posted
WillB, you seem to be confusing the Alephs and Beths. I've seen people do this before, but here is a more accurate definition of the cardinalities:
Aleph_0 is defined to be the cardinality of the smallest infinite set. You can prove that the set of natural numbers is such a set, therefore its cardinality is Aleph_0 (it is often also pronouned "Aleph-nought").
Aleph_(n+1) is defined to be the cardinality of the smallest infinite set that is greater than but not equipotent to Aleph_(n). So for instance, Aleph_1 is the next cardinality higher than Aleph_0.
Note that the subscripts (0,n,n+1) for the Alephs are not cardinal numbers themselves; they are ordinal numbers. Hence, there is an Aleph_0, Aleph_1, Aleph_2, .... and infinitely so, then Aleph_(omega) [omega is the smallest infinite ordinal number, whereas Aleph_0 is the smallest infinite cardinal number], Aleph(omega+1), Aleph(omega+2)... and infinitely so, then Aleph_(2omega) ....and on and on with infinite hierarchies of infinities.
Now, Beth_0 is defined to be the same as Aleph_0.
Beth_(n+1) is defined to be the cardinality of the power set of Beth_(n). Thus for instance Beth_1 is the power set of Beth_0, which we know is equipotent to the Continuum (Real Numbers), therefore Beth_1 is the cardinality of the real numbers.
Now, presumably the Aleph hierarchy and the Beth hierarchy overlap somehow. The problem is, within ZFC set theory you cannot determine how they overlap. One hypothesis that has been suggested is the Beth_1 = Aleph_1 (in other words, that Aleph_1 is the cardinality of the reals). This is called the Continuum Hypothesis. Your post, Will B, seemed to assume that it is true. However, it is important to note that there is some mathematical work done assuming that the Continuum Hypothesis is false. It can be shown that the Continuum Hypothesis is independent (that is, can neither be proved nor disproved) from ZFC set theory. The same applies to the more generalized version of the Continuum Hypothesis (known as GFC) which states that Aleph_n = Beth_n for all n.
Posts: 142 | Registered: Apr 2005
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posted
Where did all the marbles come from? I would like an infinite amount of Legos. Now, I wonder what the percentage of red bricks there would be...
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quote:Originally posted by Eldrad: Not quite. Going back to what King of Men said, say 2/3 of the marbles are red; then 2n/3n marbles are going to be red for every possible value of n. The thing is, when you take the limit, you're supposed to simplify the fraction as much as possible, so the n's cancel out, leaving you with only the proportion of red marbles (2/3), not the actual number of them (which would be infinite if n went off to infinity).
If you would have read my previous posts, you'd see that I already spoke of what you just pointed out. I'm not misreading anything.
Posts: 143 | Registered: Sep 2005
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posted
I pulled the name "beth" out of the air (or else the Hebrew alphabet), not knowing it was already in use . . . ok, I'll use Aleph-prime. Unless that one's taken too, in which case I'll call it zignabrump. I'm pretty sure that has no previous definition!
Posts: 1877 | Registered: Apr 2005
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Avin- I loved the 0% prime numbers example. If I remember right in my 4000 lvl math proofs class (I forgot the course name), we proved that the probability of picking an integer from the set of all real numbers was in fact 0% as well. I may be wrong though, it has been a little while.
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posted
That's correct, but for an almost different reason. In this case, the cardinality of the two sets is different, so it's much more straightforward.
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posted
This has been pretty thoroughly addressed in set theory. There are different sizes of infinity but its not intuitively obvious whether one infinite set is bigger than another infinite set.
For example the number of even integers is the same as the number of integers (both odd and even) which is the same as the number of rational numbers. The number of irrational numbers is however bigger than the number of rational numbers. What's more, the number of irrational numbers between any two rational numbers no matter how close, is greater than the number of rational numbers.
Posts: 12591 | Registered: Jan 2000
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posted
I remember when we did the proof that there were "more" irrationals than rationals. It was very confusing* - up until them, the only irrationals I ever thought about were pi, e, phi, and roots - mostly square roots, but I knew there were others.
Of course, had I thought about it, I would have realized that just the irrational roots would be a "bigger" infinite set, nevermind all the other irrationals one can make.
There are a lot of infiinite, non-repeating series of the digits 0-9.
It's just one of those blindingly obvious things one doesn't think about until confronted with it.
*By confusing I meant "upsetting to my mind's view of the number system," not hard to understand.
Posts: 26071 | Registered: Oct 2003
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posted
Hehe, Rabbit said "intuitively obvious." If you ever want to teach higher math, engineering, or sciences at a university, there's a secret requirement that you must use that phrase in your lectures. Or did you have to swear an oath to use it when you got your doctorate? Posts: 1412 | Registered: Oct 2005
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posted
Ha. My quantum professor like to tell us that "you'll show this on your homework, but don't worry its easy." HA.
Posts: 1621 | Registered: Oct 2001
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quote:Originally posted by Puppy: However, if you take a random sample anywhere in my vast field of marbles, you will find that the red marbles outnumber the green marbles two to one.
To be really clear mathematically as to what you mean by a "random sample", you need to define what you mean by random - you need to define a procedure for choosing a number at random. Traditionally, what you do is define a "probability measure", which assigns to every subset a probability in a way which is compatible with the laws of probability (in this context, this means things like the chance of picking an even number + the chance of picking prime number - the chance of picking 2 = the chance of picking a number which is either even or prime.) But here's the rub - there is no "invariant probability measure" on the natural numbers. In other words, there is no way of randomly choosing natural numbers that treats all natural numbers equally. So given a countably infinite collection of marbles, you can't use probability as a way of saying that some fraction of them are red, green, or otherwise, in a way that treats all marbles on equal footing. Of course, you can put them in an order, and say that in the given order, you alternate two red marbles and one green marble. Then, if you choose any marble in the order, the fraction of red marbles before that marble in the order will be about 2/3. But doing so puts different marbles on different footing (because they have different positions in the order.) So without specifying an order to the marbles, or some other way of treating some marbles differently than others, there's no way of talking about the fraction of marbles which is red as opposed to green. The exception to this would be if there were only finitely many red marbles, in which case it would make sense to say that the fraction of red marbles is 0, or if there were only finitely many green marbles, in which case it would make sense to say that the fraction of red marbles is 1.
quote:Or, if you arrange them in an infinite line, that line will go RED-RED-GREEN-RED-RED-GREEN, forever, into infinity.
Is that actually possible, mathematically?
As mentioned before, this makes perfect sense, mathematically. However, as someone already pointed out, when you talk about the fraction of red marbles in this way, it's not really well-behaved, in the following way: In this order, it seems perfectly clear that there are two red marbles for every green one. However, you can rearrange the marbles in line, and get two green marbles for every red one, or one thousand green marbles, or even infinitely many. (You could rearrange them so that there was one green, then one red, then two greens, then one red, then three greens, then one red, etc. Then as you went further and further along in the list, the ratio of green marbles to red marbles would eventually exceed any fixed number.)
So, while there are ways you can make sense of ratios of infinities as mentioned before, it really doesn't make sense to speak of the fraction of red marbles in an infinite collection of red and green marbles, when the red and green marbles have the same cardinality. However, there are definitely different sizes of infinity, as first discovered by Cantor, and as explained in various links already given.
If this doesn't make sense, I encourage you to think about it. There is a lot about these ideas which is very counter-intuitive and bewildering. As an example of just how non-intuitive infinity can be, I offer the following though experiment, which I stole from somewhere:
Imagine that you have a funnel which contains infinitely many ping-pong balls, and the balls are numbered 1, 2, 3, 4, 5, etc. The funnel is hanging over a barrel, so that the balls will pour into the barrel. The ping-pong balls are arranged so that the lowest numbered balls are near the bottom of the funnel, and at the beginning of our experiment, they start pouring out, in order, at an increasing rate. During the first 30 seconds of our experiment (which will only last one minute), 10 of the balls pour out, numbered 1-10. During the next 15 seconds, another 10 pour out (numbered 11-20). Then during the next 7.5 seconds, another 10 pour out (21-30). It goes on like this, with each set of 10 balls pouring out in half of the time it took the previous set, so that after 1 minute has elapsed, all of the balls have poured out of the funnel and into the barrel.
Yes, we have arranged it so that in infinite number of events have happened in a finite period of time. We can do this, because the events occur faster and faster as we go, and 1/2 minute + 1/4 minute + 1/8 minute + 1/16 minute + ... = 1 whole minute. (Just picture the second hand moving - as it moves around the clock face, it first goes half way, and then half of the remaining distance, and half of the remaining distance, and half of the remaining distance again, and so on.)
While this is going on, a tiny gremlin is chucking balls out of the barrel. During the first 1/2 minute, he chucks out ball #1. During the next 1/4 minute, he chucks out ball #2. During the next 1/8 minute, he chucks out ball #3. And so on. So, during the first 1/2 minute, 10 balls enter the barrel, and 1 ball leaves, so there are exactly 9 balls in the barrel at the 30 second mark. During the next 1/4 minute, the same thing happens, so 45 seconds into the experiment, there are 18 balls in the barrel. After 52.5 seconds, 27 balls are in the barrel. In fact, there will be 9 billion balls in the barrel exactly 1/2^1,000,000,000 second before the end of the experiment. In fact, given any large number, at some point there will be at least that many ping-pong balls in the bucket. So here is the question: after the experiment ends, when the whole minute has elapsed, how many balls are in the barrel?
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*obligatory space before hint*
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Hint: what balls are in the barrel?
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*obligatory space before solution*
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Solution: there are no balls in the barrel. Why? Because during the course of the experiment, every ball drops out of the funnel and into the barrel, and is then later chucked out by the gremlin. Take, for example, ball #287,563. During the 28,756th time interval (that is to say, during the 2^28,755th 1/2^28,756th of a minute), this particular ball will drop into the bucket. Then, during the 287,563rd time interval, the gremlin will pluck it out of the barrel and toss it over his shoulder. So any given ball is no longer in the barrel at the end of the experiment, and therefore, although at every point in the experiment the number of balls in the barrel is increasing, there are, in fact, no balls remaining at the end.
quote:Originally posted by Bokonon: So my question is, math geeks:
Which set is larger, the set of infinite sets, or the set of non-infinite sets?
-Bok
Technically speaking, neither of those are sets at all. Naive set theory (as opposed to axiomatic set theory) was ended by Russel's Paradox. If we allow all possible collections to be considered as sets, then we get get such objects as the "set of all sets", which obviously contains itself, since it is a set. We could then consider the "the set of all sets which contain themselves" and the "set of all non-self-containing sets." Russel's paradox is the question "is the set of all non-self-containing sets self containing"? This is a question with no answer, and led to the formulation of axiomatic set theory, which restricts what we consider to be a set.
Formally speaking, the objects you mentioned are "classes", and not sets. So we have a class of infinite sets, and a class of finite sets. Classes may be sets - one could speak of the class of natural numbers, which is a set - but not all classes are sets. However, classes, unlike sets, do not have cardinalities. I do not know if there is an equivalent sense of "size" for a class, but if a class is a "proper class" (is not a set), then it is bigger than any set (according to the standard mathemeticians give for two collections having the same size, which is whether there exists a 1-1 correspondance.) Since the two "sets" you gave are in fact proper classes, there are a lot of members of both, but I don't think there is a way in which you could say there is more of one than the other.
quote:Originally posted by Althai: While this is going on, a tiny gremlin is chucking balls out of the barrel. During the first 1/2 minute, he chucks out ball #1. During the next 1/4 minute, he chucks out ball #2. During the next 1/8 minute, he chucks out ball #3. And so on. So, during the first 1/2 minute, 10 balls enter the barrel, and 1 ball leaves, so there are exactly 9 balls in the barrel at the 30 second mark. During the next 1/4 minute, the same thing happens, so 45 seconds into the experiment, there are 18 balls in the barrel. After 52.5 seconds, 27 balls are in the barrel. In fact, there will be 9 billion balls in the barrel exactly 1/2^1,000,000,000 second before the end of the experiment. In fact, given any large number, at some point there will be at least that many ping-pong balls in the bucket. So here is the question: after the experiment ends, when the whole minute has elapsed, how many balls are in the barrel?
But, if we state instead that in the first 1/2 minute the gremlin throws out ball #1 and in the next 1/4 minute he throws out ball #11 and in the next 1/8 minute he throws out ball #21, ad infinitum, <edit> note, this preserves the "10 in, 1 out" per time period </edit> do we end up with an infinite number of balls in the barrel?
Posts: 2926 | Registered: Sep 2005
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posted
Yes, then we do. It's clear that when the experiment is complete, we have every ball numbered something not ending in 1. We could also end up with any desired finite number of balls in the barrel, say by having the gremlin throw out the lowest numbered ball that's not #12, #37, or #55.
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