themselves is not an extreme
view, it's perfectly standard, it's something you have to be careful to
avoid, like dividing by zero.
When we're disussing these things precision is kind of like a good
idea, sort of, more or less. When I say a sentence must not refer to itself
I mean that a sentence must not refer to itself, not that things that are
sort of like self-reference are not allowed, nor that constructions inspired
by the liar paradox are not allowed. "This sentence is false" literally
refers to itself, as does "this sentence has five words".
>Tarski's theorem tells us (roughly) that a language can't contain its own
>truth predicate, but it doesn't prohibit all sentence predicates. Rather,
>it relies on *particular* properties of the truth predicate.
>
>> No, actually I have big troubles with "This sentence has five
>>words". Because "five" means the same as "four plus one", so that
>>"this sentence has five words" and "this sentence has four plus one words"
>>should be equivalent.
>
>Why should they be equivalent? Saying that "five" means the same as "four
>plus one" does not imply that every *sentence* containing the word "five"
>will not change its meaning when "five" is replaced by "four plus one."
>Let S be the sentence "This sentence has five words" and let S' be the
>sentence "This sentence has four plus one words." Saying that "five"
>means the same as "four plus one" just says that the proposition that S
>has five words is the same as the proposition that S has four plus one
>words. It does not imply that the syntactic transformation that changes
>S to S' necessarily leaves the meaning of the sentence (i.e., the
>proposition expressed by the sentence) invariant. The meaning of the
>word "this" depends on context; if one changes the context (e.g., by
>some kind of syntactic transformation) then naturally one expects the
>meaning of the word "this" (and consequently the meaning of the sentence)
>to change.
>--
I should have been more precise. We need to talk about the distinction
between use and mention. The following sentences are true and false, respectively:
1) Five equals two plus three.
2) "Five" equals two plus three.
(The number is just a label for reference, not part of what I meant by "the
following sentence". Also I really should include some quote marks around the
entire thing, but it seems that could lead to confusion, hance I'm setting
it off with blank lines in place of quote marks.)
Sentence (2) is false because "five" is a word, not a number, while
two plus three is a number.
Here's two more sentences, again the first is true and the second
is false:
3) "Five" has four letters.
4) Five has four letters.
"Five" is in fact an English word with four letters - it happens to
be the standard name for the number five, but that's not important in (3).
On the other hand five is a number, not a word, so it doesn't "have" any
letters at all - hence (4) is false.
Now what do I mean by the use-mention distinction? Well, the word "five" is
_used_ in sentence (1) but it is not mentioned - to mention "five" would be
to say something about the word "five", but nothing in (1) says anything
about the word "five". The word "five" is mentioned in (2) but not used -
the meaning of the word "five" is irrelevant in (2), (2) says something about
the word itself, not about its meaning.
And the word "five" is mentioned in (3) but not used: In fact a reader
who knew the meaning of the word "four" but who didn't know the meaning of
the word "five" would still know that (3) was true, because the word "five"
is not used in (3), it's only mentioned (and curiously mentioned in such a way as
to make the truth of (3) evident). On the other hand "five" is used but not
mentioned in (4).
So that's the use-mention distinction; when you see a word in a sentence
sometimes the word is actually used in the sentence, so that the meaning of
the word s part of the meaning of the sentence, and sometimes the word is not
actually _used_ in the sentence, it's merely mentioned (ie the sentence says
something about the word).
When you see a word in a sentence it's clear that you need to know
whether the word is being used or mentioned or you're going to have no clue
what the sentence means. The reason self-referential sentences are disallowed
is precisely that the use-mention distinction gets blurred so as to make the
sentence unintelligible.
So on the question of the universal substitutability of the string
"three plus two" for the string "five", what I should have said that the
string "three plus two" can be substituted in any instance of a _use_ of the
word "five" - you certainly cannot substitute "three plus two" where the word
"five" is just mentioned. So the two statements
Five is one plus four
and
Three plus two is one plus four
should be equivalent, while
"Five" has four letters
and
"Three plus two" has four letters
are not required to be equivalent.
However: In the sentence "This sentence has five words" the word
"five" is used, (in addition to being mentioned, at least implicitly) -
this had better be disallowed. (You thought I'd forgotten the point<g>...)
[It happens all the time that one of my colleagues will say
Define a vector space.
as a question. A correct answer to this is "Let V=R; then V is a vector space".
That does in fact define a vector space, which is what was asked for. I don't
know that anybody's ever tried it... What's actually meant by the question is
Define "a vector space".
Now "Let V=R" doesn't fly because "R" is not the definition of the 'word'
"vector space".]
Toby Bartels
>Toby Bartels (to...@ugcs.caltech.edu) wrote:
>>The very beginning of Kevin's first post dispenses with your aspect
>>(which is a reason why it is not later emphasized).
>>Kevin was talking about that sort of omnipotence
>>which allows God to do anything that is *not* logically contradictory.
>Kevin's disposal of that aspect was based on the false premise that it
>is logically impossible to lift a rock of infinite size, since that
>would require more than an infinite lift. Therefore, Kevin concluded,
>God *can* create a rock that is too big for Him to lift, and He is not
>required to lift the rock in order to be omnipotent, because that would
>be logically impossible.
Wrong. In fact, Kevin did not dispose of it.
Some other fellows disposed of it, and Kevin merely concurred.
By `the very beginning', I meant before Kevin even mentions infinity.
>Ok, let's use your definition of omnipotence. Are you saying,
> (1) It is a logical impossibility that God can create a rock so
> big that He can't lift it, or
> (2) It is a logical impossibility that God can lift any rock He
> can possibly create, or
> (3) both, or
> (4) neither?
>Explain your answer.
I am denying (2). Now that you mention it, I also deny (1).
But I think we agree (1) is logically impossible
if `God' is defined to include omnipotence.
>>I also happen to be a fan of logic.
>>That's why I react so badly to illogical statements.
>Some of my statements were based on a different hypothesis. That does
>not make them illogical. For the sake of argument, I am now willing to
>adopt your hypothesis. Where is the illogic?
I don't think you've ever made a logically invalid argument.
But you've inserted statements in logically inappropriate places.
In particular, you used a different definition of omnipotence
than the thread had been using, without saying so.
You've also made what are IMO false statements,
apparently because you've misunderstood something;
this might also be called `illogic', I suppose.
>>For the record:
>>by claiming omnipotence when God is shown to be capable
>>of doing anything not logically contradictory,
>>the authors quoted in Kevin's original post show themselves
>>to be using a definition of omnipotence limited by the requirements of logic.
>>Kevin extended the situation a bit and found a problem with infinity,
>I didn't read it that way. I thought Kevin considered his conclusions
>regarding infinity to be a way out of the dilemma. Since (he thought)
>asking God to lift an infinitely heavy rock is asking Him to do the
>impossible, we shouldn't expect Him to be able to do it, even though He
>is omnipotent. Dilemma solved.
Well, this is how I just reread it:
Kevin stated that the explanation he quoted regarding rocks of unusual size
was satisfactory to him, indeed the best he had ever heard.
Then he analysed it further (and incorrectly, as you know)
and the issue was no longer resolved.
(But, should his errors regarding infinity be fixed,
as they were by you, the resolution would be repaired.)
>Then can God make a rock so big that He can't lift it, or not? Explain
>your answer.
It is not a logical contradiction that he should make any possible rock
and that he should lift any possible rock.
Clearly, it is a logical contradiction that he should lift any possible rock
and yet there be a possible rock that he can't lift.
(As it turns out, God doesn't exist, but this is not a logical requirement.)
-- Toby
to...@ugcs.caltech.edu
David Ullrich
>>
>>[stuff deleted]
>>
>>>> So I would in fact say that I could believe in logic or in the
>>>>I'd choose logic.
>>
>>>Oh, sure its a sentence---it's just maybe not a meaningful one. But
>>>the question is: why *isn't* it meaningful? Timothy Chow was pointing
>>>out (correctly, in my opinion) that it is the naive, unrestricted notion
>>>of "truth" that leads to difficulty. Some people think that it is the
>>>fact that the sentence "talks about itself" that somehow makes it
>>>a meaningless sentence. But that is dispelled by considering similar
>>>examples: "This sentence has five words" and "This sentence is written
>>>in English" which are self-referential but perfectly meaningful. For
>>>any well-defined property P of sentences, I believe we can make sense of
>>>"This sentence has property P". The failure of "This sentence is false"
>>>to make any sense shows that an unrestricted notion of "being true" or
>>>"being false" is *not* a well-defined property of sentences.
>
>> words". Because "five" means the same as "four plus one", so that
>> "this sentence has five words" and "this sentence has four plus
>> one words" should be equivalent.
>
>But anyway, if you don't like the locution "this sentence", it can always
>be eliminated with a certain amount of work:
>
>First, for any string X, we define the diagonalization of X to be the
>string resulting from replacing each occurrence of "..." in X by X itself,
>surrounded by quotes. So the diagonalization of "...has three words" is
>"`...has three words' has three words". Now, let G0 be the string
>
> "the diagonalization of ... has twelve words"
>
>By definition of diagonalization, the diagonalization of G0 is the
>string G:
>
> "the diagonalization of `the diagonalization of ... has twelve words'
> has twelve words"
>
>By counting, we see that G has twelve words. Therefore, the diagonalization
>of G0 has twelve words. Therefore, we conclude:
>
> the diagonalization of `the diagonalization of ... has twelve words'
> has twelve words
>
>which is just G itself. So G can be seen to assert the fact that G
>itself has twelve words. Banishing self-referential sentences is
>unnecessary and pretty much impossible, if you are going to allow
>sentences about operations on sentences.
I don't see your point here - a sentence which "can be seen
as referring to itself" is not at all the same thing as a sentence which does
in fact refer to itself. If sentences which can be interpreted as referring
to themselves are no good then for example Godel's in trouble. On the other
hand if we allow sentences that literally refer to themselves we're in big
trouble ourselves.
See note to Chow regarding "this sentence has five words".
ISI-JOAQUIN HERNANDEZ RICO
On 1 Oct 1995, Toby Bartels wrote:
> >That's why I was careful to state the dilemma in the form (omnipotent
> >God exists) ==> contradiction, rather than (God is omnipotent) ==> (God
> >can't do X). It's a preemptive strike to keep God from existing
> >before He gets a chance to nullify our logic. :-)
>
> I applaud this approach but remind you that you have succeed at it
> only for omnipotence that includes the ability to do logical impossibilities.
^^^^^^^^^^^^^^^^^^^^^^^^^^
>
>
> -- Toby
> to...@ugcs.caltech.edu
>
But then again you would have to define "omnipotence" and state that
different kinds of it exist.
Joaquin H.
Toby Bartels
>>So far, no logical contradiction has been found to result
>>from the assumption that God can do anything that's not logically impossible.
> I have a big problem with this, namely the definition of
>"logically impossible". I know what that means in certain formal
>systems, but thay have nothing to do with the real world, so surely
>something broader is intended.
I have a nice intuitive notion;
perhaps by examing the rest of your post I can convey to you its nature.
> The notion of "logically impossible" is not as well-defined as
>you think it is. Somebody mentioned the triangle with angles adding up
>to more than 180 degrees earlier - most of us would agree this is
>impossible if we're careful to define everything first.
That's exactly it. How do you define `triangle' (and the other words)?
There are definitions such that the triangle in question is possible
and others such that it isn't.
You're familiar with this state of affairs when the words refer to math;
but it also works when the words refer to the world.
> But what about time travel into the past? A lot of people would
>say that's impossible, and prove it by saying "what if you shot your
>grandfather?". But a lot of people are not convinced by this (I don't
>see why not, but that's another question...)
This is not impossible, unless you go to the trouble of defining `time travel'
to include notions of free will and the ability to change the future, etc.
> What about simultaneity? Is it logically possible for me or God
>or somebody clever to invent a gizmo that would turn on a light bulb on
>Alpha Centauri precisely when I hit a certain button here on Earth? Most
>people would see no logical contradiction there, but some people would
>asert that there is simply no such thing as simultaneity: Event A there
>and event B here may look simultaneous to one observer and not to another,
>and so the word "simultaneous" is simply meaningless.
`simultaneous' is perfectly meaningful if you define it to be.
For instance, specify a scalar field called $t$
and define simultaneity as if $t$ gave the time of an event.
> After something happens can God adjust the universe in such a way
>that it did not in fact happen? I don't mean just revise the universe so
>that it looks the same as it would if the something hadn't happened, I mean
>revise the universe so that it did not happen. Is that a logical impossibility
>or not? It certainly seems to me like it "should" count as logically
>impossible, but I'm equally certain I'm not going to be able to prove it's
>impossible.
Given that it did happen, it is logically impossible that it didn't happen.
> Or in the other direction, what about turning matter into energy?
>That's certainly impossible, except it isn't...
According the definitions I usually use of `matter' and `energy',
this is a category error. But it may not be according to yours.
-- Toby
to...@ugcs.caltech.edu
Toby Bartels
>I think there are two basic ways to handle the liar paradox, both of
>which have been discussed in this thread. Let L be the statement "This
>statement is false."
> 1. L is not a statement, and therefore the law of excluded
> middle is irrelevant, because the law of excluded middle
> applies only to statements, and not gibberish.
> 2. L is a statement, but it doesn't have a truth value. The
> law of excluded middle fails for L.
>I originally had in mind the second approach, but either one works for
>the liar paradox. Neither approach works for the heavy-rock paradox,
>however. Let P(x) be the statement "x can make a rock so large that x
>can't lift it." Is P(God) a statement? What about P(Fred)? Is either
>of those collections of words self-contradictory? Does the law of
>excluded middle fail for P(God)? For P(Fred)?
They are both statements, and they are both false.
In fact P(God) is a logical contradiction (assuming God is omnipotent).
>Whichever way you choose to look at the liar paradox, the heavy-rock
>paradox has a fundamentally different nature. P(x) is not itself a
>contradiction, and neither is "not P(x)". The contradiction appears
>when you add O(x) = "x is omnipotent."
> (1) P(x) ==> not O(x),
> (2) not P(X) ==> not O(x),
> (3) O(x).
>From these we may derive not (P(x) or not P(x)). This does not mean
>P(x) is itself a contradiction, but nevertheless a contradiction exists.
>The contradiction is not resolved by saying that O(x) does not enable x
>to do contradictory things, because that proviso does not apply to P.
It does apply to P(omnipotent being).
-- Toby
to...@ugcs.caltech.edu
Dave Seaman
Toby Bartels <to...@blend.ugcs.caltech.edu> wrote:
>Dave Seaman (a...@seaman.cc.purdue.edu) wrote in large part:
>> (2) not P(X) ==> not O(x),
>> (3) O(x).
>
>>From these we may derive not (P(x) or not P(x)). This does not mean
>>P(x) is itself a contradiction, but nevertheless a contradiction exists.
>>The contradiction is not resolved by saying that O(x) does not enable x
>>to do contradictory things, because that proviso does not apply to P.
>
>It does apply to P(omnipotent being).
If you define omnipotence as the ability to do anything that is
logically possible, and if you are sufficiently careful about nailing
down just what is logically possible, then it may be possible to avoid
the contradiction (just as you can avoid Russell's paradox by being
sufficiently careful about the definition of sets).
I don't think anyone in this thread has been nearly careful enough
about definitions to be able to say that the heavy-rock paradox is
hereby resolved.
If you say that P(God) is logically contradictory, what does that
mean?
P(God) means "God can make a rock so big that He can't lift it." Do
you really mean to say that this is logically contradictory?
We saw a (slightly flawed) model in which the statement is definitely
NOT logically contradictory -- namely Kevin's model in which God can
create an infinite rock (there being only one size of infinity in
Kevin's universe) and then can't lift it. In that model, it's the idea
of lifting an infinite rock that turns out to be contradictory, since
it would require "more than an infinite force."
Granted that higher infinites exist, it's still at least thinkable that
there might be a largest logically conceivable rock, and also a
strongest logically conceivable force. Astronomers tell us that the
largest rock is about the size of Jupiter. Anything much larger would
be a star.
Logic does not, in general, predict what will happen when the
irresistable force meets the immovable object (except, of course, that
"something's got to give.")
Therefore, I object when you single out one of the two possibilities
and say it's logically impossible, completely ignoring the fact that
the other outcome may be the impossible one.
Dave Seaman
Dave Seaman
Timothy Chow <tc...@math.lsa.umich.edu> wrote:
><>[...] let S(x) be
><>the variable sentence "You get a false sentence if you set all the
><>variables in the xth variable sentence equal to x."
>I think you miss my point, so let me play the game your way. The
>contradiction is not in S(x), but rather in R(x) = "x is the index
>number of S." S(x) is merely being used as a tool to expose the
>problem in R(x).
>
> R(x) and S(x) ==> contradiction
> R(x) and not S(x) ==> contradiction
> Therefore, not R(x).
[I posted something else and then cancelled it. If you see another
response, ignore it.]
I don't follow that second step. I think R(x) and not S(x) is
perfectly consistent. In particular, "not S(x)" means "You do not get
a false sentence when ...". This would appear to be satisfied if the
result of doing the replacement has no well-defined truth value and
therefore is not a sentence at all.
Therefore, your proof demonstrates that something that appears to be a
sentence is in fact not a sentence, while the heavy-rock paradox is not
a true paradox at all. It only amounts to showing that if you define
omnipotence in a certain way, then the set of omnipotent beings must be
empty.
Dave Seaman
Stephen Harris
>From: a...@seaman.cc.purdue.edu (Dave Seaman)
>Subject: Re: lifting a infinitely heavy rock question
>
>Logic does not, in general, predict what will happen when the
>irresistable force meets the immovable object (except, of course, that
>"something's got to give.")
>Therefore, I object when you single out one of the two possibilities
>and say it's logically impossible, completely ignoring the fact that
>the other outcome may be the impossible one.
Logic is a system for working with given propositions.
It states the connectedness or validity of the operations
or conclusions formed on the given propositions.
It does not state whether the proposition is true but
whether the argument is true (sound).
For something to be true it must have a property of existence.
The problem lies in the question. It assumes that both an
infinitely heavy rock and an infinitely strong lifter can
exist in the universe at the same time. If this premise
is false than their is no sound logical operation to perform on it.
God cannot lift himself by his bootstraps because he goes barefoot.
So the question does not have meaning as applied to existence.
If the moon is made of greencheese how many crackers will it spread?
Timothy Chow
David Ullrich <ull...@math.okstate.edu> wrote:
> The Goedel trick constructs a "sentence" which does not talk about
>anything, in fact it's not a sentence in the sense in which I was using the
>word, it's just a string of symbols. If the formal sentences in the formal
>language are given a certain interpretation then the sentence sort of appears
>to talk about itself, in particular if you assume the sentence is provable
>in the system it follows that it isn't, and conversely. But the Goedelian
>sentence does _not_ talk about itself, it doesn't talk about anything.
There are two aspects of any linguistic assertion: syntactic and semantic
("mention" and "use" if you like). The syntactic string, taken in isolation,
of course does not "talk about" anything. When you say that the Goedelian
sentence does not talk about anything, presumably you are referring to this
fact. But this is practically tautologous, e.g., in this sense, "Apples
are red" does not talk about anything either, because a purely syntactic
entity never talks about anything.
For your assertion that sentences should not talk about themselves to have
content, it must mean that the *semantic* content of a sentence should never
talk about the sentence itself. But the semantic content of the Goedelian
sentence *does* refer to itself, or at least to its syntactic representation.
If you outlaw this (as you seem to do in your subsequent discussion), then
you outlaw Goedel's proof. If you don't outlaw this, then you are allowing
the semantic content of a sentence (such as "This sentence has five words")
to include references to its syntactic representation.
Dave Seaman
Stephen Harris <mul...@azstarnet.com> wrote:
>In article <44uuu9$n...@seaman.cc.purdue.edu> a...@seaman.cc.purdue.edu (Dave Seaman) writes:
>>Logic does not, in general, predict what will happen when the
>>irresistable force meets the immovable object (except, of course, that
>>"something's got to give.")
> The problem lies in the question. It assumes that both an
> infinitely heavy rock and an infinitely strong lifter can
> exist in the universe at the same time. If this premise
> is false than their is no sound logical operation to perform on it.
Ok, so I forgot the smiley. Logic does tell us that there can't be
both an irresistible force and an immovable object. Logic does not,
however, tell us that there can't be both an infinite force and an
infinite mass. That's an entirely different proposition, and it's
clearly the one I meant, if you look back at the history of this
thread. It may be violation of physical laws, but it's not a violation
of logic.
Therefore, you fell into your own trap by reasoning about a
hypothetical that may be false.
Dave Seaman
David Ullrich
>David Ullrich (ull...@math.okstate.edu) wrote:
>
>> No, actually I have big troubles with "This sentence has five
>>words". Because "five" means the same as "four plus one", so that
>>"this sentence has five words" and "this sentence has four plus one words"
>>should be equivalent.
>
>(`five' and `four plus one' have the same meaning; but `this' has changed.)
>
>> Sentences are not allowed to talk about themselves, except this one.
>
>When designing a formal system,
>you can choose whatever you want for your propositions.
>But this sentence is still a *sentence* (and it's still true).
(In case anybody's wondering, the "except this one"
was a joke.)
When I say that sentences are not allowed to talk about themselves
I was refering to actual sentences in actual languages. Sentences in
formal languages do not talk about anything.
Not knowing much about the history of mathematics it's not clear
to me whether you're 50 yeas behind or 100. I quit.
David Ullrich
>In article <44uk35$1j...@bubba.ucc.okstate.edu>,
>David Ullrich <ull...@math.okstate.edu> wrote:
>> The Goedel trick constructs a "sentence" which does not talk about
>>anything, in fact it's not a sentence in the sense in which I was using the
>>word, it's just a string of symbols. If the formal sentences in the formal
>>language are given a certain interpretation then the sentence sort of appears
>>to talk about itself, in particular if you assume the sentence is provable
>>in the system it follows that it isn't, and conversely. But the Goedelian
>>sentence does _not_ talk about itself, it doesn't talk about anything.
>
>("mention" and "use" if you like).
The distinction between use and mention is not the same as the
distinction between syntax and semantics (which is not to say the consepts
are unrelated). For example, when a word is used in an English sentence,
(but not mentioned!) one can talk about the syntax involved in the use of
the word and one can also talk about the semantics involved in the use
of the word.
>The syntactic string, taken in isolation,
>of course does not "talk about" anything. When you say that the Goedelian
>sentence does not talk about anything, presumably you are referring to this
>fact. But this is practically tautologous, e.g., in this sense, "Apples
>are red" does not talk about anything either, because a purely syntactic
>entity never talks about anything.
>
>For your assertion that sentences should not talk about themselves to have
>content, it must mean that the *semantic* content of a sentence should never
>talk about the sentence itself. But the semantic content of the Goedelian
>sentence *does* refer to itself, or at least to its syntactic representation.
>If you outlaw this (as you seem to do in your subsequent discussion), then
>you outlaw Goedel's proof. If you don't outlaw this, then you are allowing
>the semantic content of a sentence (such as "This sentence has five words")
>to include references to its syntactic representation.
>--
This is ridiculous - I quit. The statement "Apples are red" does talk
about apples, because it is an English sentence, and English sentences
do mean things, at least they're supposed to. The "Goedel sentence" is not
an English sentence, it's a formal sentence in some formal language, and
as such it does not talk about anything. It can be interpreted as "talking
about" various things, but it does not have a meaning in and of itself.
To put it another way, the "Goedel sentence" is not a "linguistic
assertion".
You need to find a book on logic and study up on the distinction between
language and metalanguage. Let's talk about the propositional calculus for a
second - we have primitive propositions P,Q,R, etc and operators &,-,>, which
we "interpret" as "and", "not", "implies". The semantics of the language is
captured by "truth tables".
Given two formulas A and B in the language let's say that
"A tautologically implies B" if B evaluates to T for every assignment of T
and F to the primitives for which A evaluates to T. This is the same as saying
that A>B is a tautology, where a tautology is a formula which evaluates to T
for any assignment of T and F to the primitives.
Now say we have two formulas A and B. The following are both "linguistic
assertions", and they happen to be equivalent:
1) "A tautologically implies B"
2) "A>B is a tautiology".
Both (1) and (2) _say_ _something_.
On the other hand, the following is NOT an assertion, in particular
it is not equivalent to (1) and (2) - it doesn't have a chance of being
equivalent to (1) and (2) because it doesn't assert anything whatever:
3) "A>B"
Honest, I'm not making this up, find a logician and ask him about the
distinction and why it's crucially important. (If you do actually look
for a logician, please show him a verbatim copy of the above - it's
happened repeatedly in this thread that I say something and you reply
with "You seem to be saying that" followed by something that's totally
different from what I said.)
When you say that the Goedel sentence refers to itself you're
confusing (1) and (3) (rather, you're making an error of the same sort,
confusing language and metalanguage - (1) and (3) a simply a convenient
example of what I (and everybody else, by the way) means by the
distinction - (1) is a meta-statement, while (3) is actually a statement
in the language.)
*****
Or look at it this way: Let's talk about tautological equivalence
instead of tautological implication. Let's say A<>B is an abbreviation
for the formula (A>B)&(B>A) , and let's say that A and B are tautologically
equivalent if the formula A<>B is a tautology (that is, A and B evaluate
the same under any assignment of 's and F's to the primitives.)
Now, formulas in the propsitional calculus can be used to describe
electrical circuits in a natural and actually very important way. What
about the two gizmos
4) "A and B are tautologically equivalent"
and
5) "A<>B"
??? Are (4) and (5) the same? No. If we interpret formulas as electronic circuits
then (4) says something about two circuits - it says that circuit A and circuit B
will do the same thing. On the other hand (5) does not say anything about what
any given circuit will do, (5) is simply a description of a certain circuit!
Robert Blandford
> One theory I have heard on this matter is that an omnipotent God could
> make such a rock but chooses not to.
>
> In other words, an omnipotent being would have the capability of
> destroying omnipotence, but obviously wouldn't be doing so.
>
> Seems at least logically sound. Of course, if you stretch the
> definition of omnipotence far enough, there will be other
> contradictions.
Exactly in the limit an omnipotent being can do anything. Once the rock that
cannot be lifted is made, in the limit any omnipotent being may lift it.
The cycle repeats to infinity...
Interesting having to add limits to logic to make it sound. What does this
infact mean, then?
Robert.
--
robe...@cs.auckland.ac.nz
http//www.cs.auckland.ac.nz/~robert-b/
Robert Blandford
> from the assumption that God can do anything that's not logically impossible.
Either, that in it self is a contradiction or both entities are by
definition omnipotent.
> Equally obviously, omnipotence that is not limited by logic can violate logic.
We must chose the second assertion then.
> I, personally, believe neither kind of omnipotence exists in the world,
> in part for empirical reasons.
You have no reason to believe that at all.
Robert.
--
robe...@cs.auckland.ac.nz
http//www.cs.auckland.ac.nz/~robert-b/
Daryl McCullough
>The notion that sentences must not talk about themselves is not an extreme
>view, it's perfectly standard, it's something you have to be careful to
>avoid, like dividing by zero.
I would compare it instead to the prohibition against taking the square
root of a negative number. The question is whether one can make sense of
such things. It turns out that you *can* make sense of the square-root
of a negative number. You can also make sense of self-referential sentences.
Here is a simple rule that, (as far as I know) clears up any confusion
about self-referential sentences: Always interpret a sentence of the
form
this sentence has property P
to mean the same thing as:
"this sentence has property P" has property P
which isn't self-referential at all. For example:
this sentence has five words
becomes
"this sentence has five words" has five words
which is not self-referential, and true. Of course, care must be taken
to understand what sentence is being referred to as "this sentence":
The meaning is ambiguous if it could plausibly refer to more than one
sentence. But that is not a problem with self-reference, it is just a
problem with *reference*. If I say "That man is wearing a toupee", then
the meaning of my statement is ambiguous if there is more than one man
present.
[discussion of use versus mention deleted]
>However: In the sentence "This sentence has five words" the word
>"five" is used, (in addition to being mentioned, at least implicitly) -
>this had better be disallowed.
There is no reason to disallow it! I agree that it makes the analysis
more difficult, but not impossible. If somebody says
Carrot-top (who was called that because of his orange-colored hair)
just walked in the room.
it is perfectly clear what they mean. "Carrot-top" is both used and mentioned
in this sentence, which means that you can't substitute "Bill" for
"Carrot-top". You *can* paraphrase it as
Carrot-top (who was called "Carrot-top" because of his orange-colored hair)
just walked in the room.
In this paraphrase, it is clear that you can substitute "Bill" for the first
occurrence of "Carrot-top" but not for the second.
Similarly, you can expand
This sentence has five words
into
"This sentence has five words" has five words
Then "four plus one" can be substituted for the second occurrence of "five"
but not for the first.
Daryl McCullough
ORA Corp.
Ithaca, NY
Daryl McCullough
>da...@oracorp.com (Daryl McCullough) wrote:
[much stuff deleted]
>>First, for any string X, we define the diagonalization of X to be the
>>string resulting from replacing each occurrence of "..." in X by X itself,
>>surrounded by quotes...
>>Now, let G0 be the string
>>
>> "the diagonalization of ... has twelve words"
>>
>>By definition of diagonalization, the diagonalization of G0 is the
>>string G:
>>
>> "the diagonalization of `the diagonalization of ... has twelve words'
>> has twelve words"
>>
>>By counting, we see that G has twelve words. Therefore, the diagonalization
>>of G0 has twelve words. Therefore, we conclude:
>>
>> the diagonalization of `the diagonalization of ... has twelve words'
>> has twelve words
>>
>>which is just G itself. So G can be seen to assert the fact that G
>>itself has twelve words. Banishing self-referential sentences is
>>unnecessary and pretty much impossible, if you are going to allow
>>sentences about operations on sentences.
> I don't see your point here - a sentence which "can be seen
>as referring to itself" is not at all the same thing as a sentence which does
>in fact refer to itself.
In what way? A sentence never refers to *anything* except via an
interpretation.
>If sentences which can be interpreted as referring
>to themselves are no good then for example Godel's in trouble.
*That* is exactly my point---you can't forbid self-reference
without severely crippling the expressiveness of your language.
>On the other hand if we allow sentences that literally refer
>to themselves we're in big trouble ourselves.
I don't know what you mean by "literally" here. And anyway, the
problem with the Liar paradox does not require any more self-reference
than the "apparent" self-reference of Godel sentences:
Let Liar be the following sentence:
the diagonalization of `the diagonalization of .... is a false
sentence' is a false sentence
This sentence has no more (and no less) self-reference than the Godel
statement for arithmetic. Yet this sentence is every bit as paradoxical
as "This sentence is false". So the problem with the liar paradox is
not the self-reference (which I think is unavoidable in any sufficiently
expressive language) but the use of an unrestricted notion of truth or
falsity.
Toby Bartels
>The "other fellows" said that rocks, by definition, have finite sizes,
>and therefore it is sufficient that God can create any size rock and
>also can lift any size rock. The only mention of infinity was to point
>out that the collection of potential rock sizes is infinite.
>Kevin immediately paraphrased this incorrectly as saying that God can
>lift an infinitely heavy rock. As soon as Kevin allows infinitely
>heavy objects to be considered rocks, the "other fellows'" disposal of
>the problem goes out the window. Kevin's disposal of the problem was
>as I said -- God can create infinitely heavy rocks, and he can't lift
>them, but lifting them would be (he claimed) logically impossible.
I wouldn't say the other fellows' disposal
went out the window; Kevin merely thought it did.
But, as I understand his post, he never thought he had a disposal;
all he'd done was invalidate the other fellows' disposal.
He never claimed logical contradiction;
he seemed to want that the rock argument
shouldn't invalidate an omnipotent God.
Evidence: Kevin's first words after quoting the other fellows
state that he is quite pleased with their explanation.
So, when he (incorrectly) invalidates their explanation,
this should leave him with a problem.
>It's true that when you introduce higher orders of infinity, then the
>"other fellows'" disposal of the problem becomes feasible once again.
>Unlike Kevin's explanation, we once again find that an omnipotent being
>can create a rock of any size, and also can lift a rock of any size,
>and there is no contradiction. I now think this is what you were
>arguing, but it wasn't clear to me at the time.
Yes, this is basically what I was saying.
That and that, since Kevin was now essentially back
in the same position he was in after quoting the other fellows,
he didn't have a dilemma or any other problems.
-- Toby
to...@ugcs.caltech.edu
Toby Bartels
>tc...@math.lsa.umich.edu (Timothy Chow) wrote:
>>David Ullrich (ull...@math.okstate.edu) wrote:
>>> No, actually I have big troubles with "This sentence has five
>>>words". Because "five" means the same as "four plus one", so that
>>>"this sentence has five words" and "this sentence has four plus one words"
>>>should be equivalent.
>>Why should they be equivalent? Saying that "five" means the same as "four
>>plus one" does not imply that every *sentence* containing the word "five"
>>will not change its meaning when "five" is replaced by "four plus one."
>>Let S be the sentence "This sentence has five words" and let S' be the
>>sentence "This sentence has four plus one words." Saying that "five"
>>means the same as "four plus one" just says that the proposition that S
>>has five words is the same as the proposition that S has four plus one
>>words. It does not imply that the syntactic transformation that changes
>>S to S' necessarily leaves the meaning of the sentence (i.e., the
>>proposition expressed by the sentence) invariant. The meaning of the
>>word "this" depends on context; if one changes the context (e.g., by
>>some kind of syntactic transformation) then naturally one expects the
>>meaning of the word "this" (and consequently the meaning of the sentence)
>>to change.
> I should have been more precise. We need to talk about the distinction
>between use and mention.
Fine discussion snipped. I wish more people made the distinction as clearly.
But the discussion seems irrelevant, IMHO. To continue:
> In the sentence "This sentence has five words" the word
>"five" is used, (in addition to being mentioned, at least implicitly) -
>this had better be disallowed. (You thought I'd forgotten the point<g>...)
Why must it be disallowed? Yes, the sentence both uses `five'
and implicitly mentions it (through the word `this').
Why must this be a problem?
I'm well aware that many formal systems fail to include
as logical propositions strings analogous to this sentence.
But that doesn't mean all logical systems must avoid the sentence.
(And none of this changes the fact that the sentence *is* a sentence,
whether or not it is accepted as a logical proposition.)
`This sentence has five words.' and `This sentence has four words plus one.'
have different semantic contents because the word `this' has different meanings.
It is through the word `this' that any mention is made of `five' is made.
In one case, `this' mentions `five' implicitly; in another, it doesn't.
`five' and `four plus one' have the same meanings, as usual;
in one case, `five' is mentioned and in another case `four plus one' is.
What is wrong with this? Where is the contradiction? It works out fine!
-- Toby
to...@ugcs.caltech.edu
Toby Bartels
>>So far, no logical contradiction has been found to result
>>from the assumption that God can do anything that's not logically impossible.
>>Obviously, no such a priori contradiction can ever be found.
>Either, that in it self is a contradiction or both entities are by
>definition omnipotent.
??? What are both entities?
>>Equally obviously, omnipotence that is not limited by logic can violate logic.
>We must chose the second assertion then.
??? What second assertion?
>> I, personally, believe neither kind of omnipotence exists in the world,
>> in part for empirical reasons.
>You have no reason to believe that at all.
If you mean I ultimately have no purely rational justification
for believing things on the basis of empirical evidence,
I agree (although I believe many such things anyway).
If not, well, it's time for more question marks.
I'm afraid I don't know what you're alluding to in most of your post.
If there's something earlier in the thread you're thinking of,
I guess I can't find it.
-- Toby
to...@ugcs.caltech.edu
Robert Douglas Blandford
>tc...@math.lsa.umich.edu (Timothy Chow) wrote:
>>In article <44pfj7$10...@bubba.ucc.okstate.edu>,
>>David Ullrich <ull...@math.okstate.edu> wrote:
>>>tc...@math.lsa.umich.edu (Timothy Chow) wrote:
>>>>The matter isn't quite as simple as this. What your argument shows is that
>>>>the naive notion of omnipotence is self-contradictory. But the naive notion
>>>>of truth is self-contradictory (liar paradox)
>>>
>>> No, the liar paradox shows that certain constructions that appear a
>>>priori to be sentences had better not be regarded as such.
>>
>>Well, Dave Seaman seems to think that the problem lies with the law of the
>>excluded middle, so it seems your point of view is not uncontroversial.
>>
> Huh? Which problem are you referring to here by "the problem"?
>Others in this thread have suggested that God has the power to abolish the
>law of the excluded middle - I would agree that the most appropriate
>response to anyone perhaps questioning the law of the excluded middle
>is "ajhsgd ashsdn,kj wef dsf, so that UYGWd. And further, jgf sdf sd."
>I was not claiming that the question of what's a sentence and what isn't
>had anything to do with the "define omnipotence" problem, I was referring
>specifically to the liar paradox - I didn't notice Seaman saying anything
>about that. Anyway, your "the problem" seems a little ambiguous.
Another formulation of the problem, which might appeal to physicists, is the
resolution of the impact between an `immoveable' object and an `unstopable' force.
The problem is intractable if
>>In any case, let's say I grant you your resolution of the liar paradox.
>>Then my point is that one could respond to the omnipotence paradox by
>>saying, "I can either believe in logic or in the idea that 'Can God make
>>a rock too heavy for him to lift?' is a question, and of the two I'd choose
>>logic."
> If you insist - assuming we know what "God" is I don't see any
>problem with what the string "God can make a rock too heavy for him to
>lift" _means_, there's no funny self-reference going on there. Seems like
>it's either true or false to me.
Another formulation of the problem, which might appeal to physicists, is the
resolution of the impact between an infinite mass and an infinite force.
The system is intractable if both quantities are left suitable vague. By giving
some value to infinite the problem reduces to a rather simple impact. You
might like to imagine what happens when you play marbles.
Similarly, this `logical' problem is intractable given axiomatic basis of the system.
The solution of `limited omnipotence' uses exactly the same approach. The concept
of omnipotence is limited to a concept that can be described logically. What this
suggests is that the mathematics that we use might not be as general as we would
imagine.
>>Or, if I grant Dave Seaman's point of view, then I can respond to
>>the omnipotence paradox by rejecting the assumption that either "God can
>>make a rock too heavy for him to lift" or "God cannot make a rock too heavy
>>for him to lift" must be true, and not both.
>>
> Hmm. ajhsgd ashsdn,kj wef dsf, so that UYGWd. And further, jgf sdf sd.
>
>>Because the argument is an intuitive one, there are lots of different places
>>you can fiddle with the assumptions. Rejecting the concept of omnipotence
>>is not the only way out.
>>
Exactly, what is at issue here is how to fiddle with the axioms that define
logic so that this particular problem can be solved. Dave Seaman's point of
view is that he would rather not make any changes to the logical axioms
and leave the problem unsolveable.
Abolishing the law of the excluded middle allows us to respond, `I don't know'
to any predicate or sentence p.
Other solutions to the problem will fiddle the axioms either one way or the other
depending on which answer `true' or `false' seems the most appropriate.
The solution I suggested is that I suggested was that there are two systems,
the first based on the axiom `God can do anything' and the second `every
statement has a true or false value'. Nothing can be concluded from the
existence of paradoxes.
>>> I'm missing your point regarding Turing machines and Church's
>>>thesis. Do you have an example of something that you intuitively regard as
>>>an "algorithm" which cannot be realized by a Turing machine?
>>
>>No. The Church-Turing thesis asserts that no intuitive notion of procedure
>>can be any more powerful than a Turing machine algorithm. My point is that
>>the Church-Turing thesis cannot be proved or disproved logically because it
>>makes reference to the vague concept of "an intuitive notion of a procedure."
>>Similarly, the intuitive notion of omnipotence cannot be *proved* to be
>>self-contradictory; one can show that *certain* intuitions about omnipotence
>>(say, the intuition that certain formal statements X, Y, and Z accurately
>>and consistently capture the notion of omnipotence) lead to contradiction,
>>but there is always enough vagueness in the "raw" intuition to escape total
>>disproof.
> I haven't been saying that anything completely and accurately captures
>the intuitive notion of anything. If anything just the opposite: The traditional
>notion of "omnipotent" is in fact "can do anything", and yes there is some
>ambiguity there. There is no point in discussing anything with someone who
>believes in the possible existence of a being who can do things that are
>impossible, while if we revise the definition of "omnipotent" to "can do
>anything that's possible" we have serious difficulties defining the word
>"possible".
This is just one method of resolving the problem.
I like the point about the Church-Turing proof. We know there is a simple proof
that any lambda calculus is equivalent to a turing machine. Just produce a
calculus that simulates the machine and produce a turing machine that simulates
the calculus. This means that the axiomatic/intuitive basis of the two systems
can be shown to equivalent.
This means that the lambda calculus is as good as a turing machine at structuring
a program. Similarly there will be other program structuring methods which are as
good.
Robert
Toby Bartels
>>>Sentences are not allowed to talk about themselves, except this one.
>>Of course they are!
>>When designing a formal system,
>>you can choose whatever you want for your propositions.
>>But this sentence is still a *sentence* (and it's still true).
>(In case anybody's wondering, the "except this one" was a joke.)
(That's what I figured.)
> When I say that sentences are not allowed to talk about themselves
>I was refering to actual sentences in actual languages. Sentences in
>formal languages do not talk about anything.
English is an actual language.
`This sentence has five words.' is an actual sentence in that actual language.
Nothing in the development of mathematics will change this linguistic fact.
> Not knowing much about the history of mathematics it's not clear
>to me whether you're 50 years behind or 100.
What's the history of mathematics got to do with it?
We're talking about sentences in English.
I hope you don't think I was confusing sentences with formal propositions.
I was, in fact, pointing out the *difference* between them.
I figured you might be thinking that, since self referential sentences
are somewhat analogous to forbidden self referential propositions,
that this was why you thought self referential sentences weren't sentences.
I'm glad to find out this is not the case.
>I quit.
:-(
-- Toby
to...@ugcs.caltech.edu
C:DEMONSPOOLMAIL
that God and The Universe have to be consistent.
Using the strongest possible definition of omnipotency, God can do
anything, including making himself unable to do it.
Fine, as long as he can be inconsistent.
We can now try to construct some laws of nature around this. We can
say to begin with that God makes (or can alter) all the laws of nature.
So if we observe that gravity makes a miner's hat fallto the bottom of
the mine, we must assume that it is because Godhas made it so. But
although this may have happened on Monday, and every other day in the
past, we cannot use this as a prediction that the miner's hat will fall
to the bottom of the mine on Tuesday, because God could change the law
on Monday night. This is not a problem, it just means that we have a
big problem predicting things in our universe.
We can say that God has made the miner's hat fall on days adding up to
4 years out of the past 5 years. It seems as if we can say that the
probability of the miner's hat falling on Tuesday is 4/5. And that the
probability of the miner's hat falling on Wednesday is 4/5. And
Thursday. And so on, until we've had enough days of falling hat to
noticably affect our calculation of the probability - i.e. when we've
had a fair amount of days after Tuesday in proportion to the number of
days we've ever had, the probability will have changed significantly.
A way of avoiding this probability variation is to say that time
extends back infinitely (let's assume for now that God is a normal,
three-dimensional being travelling forward in time). If, up to now,
the statistical likelihood of the miner's hat falling has been 4/5,
this should be the same for Tuesday, Wednesday and for as many days
forwards you care to go. *BUT* ...
... This puts a limitation on God, i.e. that if the miner's hat has
always been "dropping for 4/5 of the time", it will now. God should be
able to alter that *now* by changing the statistical frequencies
(without having to go back in time), because he is omnipotent. The
only way he can do this is if the universe extends forward in time,
infinately. Now he can fiddle the statistics as much as he likes. Of
course, he could have got this result in other ways, but equally he
should be able to do it in any way he sees as fit, because he is
omnipotent.
In the previous example God could have, if he wanted, made it so that
the miner's hat *always* fell down the mine. If he wants to, God can
make any rule apply for as long as he wants. But can God make a rule
which applies all the time? Can he make the hat fall every time in
future, whether or not he changes his mind? (he's allowed to change his
mind, he's omnipotent). Can he make it so that, in future, he doesn't
have a choice - i.e. can he turn off his omnipotency? Of course he
can, he can do anything. Mind you, he can turn his omnipotency back on
again whenever he wants to (there IS a distinction between being
omnipotent and being able to become omnipotent, because while he is in
his "nomnipotent" state, he is restricted - if you talk about any
action he does, he is either omnipotent or nomnipotent when he does it.
We know God can switch his omnipotency on and off, but can he
"abdicate" and become permanently, irrevocably omnipotent, like a
parliament which votes itself out of existance? A truly omnipotent
being should be able to abdicate. Once he had abdicated, he would no
longer be omnipotent and it would not matter that there were things he
could not do. OK, but can he permanently abdicate, while leaving a
backdoor into omnipotency? In other words, can he say "I will no
longer be omnipotent, and it will be an irrevocable decision to become
restricted, but it will be an irrevocable decision which I can overturn
whenever I want". This is basicly the same as a temporary retirement -
If he leaves the backdoor open, then he is nomnipotent as before,
restricted, but able to rid himself of any restriction. If he doesn't
leave himself a backdoor, he will no longer be omnipotent and we
haven't got a contradiction.
We've shown that in a universe where God is omnipotent, all other laws
are only "advisory", subservient to God's omnipotence. And we've seen
how God can overcome the problem of him restricting himself, by
becoming "nomnipotent", which is different from omnipotency because
before he can exercise his powers he has to turn them back on.
An "inconsistent" God seems to be remarkably consistent.
--
David Chan (DPC...@kentroad.demon.co.uk)
------------------------------------------------------
There is a subliminal message hidden in this sentence.
David Ullrich
>>You need to find a book on logic and study up on the distinction between
>>language and metalanguage.
to which Chow replied
>Dear me...we're getting personal here, aren't we? I admit I'm no logician,
>but I've studied Tarski's original paper on the undefinability of truth, one
>undergraduate textbook on mathematical logic (Ebbinghaus, Flum, and Thomas),
>and numerous papers in the philosophy of language. Since I still haven't
>grasped the distinction between language and metalanguage after all that,
>I must be a hopeless case, and reading further books is unlikely to help me.
I was going to quit the other day, but I'd like to say that no, I wasn't
meaning to get personal. I didn't know what else to say. (Note there's been
funny problems with the news server the last few days, if you're curious about
the timing of this...)
It's happened repeatedly that I say P and you reply explaining why Q is
wrong, when Q is simply not what I said. The other day I said something about the
use/mention distinction. Because I had no idea what the guy was talking about the
first time the professor in my philosophy class (years ago) started talking about
"use versus mention", I included a tediously complete example intended to illustrate
what I meant by the distinction. In spite of this your reply sounded as though you
thought the distinction between use and mention was the same as the distinction
between syntax and semantics!
You seem convinced that the "Godel sentence" asserts its own falsehood.
That's not the sort of thing a person would complain about in informal discourse,
but in the present context it indicates a misunderstanding of the difference
between language and metalanguage. The distinction between language and metalanguage
is crucially important if we're going to get this sort of issue straight, so I tried
to explain it, evidently to no effect.
I'm not making any of this up, by the way. I'm not familiar with Tarski's
paper, but the fact that you've read it and I haven't doesn't prove to me that
you're right and I'm wrong - you say it's a paper on "the undefinability of truth".
Of course truth cannot be defined in any absolute sense, I don't see what that has
to do with the current debate. In logic we take "truth" as an undefined primitive -
ie in a formal language we have T and F, which do not MEAN "true" and "false",
they're simply T and F (which is a good thing, because then we're free to interpret
them in some other manner if we choose, like open and closed). In logic we speak of
T and F, sometimes we call them "true" and "false", but if we're careful we keep
in mind that they are in fact undefined primitives. When we're doing metalogic we
actually talk about assertions being true and false (in spite of the fact that alas
we can't quite define the words precisely). But it's assertions that are true or
false, not formulas in the language.
I'm not familiar with the undergraduate text you mention either, but I know
of plenty of undergraduate texts in logic that don't mention the language versus
metalanguage (or logic versus metalogic) question. I know a text or two that
actually deals with the logic and ignores the metalogic (how boring - actually
constructing these formal "proofs" that we talk about, for no reason that I can
see), and I know of a text or two that says a lot about the metalogic without
mentioning the word "metalogic" (because the author feels the distinction is
obvious?). Find a proof of the Incompleteness Theorem somewhere and read the
surrounding pages carefully - if you don't find a disclaimer along the lines of
"of course the sentence P does not actually assert its falsehood..." look in
another book.
Or ask someone with actual professional credibility about all of this -
again, if you say anything about anything I said please include a verbatim
quote, it _has_ in fact happened repeatedly that I say something and you appear
to think I said something totally different.
Timothy Chow
David Ullrich has advised me to ask a logician to educate me; I hope one
of the net logicians can help out. I post this because I think the issues
here are of general interest to sci.math. At least one of Ullrich and me
is suffering from common misconceptions about logic.
Caveat: I will try to represent Ullrich's position as well as I can, but
Ullrich claims that I have repeatedly misrepresented him, so you may wish
to consult his original articles (in the "lifting an infinitely heavy rock
question" thread).
David Ullrich objects to the sentence "This sentence has five words"
and claims that "Sentences should not talk about themselves." Later he
clarifies that by "sentences" he means real sentences in real languages,
and not sentences in formal languages, which he claims do not talk about
anything at all, being purely syntactic entities. Ullrich tells me that
I do not understand the distinction between language and meta-language,
from which I conjecture that he has a hierarchical model of natural
languages, in which assertions about ordinary objects count as "sentences,"
assertions about sentences count as "meta-sentences," assertions about
meta-sentences count as "meta-meta-sentences," and so on. It is illegal,
therefore, to have any kind of self-reference, because this amounts to
confusing levels in the hierarchy.
I responded that outlawing *all* forms of self-reference is an extreme
view, at least if by this one means that the semantic content of a sentence
must never refer to its syntactic structure. For this, I claim, is the
essential trick behind a Goedel sentence---i.e., the thing that we get
if we enumerate all sentences with a free variable in them and set the
variable in P(x) equal to its own index number, where P(x) = "What you get
when you set the variable in the xth sentence with a free variable equal
to x is not provable." The standard interpretation of this sentence (or
rather its formalization in, say, PA) refers to its syntactic structure.
Ullrich rejects this argument, arguing (I think) that sentences in formal
languages are not analogous to sentences in natural languages, because the
former are purely syntactic and do not mean anything (let alone refer to
anything) whereas sentences in natural languages do. However, I claim that
the analogy is actually a strong one. The string "Apples are red" can be
regarded as a purely syntactic entity; it has a standard interpretation
that English users assign to it, but there is nothing intrinsic in the
sequence of symbols that forces us to assign it this interpretation. Some
other natural language might include the same string in its list of
meaningful sentences and it might mean something totally different in that
language. Similarly, the Goedel sentence is a syntactic entity, but it
has a standard interpretation (viz., its interpretation in the standard
model), even though many alternative interpretations exist.
Given this, "This sentence has five words" is no more problematic than the
Goedel sentence. In both cases, the standard interpretation of the string
talks about the syntactic structure of the string.
There are situations where we are forced to ascend to a meta-language, e.g.,
in the case of the truth predicate (Tarski's theorem). However, this
depends on *specific* properties of the truth predicate; we are not forced
to ascend to a meta-language in the same way in the case of the Goedel
sentence, nor (if I am right about the analogy between formal and natural
languages above) in the case of the predicate "has five words." We do not
need to reject "This sentence has five words" as "confusing language and
meta-language."
So, am I confused? In particular, is the following excerpt from a previous
article of mine (referring to Ullrich's claim that the Goedel sentence does
not talk about anything) "ridiculous," as Ullrich asserts?
>The syntactic string, taken in isolation,
>of course does not "talk about" anything. When you say that the Goedelian
>sentence does not talk about anything, presumably you are referring to this
>fact. But this is practically tautologous, e.g., in this sense, "Apples
>are red" does not talk about anything either, because a purely syntactic
>entity never talks about anything.
>
>For your assertion that sentences should not talk about themselves to have
>content, it must mean that the *semantic* content of a sentence should never
>talk about the sentence itself. But the semantic content of the Goedelian
>sentence *does* refer to itself, or at least to its syntactic representation.
>If you outlaw this (as you seem to do in your subsequent discussion), then
>you outlaw Goedel's proof. If you don't outlaw this, then you are allowing
>the semantic content of a sentence (such as "This sentence has five words")
>to include references to its syntactic representation.
--
david petry
Wiener) writes:
>
>I'd say Ullrich is off his rocker. ...
>In particular, logical self-reference has a long history throughout this
>century.
CAUTION: Crackpot gibberish follows!!
Everybody who's anybody in the field of philosophical logic knows that
the following sentence is an example of a self-referential paradox:
(1) "This sentence is false"
Not so well known are the following self-referential paradoxes:
(2) "That sentence is false"
(3) "The sentence is false"
(4) "It is false"
The reason the philosophers prefer to use (1) as their premier
example of a paradox is because they have the darndest time trying
to prove to the average person that (2), (3) and (4) are also
paradoxical.
If you say to a normal, sane person, "it is false", he'll respond
"what is false?. If you say "the sentence is false", he'll ask "what
sentence is false?" If you say "that sentence is false", he'll ask
"what sentence are you refering to?"
If you say, "this sentence is false", he'll respond "I still don't
see what sentence you're refering to." But then the philosopher
can go to work using his superior intellect. He'll say, "look, you
ignorant moron, there's only one sentence around that I could
possibly be refering to, so obviously that's the one I must be
refering to. All the intelligent people in the world would immediately
agree that the sentence (1) is refering to itself. You don't want
to be an ignorant moron the rest of your life, do you?"
And with that, this normal, sane person says, "Wow, you're brilliant.
Why don't you go lock yourself in your ivory tower and spend the
rest of your life contemplating your navel and self-reference".
Matthew P Wiener
to understand natural language reference and reasoning. Classical logic
is only a small aspect of this, and attempts to shoehorn everything into
it is silly.
In particular, logical self-reference has a long history throughout this
century. The best approach is possibly Barwise's situational logic. See
his books on the subject, including THE LIAR.
--
-Matthew P Wiener (wee...@sagi.wistar.upenn.edu)
David Ullrich
themselves, you say that the Godel sentence does. I reply it does not refer to
anything, you say that its standard interpretation does. Then I attempt to clarify,
explaining that I never said you could not have a sentence S with an interpretation
V such that V refers to S: V is not referring to V, S is not referring to S, there's
no self-reference involved.
English words and standard logical symbols have standard meanings. Talking
about various models of a sentence in some formal language has no effect whatever
on the meaning of the logical connectives in the sentence, it simply changes the
referents of the variables. This is why there is a big difference between saying
the Godel sentence "means" what you say it does and saying that "Apples are red"
means what we think it does: The operation analogous to switching to a different
"interpretation" of a formal sentence is NOT redefining "apple" to mean orange,
the analogous operation would be switching to a different universe where apples
were green.
The "meaning" of the sentence "Apples are red" depends only on the meaning
of the words involved - if we imagine a universe where apples are green the sentence
still means the same thing, it's just become false. To get the sentence to mean
something different you have to change the meanings of the words (and if you're
allowed to do that in an "interpretation" then everything's gibberish). On the
other hand to get the "Godel sentence" to "mean" something different you do not
need to change the meanings of the symbols involved, switching to a different
universe (ie different interpretation) is sufficient.
There's a big difference there - this is what I mean when I assert that
"Apples are red" actually asserts something while the "Godel sentence" does not.
The meaning of "Apples are red" depends only on the meanings of the symbols,
while what you say is the "meaning" of the Godel sentence depends on the
interpretation. Now let's see if I have the netiquette straight: If you don't
see the difference you're a retard. Yeah, that's it, a compelling argument at
last. (Sorry)
Timothy Chow
David Ullrich <ull...@math.okstate.edu> wrote:
>Then I attempt to clarify, explaining that I never said you could not
>have a sentence S with an interpretation V such that V refers to S: V
>is not referring to V, S is not referring to S, there's no self-reference
>involved.
My claim is that S = "This sentence has five words" is self-referential
only in the sense that its interpretation V refers to S.
Now let me address your counterargument below.
>This is why there is a big difference between saying the Godel sentence
>"means" what you say it does and saying that "Apples are red" means
>what we think it does: The operation analogous to switching to a
>different "interpretation" of a formal sentence is NOT redefining
>"apple" to mean orange, the analogous operation would be switching to a
>different universe where apples were green.
>
>The "meaning" of the sentence "Apples are red" depends only on the meaning
>of the words involved - if we imagine a universe where apples are green the
>sentence still means the same thing, it's just become false.
Consider a Euclidean plane and a hyperbolic plane. The parallel postulate
is true in the first case and false in the other. Is this to be explained
by saying that the meaning of the assertion "some pairs of distinct lines
do not meet" is the same in both cases but just happens to be false in the
hyperbolic case? No. The assertion *means* different things in the two
cases; in particular, the meaning of the word "line" is different. (This
is clear if we look at the situation set-theoretically: the "lines" in one
case are not the same as the "lines" in the other case.) Switching to
another model *is* analogous to redefining "apple" to mean "orange."
The *meaning* of the sentence "some pairs of distinct lines do not meet"
does not depend on whether some pairs of lines in the Euclidean plane do
not, in fact, happen to meet. The operation analogous to switching to a
green-apple universe is to switch to another universe in which all pairs
of lines in the Euclidean plane do actually meet. In such a universe,
the sentence still means the same thing; it just becomes false. (Here we
see one place where there indeed *is* a disanalogy: in mathematics there
are only necessary truths and necessary falsehoods, and there is nothing
really analogous to contingent facts like the color of apples, so we can't
imagine very well a counterfactual universe where parallel lines do meet.
But that does not affect the point under discussion.)
>To get the sentence to mean
>something different you have to change the meanings of the words (and if you're
>allowed to do that in an "interpretation" then everything's gibberish). On the
>other hand to get the "Godel sentence" to "mean" something different you do not
>need to change the meanings of the symbols involved, switching to a different
>universe (ie different interpretation) is sufficient.
It is true that switching to a different interpretation is sufficient, but
this is because switching to a different interpretation amounts to changing
the meanings of some of the symbols involved. Certainly, you can't just
change the meanings of just any of the symbols without inviting chaos, but
*some* of the meanings of the symbols must change, or else you haven't
changed models at all.
>The meaning of "Apples are red" depends only on the meanings of the symbols,
>while what you say is the "meaning" of the Godel sentence depends on the
>interpretation.
Yes, I agree, because "interpretation" = "meanings of the symbols," *not*
"what the state of the universe happens to be."
I think part of the confusion comes from a Skolem's paradox problem. The
mathematical universe is *not* the same as a model of ZFC. But it is easy
to confuse the two, which is why it is easy to think that switching from
one model of ZFC to another amounts to changing the state of the universe.
To keep one's thinking straight, it really helps to at least pretend to be
a Platonist. Models of ZFC are just things hanging out in the universe,
along with other fun things like the Monster group and the nine-point
circle. The universe is just where everything mathematical lives.
>Now let's see if I have the netiquette straight: If you don't
>see the difference you're a retard. Yeah, that's it, a compelling argument at
>last. (Sorry)
You're forgiven.
Timothy Chow
>Consider a Euclidean plane and a hyperbolic plane. The parallel postulate
>is true in the first case and false in the other. Is this to be explained
>by saying that the meaning of the assertion "some pairs of distinct lines
>do not meet" is the same in both cases but just happens to be false in the
>hyperbolic case? No. The assertion *means* different things in the two
>cases; in particular, the meaning of the word "line" is different.
I realized afterwards that this way of arguing is not the most helpful,
because you might respond, "Yes, of course they mean different things;
but when you do the analogous thing with natural language by changing
the universe, meanings *don't* change, and this is why formal and natural
languages are not analogous." I sort of address this idea later in the
article, but here it is explicitly.
The analogous thing to changing the model is *not* "changing the universe"
(in the sense of considering a universe where apples are blue). Changing
the model is analogous to changing the meanings of English words. For
when you change a model, the formal word "line" suddenly means something
different. Nothing has happened to "reality"; the Euclidean lines are
still Euclidean lines and the parallel ones still don't intersect. It's
just that when we switch to a hyperbolic model, the word "line" refers to
a different mathematical object than it did before.
What changing to a blue-appled universe would be analogous to would be
changing the nature of mathematical reality so that Euclidean parallel
lines *did* intersect. The meaning of the formal sentences would remain
the same, but they would now be false.
Hope this clarifies things.
Matthew P Wiener
>wee...@sagi.wistar.upenn.edu (Matthew P Wiener) wrote:
>>I'd say Ullrich is off his rocker. [...]
>Ullrich had a great deal of difficulty formulating a suitable reply.
Well, get back in your chair first and rock yourself a spell.
David Ullrich
taken it as sarcasm and posted an obnoxious reply.
For the record, I don;t recall expressing any confusion over
which sentence is being referred to in "This sentence is false".
pe...@ix.netcom.com (david petry) wrote:
>In <45h208$o...@netnews.upenn.edu> wee...@sagi.wistar.upenn.edu (Matthew P
>Wiener) writes:
>>
>>I'd say Ullrich is off his rocker. ...
>
>>In particular, logical self-reference has a long history throughout this
>>century.
>
>
David Ullrich
>In article <45jjrs$a...@news.cis.okstate.edu>,
>David Ullrich <ull...@math.okstate.edu> wrote:
>>Then I attempt to clarify, explaining that I never said you could not
>>have a sentence S with an interpretation V such that V refers to S: V
>>is not referring to V, S is not referring to S, there's no self-reference
>>involved.
>
>My claim is that S = "This sentence has five words" is self-referential
>only in the sense that its interpretation V refers to S.
What??? You don't find any direct self-reference involved in the
words "This sentence"???
>
>Now let me address your counterargument below.
>
>>This is why there is a big difference between saying the Godel sentence
>>"means" what you say it does and saying that "Apples are red" means
>>what we think it does: The operation analogous to switching to a
>>different "interpretation" of a formal sentence is NOT redefining
>>"apple" to mean orange, the analogous operation would be switching to a
>>different universe where apples were green.
>>
>>The "meaning" of the sentence "Apples are red" depends only on the meaning
>>of the words involved - if we imagine a universe where apples are green the
>>sentence still means the same thing, it's just become false.
>
Here you claim you're about to address the argument. Then you omit
the argument and quote the conclusion.
>Consider a Euclidean plane and a hyperbolic plane. The parallel postulate
>is true in the first case and false in the other. Is this to be explained
>by saying that the meaning of the assertion "some pairs of distinct lines
>do not meet" is the same in both cases but just happens to be false in the
>hyperbolic case? No.
No, "Yes" is the right answer.
This is nonsense. Consider the sentence
Ex(Ey(x<>y))
, where "E" is an existential quantifier and "<>" is "not equals". You can
interpret this sentence in various models (more accurately, in various
structures, but let's use the popular terminology instead of the
precise terminology to avoid strings of definitions). In some models it's
true, in some models it's false, but the "E" still means "there exists"
in each model. And the meaning of the sentence is still "there exist at
least two elements in the unverse", regardless of which model you're
talking about.
>I think part of the confusion comes from a Skolem's paradox problem. The
>mathematical universe is *not* the same as a model of ZFC. But it is easy
>to confuse the two, which is why it is easy to think that switching from
>one model of ZFC to another amounts to changing the state of the universe.
Uh, by "universe" I didn't mean the real universe, or the mathematical
universe (if there is such a thing), I was referring to the underlying set
in whatever model we're talking about: A model (rather, a structure) is a
set U together with some relations, etc - by "universe" I simply meant U.
Perhaps "universe of discourse" would have been better.
>To keep one's thinking straight, it really helps to at least pretend to be
>a Platonist.
If you say so, I would disagree. Pretending to be a Platonist probably
makes it easier to prove the mean value theorem, but when we're discussing
foundational/logical issues it strikes me as a very bad idea.
>Models of ZFC are just things hanging out in the universe,
>along with other fun things like the Monster group and the nine-point
>circle. The universe is just where everything mathematical lives.
>
>>Now let's see if I have the netiquette straight: If you don't
>>see the difference you're a retard. Yeah, that's it, a compelling argument at
>>last. (Sorry)
>
>You're forgiven.
Fine. Here's a curious thing: throughout all of this I've taken
great pains to try to say exactly what I meant, and it seems to me that
you've misunderstood most of it (yes, I'm aware that you would say the same
about my understanding of your comments). But just once, in the phrase
"you're a retard", what I meant to be saying was not at all what a person
might guess just by looking at the words out of context. The curious thing
is that it seems like this is the one time you've understood exactly what
I was getting at. Go figure.
Stephen Harris
>From: David Ullrich <ull...@math.okstate.edu>
>about my understanding of your comments). But just once, in the phrase
>"you're a retard", what I meant to be saying was not at all what a person
>might guess just by looking at the words out of context. The curious thing
>is that it seems like this is the one time you've understood exactly what
>I was getting at. Go figure.
Logically, this appears to me as an example of birds of a feather flock
together. Which of course is why I am posting a reply in this discussion.
The point I would like to stress is that not all cretins are liars though
they may be retarded. Of course, I am 'figuring' from a geometric stance;
in which case the limit may approach unity as equivalency but not identity.
I hope this helps to clarify the somewhat muddy waters of complexity.
Helpfully yours,
Stephen
Robert Hill
[... much deleted ...]
> The Goedel sentence on the other hand can be viewed on 3 levels:
>
> Level 0: syntactic. The sentence is string of symbols.
>
> Level 1: semantic. The sentence has a meaning. It is a statement about
> arithmetic. Its truth or falsity is a question of
> arithmetic. At this level the sentence does not refer
> to its own syntactic structure.
>
> Level 2: meta-semantic (for want of a better word):
> We recognise that under the Goedel coding, N represents
> the Goedel sentence itself, and P represents "is not
> provable". Only at this level does the Goedel sentence
> refer to its own syntactic structure.
Sorry, I accidentally edited out the bit under level 0 or 1 where I introduced
the notation P(N) which I went on to mention in level 2. In the deleted text,
I observed that the Goedel sentence was of the form P(N) where P is an
extremely complicated predicate and N is the numeral for an extremely large
integer.
--
Robert Hill
University Computing Service, Leeds University, England
"Though all my wares be trash, my heart is true."
- John Dowland, Fine Knacks for Ladies (1600)
Alberto C Moreira
>My claim is that S = "This sentence has five words" is self-referential
>only in the sense that its interpretation V refers to S.
"This sentence has five words" is actually a left-recursive formula:
(((( ... ) has five words) has five words) has five words)
Hardly an effective formula. Should it be allowed in a theory ?
_alberto_
Timothy Chow
David Ullrich <ull...@math.okstate.edu> wrote:
>tc...@math.lsa.umich.edu (Timothy Chow) wrote:
>>My claim is that S = "This sentence has five words" is self-referential
>>only in the sense that its interpretation V refers to S.
>
> What??? You don't find any direct self-reference involved in the
>words "This sentence"???
I'm still not sure what you mean by "direct" self-reference as opposed to
any other kind of self-reference. "This sentence" refers to something at
all only insofar as the words "This sentence" mean anything. Therefore,
it is the *interpretation* of the sentence S (the interpretation that all
competent speakers of English would give it) that refers to S. If you want
to call this "direct," fine. But I would then argue that if you want to
call this kind of self-reference then the Goedel sentence is "directly"
self-referential, too. Personally, I try to avoid using the term "direct"
self-reference because the difference between "direct" and "indirect" seems
to me to be more psychological than substantive.
> Here you claim you're about to address the argument. Then you omit
>the argument and quote the conclusion.
Yes, sorry...I tried to fix this in a later article.
>>Consider a Euclidean plane and a hyperbolic plane. The parallel postulate
>>is true in the first case and false in the other. Is this to be explained
>>by saying that the meaning of the assertion "some pairs of distinct lines
>>do not meet" is the same in both cases but just happens to be false in the
>>hyperbolic case? No.
>
>No, "Yes" is the right answer.
Here we really have the crux of the matter. When I say that "some pairs of
distinct lines do not meet," the truth of the statement depends on whether
"lines" *means* "lines in the Euclidean plane" or "lines in the hyperbolic
plane." It is *not* the case that the word "lines" refers to the same thing
in both cases. How can you say that the sentences mean exactly the same
thing when they are talking about completely different objects?
> This is nonsense. Consider the sentence
>
>Ex(Ey(x<>y))
>
>, where "E" is an existential quantifier and "<>" is "not equals". You can
>interpret this sentence in various models (more accurately, in various
>structures, but let's use the popular terminology instead of the
>precise terminology to avoid strings of definitions). In some models it's
>true, in some models it's false, but the "E" still means "there exists"
>in each model.
Sure, not *everything* changes its meaning when you switch to a different
model, or else you get chaos, as I said before.
>And the meaning of the sentence is still "there exist at
>least two elements in the unverse", regardless of which model you're
>talking about.
But the meaning of the term "universe" has changed. What is a model or a
structure at all, if not an assignment of meanings to the formal strings?
If you assign exactly the same meanings, you are creating an isomorphic
model.
> Uh, by "universe" I didn't mean the real universe, or the mathematical
>universe (if there is such a thing), I was referring to the underlying set
>in whatever model we're talking about: A model (rather, a structure) is a
>set U together with some relations, etc - by "universe" I simply meant U.
>Perhaps "universe of discourse" would have been better.
In that case, there is no analogy with changing the color of apples.
Changing the color of apples is an alteration of the real universe.
> If you say so, I would disagree. Pretending to be a Platonist probably
>makes it easier to prove the mean value theorem, but when we're discussing
>foundational/logical issues it strikes me as a very bad idea.
O.K., I won't press the point. Still, non-platonists like Hartry Field will
still say that the best way of understanding mathematical statements is *as
if* they were statements about an actually existing world, even though such
a world does not "really" exist.
>is that it seems like this is the one time you've understood exactly what
>I was getting at. Go figure.
Is my track record really this bad? For example, I'm under the impression
that you disagree with some things I've said. Am I mistaken? :-)
Timothy Chow
Alberto C Moreira <alb...@moreira.mv.com> wrote:
>In article <45k5gq$1...@controversy.admin.lsa.umich.edu> tc...@math.lsa.umich.edu (Timothy Chow) writes:
>
>>only in the sense that its interpretation V refers to S.
>
>
> (((( ... ) has five words) has five words) has five words)
>
> Hardly an effective formula. Should it be allowed in a theory ?
Well, I regard "This sentence has five words" as being synonymous with
"`This sentence has five words' has five words." I believe that this is
a more accurate translation than yours, because when competent English
speakers encounter the sentence "This sentence has five words," they do
not run into an infinite regress when they try to check its truth. They
simply take the string "This sentence has five words" and count the number
of words in it.
Timothy Chow
david petry <pe...@ix.netcom.com> wrote:
>As I pointed out in my "crackpot" post a couple days ago, what
>you say is simply not true. Competent English speakers do not see
>self-reference, unless they have been specially trained to see it.
Fine...I have no quarrel with that. I'm arguing against the rejection
of "This sentence has five words" on the grounds that it is self-
referential. Your argument is stronger than mine; you would say that
English is not self-referential, and hence doesn't need to be purged
of it.
>Self-reference is a game, which is not a part of natural language.
If you read, "In Part II of this paper, we will deal with the remaining
issues," would you say, "Which paper?"
david petry
(Timothy Chow) writes:
>
>In article <45opdc$7...@ixnews7.ix.netcom.com>,
>david petry <pe...@ix.netcom.com> wrote:
>>Self-reference is a game, which is not a part of natural language.
>If you read, "In Part II of this paper, we will deal with the remaining
>issues," would you say, "Which paper?"
Consider this context: "We are in the process of writing an extremely
important paper on various social issues. In part one of this paper,
we will deal with gun control. In part II of this paper, we will deal
with the remaining issues."
In that context, there is no self-reference. So the answer to your
question is, "yes, the referent depends on the context, and hence
there is some ambiguity."
Stephen Harris
>Self-reference is a game, which is not a part of natural language.
I thought you made a good post except for the above sentence.
For example double entendre occurs often enough to have its
own name. More basically look at the structure of humor/irony/sarcasm
which is expressed verbally. There is a school called Post-Modernism.
Your statement would be truer if you replaced natural language with logic.
Cordially
Stephen
david petry
(Timothy Chow) writes:
> ... when competent English
>speakers encounter the sentence "This sentence has five words," they do
>not run into an infinite regress when they try to check its truth. They
>simply take the string "This sentence has five words" and count the number
>of words in it.
As I pointed out in my "crackpot" post a couple days ago, what
you say is simply not true. Competent English speakers do not see
self-reference, unless they have been specially trained to see it.
In every English sentence which is presumably self-referential, there
is always ambiguity about what is being referred to. If I were to
write "one plus one is two" on the blackboard, and point to it as
I said "this sentence has five words", every competent English
speaker would know that 'this sentence' refers to the sentence I wrote
on the blackboard.
The example I used previously to see the ambiguity is to compare the
sentence with 'that' replacing 'this'. Hardly anyone would say that
"that sentence has five words" is self-referential, but what is the
big difference between 'this' and 'that'? Each word is just a hint
about where to look for the referent, and 'this' suggests something
that is closer to the speaker than 'that.'
Alberto C Moreira
>"This sentence has five words" can be viewed on 2 levels:
If this is so, we have a real problem. This reminds me of Scheme:
there's a big difference between (f (g a b)) and (f '(g a b)). If you
can express both meanings by (f (g a b)) then your syntax is
ambiguous.
>Level 0: syntactic. The sentence is a string of symbols.
>Level 1: semantic. The sentence has a meaning, refers to its own syntactic
> structure, and is true. It is impossible to consider
> its truth or falsity without taking account of the fact
> that the word "This" in the sentence refers to the sentence
> itself.
What you call "syntatic" actually involves semantics different from the
semantic you call "level 1". There is no such thing as pure syntax;
there's always a semantic structure involved, even if it is implied.
There's a profound semantic difference between the following formulas:
This sentence has five letters has five letters
"This sentence has five letters" has five letters.
If the syntax can't make the difference, it should.
>The Goedel sentence on the other hand can be viewed on 3 levels:
>Level 0: syntactic. The sentence is string of symbols.
>Level 1: semantic. The sentence has a meaning. It is a statement about
> arithmetic. Its truth or falsity is a question of
> arithmetic. At this level the sentence does not refer
> to its own syntactic structure.
>Level 2: meta-semantic (for want of a better word):
> We recognise that under the Goedel coding, N represents
> the Goedel sentence itself, and P represents "is not
> provable". Only at this level does the Goedel sentence
> refer to its own syntactic structure.
Then the sentence "This sentence has five letters" has at least
three semantics and is therefore ambiguous and should be
outlawed. I still believe that, taken with "level 1" or
"level 2" semantics, the phrase embeds an infinite recursion and
isn't therefore effective; while taken with "level 0" semantics
it is a trivial statement about syntatical symbols.
Now, has anybody ever found any non-self-referencing formula
that supports Goedel's theorem ? Or does it basically say that
a consistent theory can't prove a certain class of
self-referencing statements ?
_alberto_
Toby Bartels
>tc...@math.lsa.umich.edu (Timothy Chow) writes:
>>My claim is that S = "This sentence has five words" is self-referential
>>only in the sense that its interpretation V refers to S.
> "This sentence has five words" is actually a left-recursive formula:
> (((( ... ) has five words) has five words) has five words)
> Hardly an effective formula. Should it be allowed in a theory ?
Whether or not it should be allowed in any particular theory,
it *is* in the English language, like it or not.
Also, your analysis of it is incorrect.
The formula is <(string) has five words>,
where (string) is, in fact, a five word string.
There is only one self reference.
-- Toby
to...@ugcs.caltech.edu
Stephen Harris
>The idea of "self-reference" is not essential to Godel's proof.
You need more education. That sentence has five words.
Do not take you personally. The diagonal proof is selfreferenced.
Stephen
Alberto C Moreira
>Whether or not it should be allowed in any particular theory,
>it *is* in the English language, like it or not.
We don't use every English language construct in mathematical
theories.
>Also, your analysis of it is incorrect.
>The formula is <(string) has five words>,
>where (string) is, in fact, a five word string.
>There is only one self reference.
Depends on the assumed semantics: whether or not you "quote"
the subject formula.
But if you say "This sentence cannot be proved", I propose
there's an infinite recursion in the phrase; we're not talking
about the phrase's syntax but about its semantics.
_alberto_
Timothy Chow
Bill Taylor <w...@math.canterbury.ac.nz> wrote:
>tc...@math.lsa.umich.edu (Timothy Chow) writes:
>|>
>|> Consider a Euclidean plane and a hyperbolic plane. The parallel postulate
>|> is true in the first case and false in the other. Is this to be explained
>|> by saying that the meaning of the assertion "some pairs of distinct lines
>|> do not meet" is the same in both cases but just happens to be false in the
>|> hyperbolic case? No. The assertion *means* different things in the two
>|> cases; in particular, the meaning of the word "line" is different. (This
>|> is clear if we look at the situation set-theoretically: the "lines" in one
>|> case are not the same as the "lines" in the other case.)
>
>this one go! The comments here are damn near pure sophistry, IMHO. We're
>talking about "meaning", here, which means we're talking intensional math, in
>part, not purely extensional.
Yes, I realized this ambiguity in the meaning of the word "meaning"
afterwards. Still, I think the view articulated above is a reasonable one.
>What everyone thinks the *meaning* of a (straight)
>line is perfectly clear. It's a (maximal) one-dimensional set which is locally
>geodesic. (How did Euclid put it - "lies evenly within itself", or some such
>stuttering?) The fact is, this can be interpreted (I guess!) with virtually
>identical expressions in the theory of EG and of HG.
Here's how I was looking at it. Suppose you were asked to judge whether or
not the sentence "every two lines meet in exactly one point" were true or
not. Would you not first demand that I clarify what I *meant* by a line---
a line in a Euclidean space or a line in some other space?
It does, as you astutely observe, boil down to the intensional/extensional
distinction. Consider the following sentences.
1. Every two lines in the real Euclidean plane meet in exactly one point.
2. Every two lines in the real projective plane meet in exactly one point.
3. In whatever object that we are calling a "plane" at the moment, every
two subsets that we are currently calling "lines" meet in exactly one
point.
So if we agree ahead of time that we're going to take "plane" and "line"
in the Euclidean sense, does the sentence "every two lines meet in exactly
one point" mean 1 or does it mean 3? More importantly, if we now switch
to the real projective plane, is it the case that the sentence changes
meaning from 1 to 2, or is it the case that it meant 3 before and means
3 now?
For the situation originally under discussion, namely first-order languages,
the process of picking the domain of your model is usually regarded as part
of the "semantics." This, I think, makes sense, because variables in your
first-order language don't really mean anything (even "intensionally")
until you assign a meaning to them, and the way you assign a meaning is
to specify a domain. In a natural language, the situation becomes less
clear, but the fact that in the example I give above one instinctively
asks for clarification of the *meaning* of the term "line" lends support
to the view that when we fix a model we are taking "every two lines meet
in exactly one point" to mean 1 or 2, as appropriate, rather than 3.
Bill Taylor
|>
|> Consider a Euclidean plane and a hyperbolic plane. The parallel postulate
|> is true in the first case and false in the other. Is this to be explained
|> by saying that the meaning of the assertion "some pairs of distinct lines
|> do not meet" is the same in both cases but just happens to be false in the
|> hyperbolic case? No. The assertion *means* different things in the two
|> cases; in particular, the meaning of the word "line" is different. (This
|> is clear if we look at the situation set-theoretically: the "lines" in one
|> case are not the same as the "lines" in the other case.)
Tim, it's not often I have the temerity to disagree with you, but I can't let
this one go! The comments here are damn near pure sophistry, IMHO. We're
talking about "meaning", here, which means we're talking intensional math, in
part, not purely extensional. So your comments about the models having different
"lines" is next to hogwash. What everyone thinks the *meaning* of a (straight)
line is perfectly clear. It's a (maximal) one-dimensional set which is locally
geodesic. (How did Euclid put it - "lies evenly within itself", or some such
stuttering?) The fact is, this can be interpreted (I guess!) with virtually
identical expressions in the theory of EG and of HG.
So it's logically (as good as) identical, semantically identical, and if set
theoretically different - then so much the worse for set theory!
|> The *meaning* of the sentence "some pairs of distinct lines do not meet"
|> does not depend on whether some pairs of lines in the Euclidean plane do
|> not, in fact, happen to meet.
Well now you've got me well and truly baffled! This seems to be what I just
said, more or less. Aren't you contrsdicting yourself?!
|> The mathematical universe is *not* the same as a model of ZFC. But it is
|> easy to confuse the two,
Absolutely right! (As I said: "if ... then so much the worse for set theory".)
But in the first excerpted paragraph you seem to have ignored your own dictum!
|> To keep one's thinking straight, it really helps to at least pretend to be
|> a Platonist.
Yes; though I agree with whoever it was said "it helps to be one if you want
to prove the MVT" (or whatever) "but is not so good for logic/foundations".
As I've said before, I tend to be a realist about discrete math, uncertain
about analysis, and almost a formalist about (hyper-countable) set theory.
If you feel I've misinterpreted you, or if it's worth anothrer response, please
do so. It seems to me you've definitely contradicted yourself quite badly!
-------------------------------------------------------------------------------
Bill Taylor w...@math.canterbury.ac.nz
-------------------------------------------------------------------------------
The chief difference between mathematics and physics is that
in mathematics we have much more direct contact with reality.
-------------------------------------------------------------------------------
Alberto C Moreira
[lots of good words deleted...]
>The idea of "self-reference" is not essential to Godel's proof.
In which case it should be easy to find a non-self-referencing statement
in every theory which is true but non-provable. But if there are none,
the point remains: Goedel's theorem states only that a consistent theory
will fail to prove a certain class of self-referencing statements; which
statements are sort of debatable to begin with!
_alberto_
Toby Bartels
>to...@ugcs.caltech.edu (Toby Bartels) writes:
>>How would you react to
>>`The very sentence you are reading or hearing right now,
>>the one referred to by the very words you're now interpreting,
>>which refers in the most direct possible manner to itself,
>>has 34 words.'? Is that clear enough?
>I would say to myself, "Gee, I'm going to have to think about
>that one. Let's see .. 'the very sentence you are reading or
>hearing' .. hmm, I'm not reading or hearing a sentence right
>at the moment; I'm thinking. It's not obvious what that
>refers to ..".
So we see that, clear as I tried to make it, it is still ambiguous.
Will you agree that it is *less* ambiguous,
that it stands a better chance of being interpreted as I desire,
than `This sentence has five words.'?
>I would still classify it as a game.
Of course it's a game! So what?
Now that we've established once more that we are playing a game,
I will perhaps be forgiven for indulging in the exercise,
which I know beforehand ultimately to be futile,
to find an unambiguous, self referential, English sentence.
`By `this sentence', I mean the sentence in which the words
`this sentence' appear; this sentence has eighteen words.'
is my next candidate.
If the first clause is interpreted as I intend it,
the ambiguity you've found in the second clause is gone.
However, there is ambiguity in the first clause,
and other ambiguities remain in the second clause.
But I think it is less ambiguous than `This sentence has five words.',
don't you?
-- Toby
to...@ugcs.caltech.edu
just kidding!
Toby Bartels
>I guess my point is that self-reference is always associated with
>ambiguities in language, and any attempt to remove the ambiguities
>will destroy the self-reference.
A language such as English is too complex and ill defined
for us every to remove all ambiguity.
But let us restrict the subject of English enough that we can now do so.
The string `this sentence' is to be interpreted as a noun phrase
refering to the sentence in which the string appears.
The string `has five words' is to be interpreted as a predicate
which assigns truth to precisely those strings which consist of five words
and which assigns falsehood to anything which is not a string.
Now, if the grammatical structure is made strict enough
that no string can appear in more than or less than one sentence,
and if punctuation is also pedantically defined,
the above *should* be pretty unambiguous.
It's not, because I explained it in English,
but the fault lies with the explanation.
There is no ambiguity in the pseudoEnglish I've created,
in which `This sentence has five words.' is self referential (and true).
I don't mean to imply that any English sentence is unambiguous;
on the contrary, they all are, including the self referential ones.
But your statement that ambiguity is necessary for self reference is false.
-- Toby
to...@ugcs.caltech.edu
Matthew P Wiener
>As I pointed out in my "crackpot" post a couple days ago, what
>you say is simply not true. Competent English speakers do not see
>self-reference, unless they have been specially trained to see it.
That is nonsense. Children can be amused with self-referential word
games quite easily.
>In every English sentence which is presumably self-referential, there
>is always ambiguity about what is being referred to.
So what? There is always context. This, in fact, is the heart of
the Barwise-Etchemendy analysis of the liar's paradox.
> If I were to
>write "one plus one is two" on the blackboard, and point to it as
>I said "this sentence has five words", every competent English
>speaker would know that 'this sentence' refers to the sentence I wrote
>on the blackboard.
Correct.
>The example I used previously to see the ambiguity is to compare the
>sentence with 'that' replacing 'this'. Hardly anyone would say that
>"that sentence has five words" is self-referential, but what is the
>big difference between 'this' and 'that'? Each word is just a hint
>about where to look for the referent, and 'this' suggests something
>that is closer to the speaker than 'that.'
Indeed. You answered your question. And when there is no specified
context, "this" boils down to the only sentence in sight.
>Self-reference is a game, which is not a part of natural language.
A self-made phoney stance.
Matthew P Wiener
> Now, has anybody ever found any non-self-referencing formula
> that supports Goedel's theorem ? Or does it basically say that
> self-referencing statements ?
There exist model theoretic proof of Peano incompleteness.
david petry
Bartels) writes:
>>I would still classify it [a self-referential sentence] as a game.
>Of course it's a game! So what?
Here's the point I'm making.
The rules we use to determine referents contain ambiguities and
uncertainties. Self-reference hides within those ambiguities and
uncertainties.
If we very carefully analyse self-reference, in the sense that
we remove all ambiguity, the self-reference disappears.
You seem to know that, so we're not really disagreeing with each
other.
Matthew P Wiener
>In <mulcybr.18...@azstarnet.com> mul...@azstarnet.com (Stephen Harris)
>writes:
>>
>>>Self-reference is a game, which is not a part of natural language.
>>
>> For example double entendre occurs often enough to have its
>> own name. More basically look at the structure of humor/irony/sarcasm
>Well, I guess you're right, but what you say is not incompatible with
>what I said. Double entendre, humor, irony, and sarcasm are all games.
And also, double entendre, humor, irony, and sarcasm all also all part
of natural language. You'll have to come up with a better comment to
defend.
>I guess my point is that self-reference is always associated with
>ambiguities in language, and any attempt to remove the ambiguities
>will destroy the self-reference.
This is clearly false. You have given examples where the removal of
ambiguities by the creation of unintended references destroyed the
self-reference. So what?
Consider Quine's context-free sentence:
"Appended to its own quotation yields a falsehood" appended to its own
quotation yields a falsehood.
Matthew P Wiener
>In <45trt4$9...@gap.cco.caltech.edu> to...@avarice.ugcs.caltech.edu (Toby
>Bartels) writes:
>>How would you react to
>>`The very sentence you are reading or hearing right now,
>>the one referred to by the very words you're now interpreting,
>>which refers in the most direct possible manner to itself,
>>has 34 words.'? Is that clear enough?
>I would say to myself, "Gee, I'm going to have to think about
>that one. Let's see .. 'the very sentence you are reading or
>hearing' .. hmm, I'm not reading or hearing a sentence right
>at the moment; I'm thinking. It's not obvious what that
>refers to ..".
>I would still classify it as a game.
The only game is your dodging. You seem to have trained yourself
to *refuse* to see self-reference.
david petry
writes:
>pe...@ix.netcom.com (david petry) writes:
>
>>I guess my point is that self-reference is always associated with
>>ambiguities in language, and any attempt to remove the ambiguities
>>will destroy the self-reference.
>But your statement that ambiguity is necessary for self reference is false.
Obviously you believe the following sentence is unambiguously
self-referential.
(1) This sentence has five words.
Is the following sentence unambiguously self-referential?
(2) It has four words.
If you explain carefully why or why not, I might continue arguing
with you. If you don't, I will not argue with you.
david petry
P Wiener) writes:
>
>In article <45rahj$e...@ixnews6.ix.netcom.com>, petry@ix (david petry)
writes:
>>I guess my point is that self-reference is always associated with
>>ambiguities in language, and any attempt to remove the ambiguities
>>will destroy the self-reference.
>This is clearly false. You have given examples where the removal of
>ambiguities by the creation of unintended references destroyed the
>self-reference. So what?
The point is that the process of determining references is a process
of guessing what the creator of the sentence intended. The intended
reference cannot be unambiguously specified within the sentence
itself.
>Consider Quine's context-free sentence:
>"Appended to its own quotation yields a falsehood" appended to its own
>quotation yields a falsehood.
First of all, the average competent English speaker would have a great
deal of difficulty making sense out of that sentence. For one thing,
the idea of a quoted sentence clause's quotation is not a standard
concept.
Secondly, if you think I could not concoct a context in which that
sentence has another interpretation, you are wrong. It would be a
convoluted context, simply because the sentence is so convoluted.
But apparently you wouldn't consider that to be proof of anything, so
I won't bother doing it.
david petry
P Wiener) writes:
>
>In article <45opdc$7...@ixnews7.ix.netcom.com>, petry@ix (david petry)
writes:
>>In every English sentence which is presumably self-referential, there
>>is always ambiguity about what is being referred to.
>
>So what? There is always context.
Not true. When we take a sentence out of context and discuss it,
there is no longer any context.
So when discussing the sentence "this sentence is nice", the most
reasonable assertion is that when taken out of context and discussed
as we are doing, "this sentence" must be regarded as a free
variable. It just like the 'x' in the sentence "x < 3". When
taken out of context, it's not meaningful to ask whether "x<3" is
a true sentence or not, and likewise, when taken out of context, it's
not meaningful to ask whether "this sentence is nice" is true or not.
Toby Bartels
>The rules we use to determine referents contain ambiguities and
>uncertainties. Self-reference hides within those ambiguities and
>uncertainties.
>If we very carefully analyse self-reference, in the sense that
>we remove all ambiguity, the self-reference disappears.
>You seem to know that, so we're not really disagreeing with each
>other.
But I don't know that.
I do know this:
While the English language contains apparently self referential sentences,
it contains no unambiguous sentences;
in particular, it contains no unambiguously self referential sentences.
But I also know this:
One can construct a language which contains no ambiguous sentences
and yet which nevertheless contains self referential sentences.
-- Toby
to...@ugcs.caltech.edu
Alberto C Moreira
[some deleted...]
>Are you suggesting something along the lines of
>that an English language construct is meaningless
>unless we use it in mathematical theories?
No, I'm saying that many English language constructs lack
the necessary rigour and exactness to be useful in a
mathematical theory. That's why we use symbols.
[more deleted...]
>The noun phrase `this sentence' refers (in this case)
>to the string `This sentence cannot be proved.'.
>There is still only one necessary recursion.
As stated, the phrase is ambiguous. The semantics
of "this" or "this sentence" are left for you and me to
guess, and we guessed two different things.
Yet I'm saying that if we use "this" in the "quoted"
sense, the phrase is trivial and hardly worth a
second look.
>`This sentence cannot be proved.', as I interpret it,
>means that the string `This sentence cannot be proved.'
>is not a theorem in any formal system (which is false).
>(If I encountered the sentence in a more specific context,
>such a discussion of a particular formal system,
>I would interpret it more strictly.)
You need to define your semantics first. There's no
such thing as pure syntax; your theory will be only
as good as its semantics. If your semantics stops
at symbols and strings, then you're talking only
about symbols and strings and nothing else. But if
you want to go beyond symbols and strings, your
semantics must be defined - and very formally, too.
Switching on the fly between syntax and ill-defined
semantics seems to me as a magician's trick.
I still say there's a world of difference between
" 'this sentence has five words' has five words"
and "this sentence has five words has five words".
_alberto_
>-- Toby
> to...@ugcs.caltech.edu
Matthew P Wiener
>In <463vm8$m...@netnews.upenn.edu> wee...@sagi.wistar.upenn.edu (Matthew
>P Wiener) writes:
>>In article <45rahj$e...@ixnews6.ix.netcom.com>, petry@ix (david petry)
>writes:
>>>I guess my point is that self-reference is always associated with
>>>ambiguities in language, and any attempt to remove the ambiguities
>>>will destroy the self-reference.
>>This is clearly false. You have given examples where the removal of
>>ambiguities by the creation of unintended references destroyed the
>>self-reference. So what?
>The point is that the process of determining references is a process
>of guessing what the creator of the sentence intended.
This is true of all use of natural language.
> The intended
>reference cannot be unambiguously specified within the sentence itself.
Uh, so what? You are now contradicting yourself about the role of
removing ambiguity. As it is, self-reference arises when a reader
parses the sentence, *removes any pronoun or similar ambiguity*, and
comes up with the sentence in question.
Your counterexamples, involved specialized contexts, while cute, were
entirely artificial.
>>Consider Quine's context-free sentence:
>>"Appended to its own quotation yields a falsehood" appended to its own
>>quotation yields a falsehood.
>First of all, the average competent English speaker would have a great
>deal of difficulty making sense out of that sentence.
Irrelevant. You parsed it, right?
> For one thing,
>the idea of a quoted sentence clause's quotation is not a standard
>concept.
Irrelevant. You parsed it, right?
>Secondly, if you think I could not concoct a context in which that
>sentence has another interpretation, you are wrong.
Good luck.
> It would be a
>convoluted context, simply because the sentence is so convoluted.
Good luck.
>But apparently you wouldn't consider that to be proof of anything, so
>I won't bother doing it.
Good luck.
Matthew P Wiener
>In <463vvb$m...@netnews.upenn.edu> wee...@sagi.wistar.upenn.edu (Matthew
>P Wiener) writes:
>>>I would still classify it as a game.
>>The only game is your dodging. You seem to have trained yourself
>>to *refuse* to see self-reference.
>My dodging is only one of the games being played here.
Liar.
>I can certainly "see" self-reference in the sense that I can play the
>game of self-reference if I want to.
OK, so you're not a nutter.
>Everyone who posts in sci.math is a moronic liar.
Me too!
Matthew P Wiener
>In <463uv0$m...@netnews.upenn.edu> wee...@sagi.wistar.upenn.edu (Matthew
>P Wiener) writes:
>>In article <45opdc$7...@ixnews7.ix.netcom.com>, petry@ix (david petry)
>writes:
>>>In every English sentence which is presumably self-referential, there
>>>is always ambiguity about what is being referred to.
>>So what? There is always context.
>Not true. When we take a sentence out of context and discuss it,
>there is no longer any context.
False. The context is just not in front of our noses. Consider a
paper that begins, "This paper will prove ..."
>So when discussing the sentence "this sentence is nice", the most
>reasonable assertion is that when taken out of context and discussed
>as we are doing, "this sentence" must be regarded as a free
>variable. It just like the 'x' in the sentence "x < 3".
Not at all. The "this" kills that interpretation.
> When
>taken out of context, it's not meaningful to ask whether "x<3" is
>a true sentence or not, and likewise, when taken out of context, it's
>not meaningful to ask whether "this sentence is nice" is true or not.
True but irrelevant.
Alberto C Moreira
>In article <alberto.47...@moreira.mv.com>,
> Not really. You don't need self-reference (the fixed-point lemma)
> to prove Goedel's theorem. (Not that there is anything wrong if you do).
If the proof is correct, there should be no difficulty finding one
unprovable statement in every consistent theory. But if every
such unprovable statement contains a self-reference, then I
contend that the "philosophical" content of Goedel's theorem
is greatly overstated.
> The global view is more illuminating. Having coded statements by
> integers, the set Th(T) of statements provable from the Peano axioms T
> is a recursively enumerable (Sigma-0-1) set of integers. [It is of the
> form Ex R(x,y), R recursive; Sigma-0-2 means Ex Ay R(x,y,z), and so on].
> The set A of true statements of arithmetic is not Sigma-0-1; in fact
> it is not Sigma-0-n for any n, i.e. it is not arithmetical. By the way,
> every level of Sigma-0-n contains "new" sets.
Once more you confirm my point. Anything not Turing-computable
must be proved so by diagonalization. And diagonalization does
necessarily involve self-reference. Hence, I could say that Goedel's
argument isn't but a rehash of Cantor's diagonalization argument.
> So Th(T) cannot be the same as A.
> This generalizes to other consistent T's.
I hear you and I don't disagree. Yet the point remains: if we can't
find in every consistent theory an unprovable statement that doesn't
in some way conjure diagonalization or contains the phrase
"this sentence" in it, then I maintain that Goedel's theorem isn't
the bullwark it is touted to be, but just another way of saying that
recursion in logical formulas is a dangerous toy.
_alberto_
david petry
(Alberto C Moreira) writes:
>I contend that the "philosophical" content of Goedel's theorem
>is greatly overstated.
Well, some people think the philosophical content of philosophy is
greatly overstated.
>And diagonalization does necessarily involve self-reference.
Is that a fact?
Presumably you know how to use the diagonalization argument to
show the existence of transcendental numbers, and you probably
are aware that the argument can be used to concretely construct
transcentdental numbers.
Would you show us how self-reference comes in to play in the
construction of transcendental numbers via the diagonalization
argument?
Toby Bartels
>to...@wrath.ugcs.caltech.edu (Toby Bartels) writes:
>>But your statement that ambiguity is necessary for self reference is false.
>Obviously you believe the following sentence is unambiguously
>self-referential.
>(1) This sentence has five words.
In the ad hoc language I created in the post you quoted,
(1) is unambiguously self referential.
In English, (1) is ambiguous.
>Is the following sentence unambiguously self-referential?
>(2) It has four words.
(a) In the ad hoc language I created in the post you quoted, (2) is nonsense.
(b) In English, (2) is ambiguous.
(c) In the ad hoc language created from the ad hoc language
I created in the post you quoted by adding that `it' is
a noun phrase which refers to the sentence in which it appears,
(2) is unambiguously self referential.
>If you explain carefully why or why not, I might continue arguing
>with you. If you don't, I will not argue with you.
(a) `it' is undefined in the ad hoc language I created in the post you quoted.
(b) English is so complicated a language that nothing is unambiguous in it.
(c) In the ad hoc language created from the ad hoc language
I created in the post you quoted by adding that `it' is
a noun phrase which refers to the sentence in which it appears,
`it' refers to the sentence in which it appears.
Since `It has four words.' contains `it',
`It has four words.' refers to the sentence in which appears what it contains,
which is itself.
If the above is not unambiguous enough for you,
let me know and I'll try to make it clearer. :-)
-- Toby
to...@ugcs.caltech.edu
Dr D F Holt
to...@avarice.ugcs.caltech.edu (Toby Bartels) writes:
>How would you react to
>`The very sentence you are reading or hearing right now,
>the one referred to by the very words you're now interpreting,
>which refers in the most direct possible manner to itself,
>has 34 words.'? Is that clear enough?
>I rather suspect most competent English speakers and readers
>will understand it, interpret it roughly as I do, and even consider it true
>(assuming they don't miscount), at the time they read or hear it.
>Of a certainty some will.
>
On the contrary, very few people will do this.
They will have to start again and re-scan the sentence, counting
the words as they go along, in order to ascertain its truth.
One of the possible objections to this kind of self-referential statement is
that it cannot possibly be interpreted and have its truth value evaluated
on a single scan. Two scans are necessary.
Derek Holt.
Alberto C Moreira
>Would you show us how self-reference comes in to play in the
>construction of transcendental numbers via the diagonalization
>argument?
No. You show me how to prove that any uncountably infinite number
exists without using diagonalization.
That, plus the fact that the existence of even a countable infinite set
has to be axiomatized, gives me a sweet taste of fairyland. Just for
one brief moment, drop the axiom of infinity and see how much of
your majestic building still stands up.
_alberto_
Toby Bartels
>to...@ugcs.caltech.edu (Toby Bartels) writes:
>>Are you suggesting something along the lines of
>>that an English language construct is meaningless
>>unless we use it in mathematical theories?
> No, I'm saying that many English language constructs lack
> the necessary rigour and exactness to be useful in a
> mathematical theory. That's why we use symbols.
Good; I agree :-). But then this is irrelevant to the thread.
This thread is a discussion of English sentences and their self reference,
not of their use in mathematical theories.
>>The noun phrase `this sentence' refers (in this case)
>>to the string `This sentence cannot be proved.'.
>>There is still only one necessary recursion.
> As stated, the phrase is ambiguous. The semantics
> of "this" or "this sentence" are left for you and me to
> guess, and we guessed two different things.
Yes; that's the way it works with English.
In the semantics I happen to guess, they refer to the sentence they're in.
That's one recursion.
You might guess a semantics in which they refer to something else.
If that's also a recursion, it's still just one.
> Yet I'm saying that if we use "this" in the "quoted"
> sense, the phrase is trivial and hardly worth a
> second look.
If the "quoted" sense is the sense you quoted from my post,
I agree it is trivial. I also think that, in that sense,
it's self referential (leading to one recursion).
Do you agree?
>>`This sentence cannot be proved.', as I interpret it,
>>means that the string `This sentence cannot be proved.'
>>is not a theorem in any formal system (which is false).
>>(If I encountered the sentence in a more specific context,
>>such a discussion of a particular formal system,
>>I would interpret it more strictly.)
> You need to define your semantics first.
I did. And I told you what they mean in this case.
If you want more details on what they are, just ask.
> There's no
> such thing as pure syntax; your theory will be only
> as good as its semantics.
We're not discussing theories; we're discussing English.
But this is still true - my use of English requires semantics.
Are you happy with the semantics I used with `This sentence cannot be proved.'?
> If your semantics stops
> at symbols and strings, then you're talking only
> about symbols and strings and nothing else. But if
> you want to go beyond symbols and strings, your
> semantics must be defined - and very formally, too.
They don't always need to be defined very formally
in every use of a natural language.
(But they do when used to describe math, of course.)
> Switching on the fly between syntax and ill-defined
> semantics seems to me as a magician's trick.
I don't think I've been doing this.
I've tried to make my semantics very clear.
> I still say there's a world of difference between
> " 'this sentence has five words' has five words"
> and "this sentence has five words has five words".
Can the second sentence be rephrased as
`That this sentence has five words has five words.'?
The reason I wish to do this is that `this sentence has five words'
is not an English noun phrase, while `that this sentence has five words' is.
If I understand them correctly, there is indeed a big difference.
The second discusses the meaning of the words discussed by the first.
-- Toby
to...@ugcs.caltech.edu
Timothy Chow
< has to be axiomatized, gives me a sweet taste of fairyland. Just for
< one brief moment, drop the axiom of infinity and see how much of
< your majestic building still stands up.
Quite a bit, actually. We can drop set theory entirely and still get a long
way with, for example, PA. I also wouldn't be surprised if one could recover
just about everything via category theory, without reference to sets at all,
let alone infinite sets.
--
Tim Chow tc...@umich.edu
Matthew P Wiener
>In <4638ij$4...@gap.cco.caltech.edu> to...@wrath.ugcs.caltech.edu (Toby
>Bartels) writes:
>>>I would still classify it [a self-referential sentence] as a game.
>>Of course it's a game! So what?
>Here's the point I'm making.
>The rules we use to determine referents contain ambiguities and
>uncertainties. Self-reference hides within those ambiguities and
>uncertainties.
False.
>If we very carefully analyse self-reference, in the sense that
>we remove all ambiguity, the self-reference disappears.
False.
The examples you gave to this effect consisted of you introducting
contexts where the ambiguity was pointedly present. That proves
nothing but your own willful blindness.
For example, if you open up Vann McGee TRUTH, VAGUENESS, AND PARADOX
(Hackett, 1991) to the preface, it begins "This book is an investigation
into the logic of truth." There is nothing ambiguous about which book
McGee is referring to.
david petry
Wiener) writes:
>For example, if you open up Vann McGee TRUTH, VAGUENESS, AND PARADOX
>(Hackett, 1991) to the preface, it begins "This book is an investigation
>into the logic of truth." There is nothing ambiguous about which book
>McGee is referring to.
What you say is true *if* you are confident that you know exactly what
the context of the sentence is. For example, if I were to open the
book up to the preface and showed you that one sentence and not let
you read anything else to determine the context, then you could not
be sure which book the sentence is referring to.
I'm not saying anything more than that.
Bill Taylor
|>
|> "Appended to its own quotation yields a falsehood" appended to its own
|> quotation yields a falsehood.
Can someone tell me why this version is thought to be any more sexy than
the original "This statement is false" ?
-------------------------------------------------------------------------------
Bill Taylor w...@math.canterbury.ac.nz
-------------------------------------------------------------------------------
Since we got a video player, I hardly ever read any more.
I used to be quite a bookworm. Now I'm more of a tapeworm.
-------------------------------------------------------------------------------
Toby Bartels
>wee...@sagi.wistar.upenn.edu (Matthew P Wiener) writes:
>>For example, if you open up Vann McGee TRUTH, VAGUENESS, AND PARADOX
>>(Hackett, 1991) to the preface, it begins "This book is an investigation
>>into the logic of truth." There is nothing ambiguous about which book
>>McGee is referring to.
>What you say is true *if* you are confident that you know exactly what
>the context of the sentence is. For example, if I were to open the
>book up to the preface and showed you that one sentence and not let
>you read anything else to determine the context, then you could not
>be sure which book the sentence is referring to.
>I'm not saying anything more than that.
Really!?!?!?
I could have *sworn* you were saying ambiguity was necessary for self reference.
My newsreader must be on the blink again.
You've done a good job of finding ambiguity
in every self referential English sentence
with which you've been presented.
But that shows nothing about self reference,
since it's completely explained by the nature of English.
-- Toby
to...@ugcs.caltech.edu
Stephen Harris
>> Once more you confirm my point. Anything not Turing-computable
>> must be proved so by diagonalization. And diagonalization does
>> necessarily involve self-reference. Hence, I could say that Goedel's
> Hmm, what is a general definition of diagonalization?
> In any case, consider: if all tape cells are blank, halt in state No.
> If there is a stroke anywhere, halt in State Yes. No Turing Machine can
> do this, and the proof does not seem to need the D word.
This analogy is a bit literal...(suppose that someone does not catch the
pun in the above prase) Does that mean no self-reference was involved
because it was not perceived? The English language contains expressions
for this:The joke went over his head. The humor/sarcasm was lost on hir.
My point is that the analagous situation is when the computer continues
when assigned to find a solution to an unknowable question.
This brings up the Law of the Excluded Middle. Aristotelean Logic has
A and notA and together this is the sum total of the possibilities. A'
A and ~A always stand in relationship to each other because each is
always what the other is not. Takes self-reference to see what it not.
The relational connection passes through A', all that is.
> Cantor proved "every sequence misses a real" by a_n(n) + 1. I can't
> in good faith call this a self-reference... I wouldn't mind doing so, if
> someone gave an argument that allowed me to see it!
> Yes, the Hierarchy property of Sigma-0-n is usually proven by a
> definable version of Cantor's idea (universal sets). Am I missing some-
> thing? Is there an objection?
Cantor's universal sets continually expand to sets of meta,meta,meta,...
which is the same idea of the d. proof he used. To posit there are such
sets for the d. proof and then that aren't such sets (one that is the
conclusion) is inconsistent which is the meaning of Godels U.
>> argument isn't but a rehash of Cantor's diagonalization argument.
Both Cantor and Godel use the d. argument for their proof. This is the Law
of the excluded middle in action. That is why CH is important. Knowing
the correct boundary to establish relationship origin. However, the central
part of Cantor's use of the d. was never proven, well-ordered and
well-structured. So it must be regarded axiomatically, which means a self
evident truth. GIT's or U was an easy extension of Skolem's (1920 paper)
Both Kolmogorov and Chaitin; Solomanoff a bit earlier easily extended U
to all formal systems. Meaning they are all random. This seems to me to
be a more than trivial philosophical realization. It is just about a game
theory approach to the ethical living value of being open-minded.
> Again, it seems Cantor's idea by itself is not enough to
prove Goedel's> theorem. Does anyone see otherwise?
Goedel's theorem is already proven the same way as Cantor. If you
can disprove Goedel you have simultaneously disproven Cantor.
The proofs are consistent with each other, same axioms and theorems.
Cordially,
Stephen
Toby Bartels
>wee...@sagi.wistar.upenn.edu (Matthew P Wiener) writes:
>|>"Appended to its own quotation yields a falsehood" appended to its own
>|>quotation yields a falsehood.
>Can someone tell me why this version is thought to be any more sexy than
>the original "This statement is false" ?
`This statement is false.' refers to `This statement is false.'.
In other words, it refers to itself. Some people think that's unsexy.
``appended to its own quotation yields a falsehood.'
appended to its own quotation yields a falsehood.'
refers to `appended to its own quotation yields a falsehood.'.
In other words, it doesn't refer to itself.
Some people think that's sexy.
-- Toby
to...@ugcs.caltech.edu
david petry
>>|> "Appended to its own quotation yields a falsehood" appended to its own
>>|> quotation yields a falsehood.
>>Can someone tell me why this version is thought to be any more sexy than
>>the original "This statement is false" ?
>because not even david petry is shameless enough to deny its
>unambiguous self-referentiality.
False.
james dolan
>wee...@sagi.wistar.upenn.edu (Matthew P Wiener) writes:
>|>
>|> Consider Quine's context-free sentence:
>|>