Infinite

[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 [latex] n \in [0,1) [/latex] 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.