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Infinite

The same degree of infinity as their are owles. Did I miss the infinite owl part of the commentary?
 
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.
 
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
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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.
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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.
 
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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....
 
Overthinker said:
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.
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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.
 
Overthinker said:
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....
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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.
 
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Not to mention it is based on an arbitrary point in time that is the basis of the dates.
 
Cabbage said:
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.
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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.
 
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.
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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.
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(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.
 
Cabbage said:
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.
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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.
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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.
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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.

Proof of equality is left as an exercise to the reader ;) .
 
Jekyll said:
I misread that as well the first time I looked at it. Read it again.
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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).
Jekyll said:
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.

Proof of equality is left as an exercise to the reader ;) .
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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").
 
mummymonkey said:
If there are an infinte number of owls; how many owl eyes are there?
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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.
 
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.)
 
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