Infinite

[mummymonkey]

If there are an infinte number of owls; how many owl eyes are there?

[EternalSceptic]

If there are an infinte number of owls; how many owl eyes are there?

Aleph 0 (infinite)

[Michael Redman]

If there are an infinte number of owls; how many owl eyes are there?That depends. How many owls can you fit on one tree?

[fuelair]

If there are an infinte number of owls; how many owl eyes are there?
an infinite number.:)

[Gregory]

The same degree of infinity as their are owles. Did I miss the infinite owl part of the commentary?

[Tumblehome]

I suppose an infinite amount of anything physical is impossible because it would take an infinite amount of matter, but I gather that isn't your point, to which I don't have an answer.

[rjh01]

Have fun with infintity!
infintity * 2 = infintity. Now we divide both sides by infintity.
1 * 2 = 1. Now where is my $1m?

[Cabbage]

I'm thinking this is in reference to this part of today's commentary:

http://www.randi.org/jr/2007-05/050307.html#i9

In that case, since the thread's already here, I thought I'd point out a couple of mistakes in that section.

First:Aleph Null, I seem to remember, is the number of single numbers, Aleph One is the number of points on a line, Aleph Two is the number of possible lines

This is a common mistake that I've posted about before. If "single numbers" means "integers", then the statement about Aleph Null is correct.

Aleph One, on the other hand, is the first infinite cardinal greater than Aleph Null. It can be characterized as being the cardinality of unique ways (up to order isomorphism) of well ordering a countably infinite set.

The "number of points on a line" is an infinite cardinal (we'll call it "c") greater than Aleph Null. This implies that c is at least Aleph One. However, it does not imply that c=Aleph One. This is known as the Continuum Hypothesis (CH). CH is independent of ZFC set theory, which means that both CH and not CH are consistent with ZFC. Many set theorists believe that c is much larger than Aleph One.

(Edited here to add that I believe I've heard this quote from "One, Two, Three,...Infinity" before, and so the mistake seems to be due to Gamow rather than Randi).

The other mistake:While the number of planets capable of supporting life in the universe is vast by standards of human intuition, the chance of us finding life is the product of the number of planets times the probability of life starting given such a planet.

This implies that if, say, the number of planets capable of supporting life is one billion and the probability for a specific planet to have life is one millionth, then the probability of life on any planet is 1000! Obviously, that doesn't work.

Let N be the number of planets to consider and let p be the (constant) probability of life on any one of those planets. The probability of NO life on a given planet is then 1-p. The probability of NO life on EVERY planet is then (1-p)^N. And so the probability of life on at least one planet is 1 - (1 - p)^N.

[Slimething]

The answer is 42 and something about fish. Next!

[rjh01]

Stop quoting the bible or this thread will be split into the religion forum.

[Overthinker]

Not purely mathematical, or at least not spatially, but I think when talking about life on other planets we need to also think about the time factor - intelligent life, as so many museums love to remind us, has existed on Earth for a heartbeat. Even granted that the odds of life on an extrasolar M-class (okay, Earth-like) planet are X, we need to remember that that is X throughout a considerable period of time, so as Clarke has said )if I remember correctly), we may meet apes or angels, but never men.

Oh, and talking of numbers, that 02:03:04, 05-06-07 is not only accurate only for Americans, who inexplicably go medium-small-large in their ordering of dates, but will in fact be repeated - in 3007. And has happened in 1007, and technically 7 (also 107 I would image) although I rather doubt anyone then noticed. And lets not get into BC either....

[wahrheit]

02:03:04, 05-06-07 is not only accurate only for Americans, who inexplicably go medium-small-large in their ordering of dates, but will in fact be repeated - in 3007.

I also noticed this, since we over here across the big pond will have that "magic" time and date in four weeks, on June 5th.

[fuelair]

Not purely mathematical, or at least not spatially, but I think when talking about life on other planets we need to also think about the time factor - intelligent life, as so many museums love to remind us, has existed on Earth for a heartbeat. Even granted that the odds of life on an extrasolar M-class (okay, Earth-like) planet are X, we need to remember that that is X throughout a considerable period of time, so as Clarke has said )if I remember correctly), we may meet apes or angels, but never men.

Oh, and talking of numbers, that 02:03:04, 05-06-07 is not only accurate only for Americans, who inexplicably go medium-small-large in their ordering of dates, but will in fact be repeated - in 3007. And has happened in 1007, and technically 7 (also 107 I would image) although I rather doubt anyone then noticed. And lets not get into BC either....
actually, 2107 is also '07 in that way of doing it as is 2207, etc. happens every one hundred years , not just every thousand and definitely not not ever again.

[cyborg]

Not to mention it is based on an arbitrary point in time that is the basis of the dates.

[Jekyll]

Aleph One, on the other hand, is the first infinite cardinal greater than Aleph Null. It can be characterized as being the cardinality of unique ways (up to order isomorphism) of well ordering a countably infinite set.

Given that the cantor set is isomorphic to the real numbers and of size P(Aleph null), I think the bit in bold is wrong.
This fits in with the generalised version of the continuum hypothesis being: P(Nx)=Nx+1.

http://mathworld.wolfram.com/ContinuumHypothesis.html

Apart from that, good post.

[Cabbage]

Given that the cantor set is isomorphic to the real numbers and of size P(Aleph null), I think the bit in bold is wrong.
If by "isomorphic" you mean "has the same cardinality as", that's right. The following sets all have the same cardinality:

The power set of the natural numbers

The set of real numbers

The Cantor set

All of these sets have cardinality 2^(Aleph Null). But how big is 2^(Aleph Null)? That's the problem.

About Aleph-One:Aleph-1 is...equal to the cardinality of the set of countable ordinal numbers.
(from http://mathworld.wolfram.com/Aleph-1.html )

Which is just a different way of saying what I said, since the set of countable ordinal numbers corresponds exactly to the unique ways of well ordering a countable set.

So on one hand we have the cardinality of the reals 2^(Aleph Null) (commonly referred to as c). We know this is bigger than Aleph Null, but we don't know how big it is.

On the other hand, we have Aleph-One, which is the next largest cardinal after Aleph Null.

The continuum hypothesis (CH) is the claim that c=Aleph-One, but CH can't be proven and is independent of ZFC set theory.

Without CH there's no guarantee that c=Aleph-One. Aleph-One is still Aleph-One, but it's possible for c to equal (almost) anything strictly greater than Aleph-One: c could be Aleph-Two, Aleph-Three, Aleph-53489527, or much much bigger.

[Jekyll]

About Aleph-One:
(from http://mathworld.wolfram.com/Aleph-1.html )

Aleph-1 is...equal to the cardinality of the set of countable ordinal numbers.

I misread that as well the first time I looked at it. Read it again.

Aleph-1 is the set theory symbol for the smallest infinite set larger than aleph_0 , which in turn is equal to the cardinality of the set of countable ordinal numbers.

Which is just a different way of saying what I said, since the set of countable ordinal numbers corresponds exactly to the unique ways of well ordering a countable set.
The well orderings of a countable set are at least of size c.
Take the set of all well orderings of N.
For each element in this set, throw out all members of this well-ordering that don't begin with a 1. You'll be left with something that looks like <11, 103,19157327,1,.......>
Throw away the inital 1 to get <1, 03,9157327,,.......>.
Now glue all these digits together preceded by a '0.' .
In the example I used you'd get 0.1 03 9157327 ....

This mapping from the set of well orderings to [0,1) is obviously onto, as given a n \in [0,1) it is trivial to generate a well ordering which would be mapped onto it. Hence (the number of well orderings on N) is >=c.

Proof of equality is left as an exercise to the reader ;) .

[Cabbage]

I misread that as well the first time I looked at it. Read it again.
Yeah, I see now the "which in turn". Grammatically, it does seem to be saying that aleph null is the cardinality of the countable ordinals. If that's what the sentence is saying, it's wrong. The cardinality of the countable ordinals is aleph one. Aleph null is the cardinality of the finite ordinals, not the countable ordinals.

In general, consider a set of cardinality aleph_n. The cardinality of unique ways of well ordering such a set will always be aleph_(n+1).

Take the set of all well orderings of N.
For each element in this set, throw out all members of this well-ordering that don't begin with a 1. You'll be left with something that looks like <11, 103,19157327,1,.......>
Throw away the inital 1 to get <1, 03,9157327,,.......>.
Now glue all these digits together preceded by a '0.' .
In the example I used you'd get 0.1 03 9157327 ....

This mapping from the set of well orderings to [0,1) is obviously onto, as given a n \in [0,1) it is trivial to generate a well ordering which would be mapped onto it. Hence (the number of well orderings on N) is >=c.

Proof of equality is left as an exercise to the reader ;) .
It looks like you're talking about all the different permutations of N. This is definitely different from talking about the set of different ways of well ordering the natural numbers.

Say we have the following orders on N = {0,1,2,3,...}:

First the standard order: (0,1,2,3,...)

Now some variations:

(1,0,3,2,5,4,6,7,...)

(9,8,7,6,5,4,3,2,1,0,19,18,17,16,...)

(0,1,3,2,4,5,7,6,...)

These are all permutations of N. By that I mean their order structures are all the same. In each case there's a first element, a second element, a third element, and so on.... As ordered sets they are all isomorphic, because they all have identical order structures.

The number of permutations of N is definitely c=2^(aleph null), I don't disagree with that. Extending the list from my previous post, all of the following sets have the same cardinality:

The power set of the natural numbers

The set of real numbers

The Cantor set

The set of permutations of natural numbers.

All of these have cardinality c=2^(aleph null).

However, this is distinct from what I was saying about aleph one and the number of unique ways of well ordering the natural numbers (up to order isomorphism).

As I mentioned earlier, the above permutations all have the same order structure. Together, they are only describing one way of well ordering the natural numbers (because they are all order isomorphic).

Some different ways of well ordering the natural numbers:

(1,2,3,...,0)

This is not order isomorphic with the standard ordering because there is one element (0) that is preceded by infinitely many elements. This is not the case in the standard ordering of N.

Another way would be (0,1,2,3,5,6,7,...,4). This, of course, is order isomorphic to the previous one, so I'm not particularly interested in it. I'm only interested in the structure, not which element plays which particular role in the structure.

Other (order isomorphic) unique ways of ordering the natural numbers:

(2,3,4,5,...,0,1)

Distinct from all previous ones because there are now two elements (0 and 1) that are each preceded by infinitely many elements.

More:

(3,4,5,6,7,...,0,1,2)

(1,3,5,7,...,2,4,6,8,10,...) (all the odds (in standard order) followed by all the evens)

(1,3,5,7,9...2,6,10,14,18,...4,12,20,28,36,...8,24 ,40,56,72,...) (all the odds, followed by 2*each odd, followed by (2^2)*each odd,....)

These all have unique order structures. The cardinality of the collection of all such unique well orderings of N is Aleph One.

So the cardinality of permutations of N is c=2^(aleph null), while the cardinality of (order structure-unique) well orderings of N is aleph one.

These aren't necessarily the same. The continuum hypothesis says 2^(aleph null)=aleph one, but the continuum hypothesis may be false.

What we do know:

Aleph one is the next cardinal immediately after Aleph null. Aleph one is also the cardinality of order structure-unique well orderings of the natural numbers.

The cardinality of the reals = cardinality of power set of natural numbers = cardinality of Cantor set = cardinality of set of permutations of natural numbers = ... (and so on). This cardinality is referred to as c, which equals 2^(aleph null).

We also know c = 2^(aleph null) is greater than or equal to aleph one.

What we don't know:

Whether or not c actually equals aleph one. c might be much bigger.

By the way, I've always liked this website:

http://www.ii.com/math/ch/

Particularly because it has a specific section regarding common misconceptions about CH:

http://www.ii.com/math/ch/#confusion

(Re-finding this page just now, I realize it's what I was thinking of when I claimed earlier, "...I believe I've heard this quote from 'One, Two, Three,...Infinity' before").

[pauldmin]

If there are an infinte number of owls; how many owl eyes are there?

Please remember that infinite is not a fixed constant but merely an unapproachable limit. In a lot of calculations you can say that an expression tends towards a fixed value as the variable tends to infinite, but that does not imply that the variable will actually reach infinite.

To state that there are an infinite number of owls is impossible, but you could say that the number of owls approaches infinite. Therefore, the number of owl eyes also approaches infinite.

Of course, some numbers may approach infinite faster than others, but that is simply a matter of Order.

[Gregory]

That's true in calculus; I think it would be proper to view the infinite ordinals as "fixed constants," though-you can even do arithmatic with them.

(Which are what all the posts above us are about.)

[mummymonkey]

Please remember that infinite is not a fixed constant but merely an unapproachable limit. In a lot of calculations you can say that an expression tends towards a fixed value as the variable tends to infinite, but that does not imply that the variable will actually reach infinite.

To state that there are an infinite number of owls is impossible, but you could say that the number of owls approaches infinite. Therefore, the number of owl eyes also approaches infinite.

Of course, some numbers may approach infinite faster than others, but that is simply a matter of Order.So you can never have an infinite amount of anything? Does that mean the universe can not then be infinite? (the previous maths stuff was way over my head)

[Zep]

.999~=1

Or so I'm told! ;)

[Jekyll]

However, this is distinct from what I was saying about aleph one and the number of unique ways of well ordering the natural numbers (up to order isomorphism).

Ah ha. Thank you.

[Gregory]

So you can never have an infinite amount of anything? Does that mean the universe can not then be infinite? (the previous maths stuff was way over my head)

Well, the total amount of energy/matter in the universe is finite (I think--not a physicist), so no--you can't have an infinite amount of anything. But that doesn't imply that the universe is finite, just that it's mostly empty.

[arthwollipot]

Not to mention it is based on an arbitrary point in time that is the basis of the dates.

And where on the planet you happen to be. At 02:03:04 05/06/07 in Florida (where, presumably, Randi is posting from), it will be 16:03:04 06/05/07 here in Canberra.

Note that I've also changed the date format here so that it's more confusing.

[Nova Land]

If there are an infinite number of owls, how many owl eyes are there?


Ah! You should have posted this over in puzzles, rather than here. But at least this gives me the opportunity to do one of my favorite things: disagree with virtually everyone who has posted.

Aleph 0 (infinite)


Wrong.

an infinite number.:)


Wrong.

The same degree of infinity as their are owles. Did I miss the infinite owl part of the commentary?


Wrong.

Have fun with infinity!
infintity * 2 = infintity. Now we divide both sides by infintity.
1 * 2 = 1. Now where is my $1m?


Wrong.

If by "isomorphic" you mean "has the same cardinality as", that's right. The following sets all have the same cardinality...

[snip]

... Which is just a different way of saying what I said, since the set of countable ordinal numbers corresponds exactly to the unique ways of well ordering a countable set.

So on one hand we have the cardinality of the reals 2^(Aleph Null) (commonly referred to as c). We know this is bigger than Aleph Null, but we don't know how big it is....

[snip]...

Without CH there's no guarantee that c=Aleph-One. Aleph-One is still Aleph-One, but it's possible for c to equal (almost) anything strictly greater than Aleph-One: c could be Aleph-Two, Aleph-Three, Aleph-53489527, or much much bigger.


Impressively complicated, but wrong.

... The well orderings of a countable set are at least of size c.
Take the set of all well orderings of N.
For each element in this set, throw out all members of this well-ordering that don't begin with a 1. You'll be left with something that looks like <11, 103,19157327,1,.......>
Throw away the inital 1 to get <1, 03,9157327,,.......>.
Now glue all these digits together preceded by a '0.' .
In the example I used you'd get 0.1 03 9157327 ....

This mapping from the set of well orderings to [0,1) is obviously onto, as given a n \in [0,1) it is trivial to generate a well ordering which would be mapped onto it. Hence (the number of well orderings on N) is >=c...


Also impressive, and also wrong.

...

In general, consider a set of cardinality aleph_n. The cardinality of unique ways of well ordering such a set will always be aleph_(n+1).

It looks like you're talking about all the different permutations of N. This is definitely different from talking about the set of different ways of well ordering the natural numbers.

Say we have the following orders on N = {0,1,2,3,...} ...

[snip]
... Some different ways of well ordering the natural numbers:

(1,2,3,...,0)

This is not order isomorphic with the standard ordering because there is one element (0) that is preceded by infinitely many elements. This is not the case in the standard ordering of N.

Another way would be...

[snip]

... More:

(3,4,5,6,7,...,0,1,2)

(1,3,5,7,...,2,4,6,8,10,...) (all the odds (in standard order) followed by all the evens) ...

[snip]

...
These aren't necessarily the same. The continuum hypothesis says 2^(aleph null)=aleph one, but the continuum hypothesis may be false.

What we do know:

Aleph one is the next cardinal immediately after Aleph null. Aleph one is also ... [snip]


Even more complicated, but still wrong.

Please remember that infinite is not a fixed constant but merely an unapproachable limit. In a lot of calculations you can ... [snip]


Wrong.

All of you are over-looking the obvious and making an unwarranted assumption. No amount of erudite disposition on infinity will change that.

It doesn't matter what kind of infinity we are talking about -- countably infinite, uncountably infinite, aleph null, or whatever. That is because the answer to the question of how many eyes there are in a set of infinite owls is not infinity.

Now, I will grant that it is possible for there to be an infinite number of owl eyes on infinite owls. B t that is not the only possibility. It is also possible that there are only a finite number of eyes. Any finite number you care to name is a possibility -- and, given the lack of information in the puzzle, as reasonable a possibility as an infinite number.

Let me make this simple by taking only one owl to start with. This owl was in a fight in its youth, and one of its two eyes was plucked out. How many eyes does this owl have? I think any reasonable person will answer, one. It is a one-eyed owl. You don't even have to be reasonable to know that. Heck, even I would say it has one eye.

Okay, we've established the principle. Now let's consider the actual case I'm interested in. Imagine an owl which had an unfortunate accident (was flying with scissors, despite its mother's warning never to do this) and poked out both its eyes. How many eyes does this owl have? Answer: if an owl which has lost one of two eyes is a one-eyed owl, then an owl which has lost both of its eyes is a no-eyed owl. The number of eyes it has is zero.

In order for there to be an infinite number of eyes on an infinite number of owls, it is necessary that an infinite number of owls have eyes. But in the problem as stated, there is nothing that requires any of the owls to have eyes.

If the puzzle stated that each owl has at least one eye, the discussion of an infinite number of eyes would be justified. Or, if it stated that there are only a finite number of owls lacking both eyes, then again talk of infinite eyes would be justified. But that was not stated, and cannot simply be assumed.

Therefore the correct answer to the question of how many eyes an infinite number of owls have is: We don't know. There is not enough information.

[GreedyAlgorithm]

In order for there to be an infinite number of eyes on an infinite number of owls, it is necessary that an infinite number of owls have eyes.
Wrong.
If an owl with poor eyesight purchases and wears eyeglasses, any reasonable person would say the owl has four eyes. (If a one-eyed owl wore glasses, we would say the owl had two or three eyes.) And if I saw an owl with two eyes, and two pairs of glasses I would say the owl had six eyes. Suppose that there could be a set of infinite owls. It is no large stretch to then postulate a single owl wearing an infinite number of glasses, hobnobbing with an infinite number of eyeless owls. In that case we would have an infinite number of eyes on an infinite number of owls while only finitely many owls (in fact just one) have eyes.

Edit: That last "have" bugs me. I think it's right even though we know 'owls' refers to one owl, but it still bugs me.

[BillyJoe]

Except that the owl with both eyes poked out by scissors still has two eyes, useless as they are; and even the owl with one eye plucked out still has two eyes, it's just that his other eye lies rotting somewhere, but whose eye is it other than his, no one else can claim possession of it; even if they eat it it is still not their eye, maybe their food but not their eye....

[BillyJoe]

...and if a hare loses his spectacles, we may not need a royal commission into it because he may just have a spare pair.

[Nova Land]

If an owl with poor eyesight purchases and wears eyeglasses, any reasonable person would say the owl has four eyes...


No, that isn't true. Some reasonable people, speaking colloquially, might call a person (or owl) who wears glasses "four-eyes". But if you asked people to count how many eyes the person (or owl) has, most reasonable people would count two. Because -- as Lincoln said about tails and dogs legs -- calling lenses in eyeglasses "eyes" doesn't make them eyes.

A person with an eye missing has one eye. An owl with an eye missing has one eye. A person with two eyes, wearing glasses, has two eyes.

It's important to treat a puzzle like this with the proper respect. Semantic quibbling of the kind you're indulging in is the kind of thing I'd expect from someone like BillyJoe, not a serious poster such as yourself.

[Nova Land]

Uh oh...

Except that the owl with both eyes poked out by scissors still has two eyes, useless as they are...


BillyJoe! What an unexpected pleasure to see you? What are you doing over here? Meg has a new Catch Phrase up and I just assumed you'd be busy working on that.

Well. Much as I hate to disagree with you, you are incorrect. An owl which is missing its eyes has no eyes. What it has is two eye sockets. Eye sockets are not the same as eyes, any more than light sockets are the same as light bulbs.

[Steven Howard]

Except that the owl with both eyes poked out by scissors still has two eyes, useless as they are; and even the owl with one eye plucked out still has two eyes, it's just that his other eye lies rotting somewhere, but whose eye is it other than his, no one else can claim possession of it; even if they eat it it is still not their eye, maybe their food but not their eye....

That's just silly. If an owl loses an eye, how can you say he still has it?
If you lose something, then by definition you don't have it any more.

[Slimething]

What if each owl owns a potato? (Before you say it, yes, owls can own potatoes!)

[Beleth]

I just hope each owl doesn't dicks his own potatoes.

[BillyJoe]

That's just silly. If an owl loses an eye, how can you say he still has it?
If you lose something, then by definition you don't have it any more.


But, if you lose something, it is still yours; otherwise, if I find your wallet, I will not return it if you think it's no longer yours.

[BillyJoe]

BillyJoe! What an unexpected pleasure to see you? What are you doing over here? Meg has a new Catch Phrase up and I just assumed you'd be busy working on that.


Why, thank you, Nova, but I ran her puzzle through the hexadecimal translator and it came up with nonsense, ’ˆ¥•3 to be precise, and that's worse than original puzzle.

Well. Much as I hate to disagree with you, you are incorrect. An owl which is missing its eyes has no eyes. What it has is two eye sockets. Eye sockets are not the same as eyes, any more than light sockets are the same as light bulbs.


Then I misunderstood you, possibly because of your imprecision, I don't know what else. When you poke out your car's headlights, you don't have two empty sockets, just headlights that are poked in and smashed, as it were, and pretty well useless, but you still actually have two headlights.

[Gregory]

Ah! You should have posted this over in puzzles, rather than here. But at least this gives me the opportunity to do one of my favorite things: disagree with virtually everyone who has posted.




Wrong.




Wrong.




Wrong.




Wrong.




Impressively complicated, but wrong.




Also impressive, and also wrong.




Even more complicated, but still wrong.




Wrong.

All of you are over-looking the obvious and making an unwarranted assumption. No amount of erudite disposition on infinity will change that.

It doesn't matter what kind of infinity we are talking about -- countably infinite, uncountably infinite, aleph null, or whatever. That is because the answer to the question of how many eyes there are in a set of infinite owls is not infinity.

Now, I will grant that it is possible for there to be an infinite number of owl eyes on infinite owls. B t that is not the only possibility. It is also possible that there are only a finite number of eyes. Any finite number you care to name is a possibility -- and, given the lack of information in the puzzle, as reasonable a possibility as an infinite number.

Let me make this simple by taking only one owl to start with. This owl was in a fight in its youth, and one of its two eyes was plucked out. How many eyes does this owl have? I think any reasonable person will answer, one. It is a one-eyed owl. You don't even have to be reasonable to know that. Heck, even I would say it has one eye.

Okay, we've established the principle. Now let's consider the actual case I'm interested in. Imagine an owl which had an unfortunate accident (was flying with scissors, despite its mother's warning never to do this) and poked out both its eyes. How many eyes does this owl have? Answer: if an owl which has lost one of two eyes is a one-eyed owl, then an owl which has lost both of its eyes is a no-eyed owl. The number of eyes it has is zero.

In order for there to be an infinite number of eyes on an infinite number of owls, it is necessary that an infinite number of owls have eyes. But in the problem as stated, there is nothing that requires any of the owls to have eyes.

If the puzzle stated that each owl has at least one eye, the discussion of an infinite number of eyes would be justified. Or, if it stated that there are only a finite number of owls lacking both eyes, then again talk of infinite eyes would be justified. But that was not stated, and cannot simply be assumed.

Therefore the correct answer to the question of how many eyes an infinite number of owls have is: We don't know. There is not enough information.


Did you mistake this for the puzzle forum? It isn't.