Interesting. Really, all the various forms of constructivism are interesting, even if you don't happen to believe in the philosophy behind it. My person field of research is computability theory, which studies, in part, a level right above constructivism. I hadn't heard of finitism, nor do I know of what sort of results you get, besides just denying lots of things exist, but now I'm curious.
However, I don't really see what finitism has to do with your complaints about sizes of infinities. If I read it correctly, a finitist would simply deny the existence of "the set of primes." Although, the wikipedia article does not go into what sort of operations can be used the the finite number of steps. (Is power set allowed? I don't know.) What's more, it looks like Classical Finitists would say that countably infinite sets (such as primes or integers) exist. But I don't see that they would distinguish the sizes. Or am I missing something?
I'm sorry, but I have absolutely no idea what you mean when you say:
The introduction of 'infinity' into this allows for time to be factored out of consideration. That's perfectly fine and I'm sure there's a lot of useful discoveries that have come out of that.
As for the rest of your points, I'm also a little unsure what you are getting at.
But, infinity is merely a concept. You can ascribe it to both of these if you wish, but I simply think that loses some information about the relation.
While infinity is a concept, I don't see what that has to do with anything. Many would argue that the number 4 is merely a concept. That does not mean we don't prove true facts about it, as well as use it successfully. While I don't quite understand what you mean by ascribing infinity to the sets and losing information, I would point out that it's not as if by saying that both the natural numbers and the primes have cardinality aleph_0, that we are prohibited from saying other things.
Maybe another concept is called for? Or, maybe I'm ignorant of the existence of this other concept. I'm sure the mathematicians at the time of Cantor would've thought of this.
How about the concept of "have equal cardinality but one is a proper subset of the other." Or how about the "the first is the natural number and the second is the prime numbers, and I like bacon" relation. Both of these are definitely true of the natural numbers and the primes. They contain plenty of information besides the cardinality of the sets. I suspect that the only reason that there is not a name for these relations is that they are not particularly useful.
I'm not entirely sure what you would be looking for. Besides the cardinal numbers, another system for dealing with infinite sets is the ordinal numbers. However, the ordinal corresponding to the set of primes is the same as the ordinal corresponding to the set of natural numbers (both are omega).
Really, I don't see why such a system would be needed, even if it were possible. What's wrong with saying that yes, both the set of primes and the set of natural number (or integers, if you prefer) are infinite, but it turns out that the primes are a proper subset of the natural numbers. That seems to capture more information that saying the the set of natural numbers are "bigger" than the set of primes.
The more I think about it, the more I wonder why the surprise is that one infinite set is not larger than another infinite set. For me, when I first ran into this stuff, I was much more surprised that there might be different sizes of infinity. If anything, it is that which goes against our intuition. Thoughts?
So that you don't sound too ignorant when you talk about mathematical induction, let me clarify something for you. Mathematical induction can be used to prove a property of every member of an infinite set. It can not, in general, prove a property about the infinite set itself (without going into transfinite induction, which is tricky, and definitely depends on a very precise notion of infinity).
In your thought experiment, you "inductively" proved that for every natural number n, the number of natural numbers less than n is greater than the number of primes less than n. And this is true. However, it takes a (incorrect) leap to go from that to say that the entire set of natural numbers is larger than the entire set of primes. See the difference?
Maybe what you are looking for, instead of cardinality, is the "is a proper subset of" relation. For it is certainly true that the primes are a proper subset of the integers -- there are integers which are not primes, but every prime is an integer. And while it is certainly useful to have the proper subset relation in math (and thus, why we have it), it is very different than cardinality.
Cardinality is a measure of the size of the set, not necessarily the relationship to other sets. In fact, consider the sets A = {1, 2, 3,...} and B = {-1, -2, -3,...}. Neither is a subset of the other. Yet we can still compare them by saying that they have the same size (cardinality).
Perhaps why all this cardinality business seems a little counter-intuitive is because often in common language when we say one collection is larger than the other, we aren't entirely clear about in which sense we mean: size or proper subset. Indeed, for any finite sets A and B, if A is a proper subset of B, then A has smaller cardinality than B. This does not necessarily hold if A and B are infinite.
Yes, not every correspondence between integers and primes will be 1-1, but that doesn't mean there isn't one which is. First, consider the following 1-1 correspondence between the positive integers and the primes: 1 -> 2, 2 -> 3, 3 -> 5, 4 -> 7, 5 -> 11, etc. That is, to the nth positive integer, we associate the nth prime.
Now to get a 1-1 correspondence between the primes and the integers, we simply need a 1-1 correspondence between the integers and the positive integers: 0 -> 1, 1 -> 2, -1 -> 3, 2 -> 4, -2 -> 5, etc. Then clearly going from integer to positive integer to prime gives a 1-1 correspondence.
Isn't math fun?!
What do you mean by "more integers than primes"? Since both are infinite, the standard way of comparing sizes is to ask whether there is a 1-1 correspondence between the two sets. And there is. So in the cardinality sense, there are exactly as many primes as there are integers.
And by the way, cardinality is extremely precise.
Kile is a LaTeX editor. But if you are currently writing your TeX as a text file and compiling it by hand, then it's worth checking out. If you are not on Linux, then TeXnicCenter for Windows also works well.
Even with Kile, there are things that are quite challenging to do with LaTeX. But I always liked that. Yet another thing for me to feel superior for being able to do.
I agree. So the next thing to think about is how this is going to get fixed. While I would rather see our Senators spending time on more important matters, they do have a point: ratings should be consistent. And there should be games rated AO out there. The ESRB should rate lots of games AO, and game developers should take a stand and stick with it. Eventually Sony and Nintendo, as well as Walmart and the like, will come around and see that there is a market for adult entertainment (and not just the sexy kind). Then everyone wins.
One reason: no way to prosecute! Until we have brain scanners, such a law would be a waste of time. But this is besides the point anyway. Attempted copyright infringement is NOT a thought crime. There is a big difference between thinking about downloading a dvd (or even really really wanting to) and actually trying to download it. Even if the police stop you before you finish the download, you were definitely doing something wrong - just not doing it very well.
That being said, I hope this legislation gets laughed out of congress asap.
While this site might take libel comments to an extreme, it is not as if this is the first site to publicly post negative (and likely false) statements about named individuals. Consider: http://www.myprofessorsucks.com/. Some of the bad reviews of professors are really bad; "he suggested I come over to the grad dorms to talk about raising my grade," for instance. Any employer reading that might think twice about hiring that professor.
Of course, this issue will go away soon enough. We just need to wait until law firms realize that these postings cannot be taken as a reliable source of information about prospective employees. I suspect universities already feel that way about MyProfessorSucks.com.
But we are not talking about giving libraries and schools the right to keep us from waisting time. Instead, this bill would take away their right to let us. And in so doing, take away our rights to do the same. That's a huge difference.
The problem is, the iPod is TOO random. True randomness guarentees the possibility that a single song can be played 10 times back to back to back to...to back. And nobody wants that when listening to music!
The solution? Random for music should mean something different than random. The next track should be different than the one currently playing. Different genre, different artest, different tempo, different key, different . And which different xxxx should not be random either. After a death metal song, I don't want sludge metal (well, I do, but not on random mode). The iPod should pick a song that is truely different than the one I am currently listening to.
The point it, in user applications, nobody every wants real randomness. So why do we keep getting it?
By the way, it sounds like you have a beef with the fact some things are illegal in your state (I'm going to hope the beef is with private gambling and NOT child molestation). Your problem shouldn't be with the laws being enforced, but the laws themselves.
This is exactly the point! There are some unjust laws. Always have been, and what's worse, there always will be. So if we allow complete surveillance of our lives by those who make the laws, there is potential for those lawmakers to make any law they desire. And once the law is made, there would be no way of changing it. Even if you don't believe that the government of today is corrupt (and I would agree with you), there is no reason to believe that government will never be so.
Slashdot Mirror
Careful! For that test to work, your palms must be facing out! Otherwise you are apt to confuse the candidates. Damn you voting fraud!
Interesting. Really, all the various forms of constructivism are interesting, even if you don't happen to believe in the philosophy behind it. My person field of research is computability theory, which studies, in part, a level right above constructivism. I hadn't heard of finitism, nor do I know of what sort of results you get, besides just denying lots of things exist, but now I'm curious.
However, I don't really see what finitism has to do with your complaints about sizes of infinities. If I read it correctly, a finitist would simply deny the existence of "the set of primes." Although, the wikipedia article does not go into what sort of operations can be used the the finite number of steps. (Is power set allowed? I don't know.) What's more, it looks like Classical Finitists would say that countably infinite sets (such as primes or integers) exist. But I don't see that they would distinguish the sizes. Or am I missing something?
The introduction of 'infinity' into this allows for time to be factored out of consideration. That's perfectly fine and I'm sure there's a lot of useful discoveries that have come out of that.
As for the rest of your points, I'm also a little unsure what you are getting at.
But, infinity is merely a concept. You can ascribe it to both of these if you wish, but I simply think that loses some information about the relation.
While infinity is a concept, I don't see what that has to do with anything. Many would argue that the number 4 is merely a concept. That does not mean we don't prove true facts about it, as well as use it successfully. While I don't quite understand what you mean by ascribing infinity to the sets and losing information, I would point out that it's not as if by saying that both the natural numbers and the primes have cardinality aleph_0, that we are prohibited from saying other things.
Maybe another concept is called for? Or, maybe I'm ignorant of the existence of this other concept. I'm sure the mathematicians at the time of Cantor would've thought of this.
How about the concept of "have equal cardinality but one is a proper subset of the other." Or how about the "the first is the natural number and the second is the prime numbers, and I like bacon" relation. Both of these are definitely true of the natural numbers and the primes. They contain plenty of information besides the cardinality of the sets. I suspect that the only reason that there is not a name for these relations is that they are not particularly useful.
Or did you have something else in mind?
I'm not entirely sure what you would be looking for. Besides the cardinal numbers, another system for dealing with infinite sets is the ordinal numbers. However, the ordinal corresponding to the set of primes is the same as the ordinal corresponding to the set of natural numbers (both are omega).
Really, I don't see why such a system would be needed, even if it were possible. What's wrong with saying that yes, both the set of primes and the set of natural number (or integers, if you prefer) are infinite, but it turns out that the primes are a proper subset of the natural numbers. That seems to capture more information that saying the the set of natural numbers are "bigger" than the set of primes.
The more I think about it, the more I wonder why the surprise is that one infinite set is not larger than another infinite set. For me, when I first ran into this stuff, I was much more surprised that there might be different sizes of infinity. If anything, it is that which goes against our intuition. Thoughts?
So that you don't sound too ignorant when you talk about mathematical induction, let me clarify something for you. Mathematical induction can be used to prove a property of every member of an infinite set. It can not, in general, prove a property about the infinite set itself (without going into transfinite induction, which is tricky, and definitely depends on a very precise notion of infinity).
In your thought experiment, you "inductively" proved that for every natural number n, the number of natural numbers less than n is greater than the number of primes less than n. And this is true. However, it takes a (incorrect) leap to go from that to say that the entire set of natural numbers is larger than the entire set of primes. See the difference?
Maybe what you are looking for, instead of cardinality, is the "is a proper subset of" relation. For it is certainly true that the primes are a proper subset of the integers -- there are integers which are not primes, but every prime is an integer. And while it is certainly useful to have the proper subset relation in math (and thus, why we have it), it is very different than cardinality.
...} and B = {-1, -2, -3, ...}. Neither is a subset of the other. Yet we can still compare them by saying that they have the same size (cardinality).
Cardinality is a measure of the size of the set, not necessarily the relationship to other sets. In fact, consider the sets A = {1, 2, 3,
Perhaps why all this cardinality business seems a little counter-intuitive is because often in common language when we say one collection is larger than the other, we aren't entirely clear about in which sense we mean: size or proper subset. Indeed, for any finite sets A and B, if A is a proper subset of B, then A has smaller cardinality than B. This does not necessarily hold if A and B are infinite.
Yes, not every correspondence between integers and primes will be 1-1, but that doesn't mean there isn't one which is. First, consider the following 1-1 correspondence between the positive integers and the primes: 1 -> 2, 2 -> 3, 3 -> 5, 4 -> 7, 5 -> 11, etc. That is, to the nth positive integer, we associate the nth prime.
Now to get a 1-1 correspondence between the primes and the integers, we simply need a 1-1 correspondence between the integers and the positive integers: 0 -> 1, 1 -> 2, -1 -> 3, 2 -> 4, -2 -> 5, etc. Then clearly going from integer to positive integer to prime gives a 1-1 correspondence. Isn't math fun?!
What do you mean by "more integers than primes"? Since both are infinite, the standard way of comparing sizes is to ask whether there is a 1-1 correspondence between the two sets. And there is. So in the cardinality sense, there are exactly as many primes as there are integers. And by the way, cardinality is extremely precise.
Kile is a LaTeX editor. But if you are currently writing your TeX as a text file and compiling it by hand, then it's worth checking out. If you are not on Linux, then TeXnicCenter for Windows also works well. Even with Kile, there are things that are quite challenging to do with LaTeX. But I always liked that. Yet another thing for me to feel superior for being able to do.
I agree. So the next thing to think about is how this is going to get fixed. While I would rather see our Senators spending time on more important matters, they do have a point: ratings should be consistent. And there should be games rated AO out there. The ESRB should rate lots of games AO, and game developers should take a stand and stick with it. Eventually Sony and Nintendo, as well as Walmart and the like, will come around and see that there is a market for adult entertainment (and not just the sexy kind). Then everyone wins.
One reason: no way to prosecute! Until we have brain scanners, such a law would be a waste of time. But this is besides the point anyway. Attempted copyright infringement is NOT a thought crime. There is a big difference between thinking about downloading a dvd (or even really really wanting to) and actually trying to download it. Even if the police stop you before you finish the download, you were definitely doing something wrong - just not doing it very well.
That being said, I hope this legislation gets laughed out of congress asap.
While this site might take libel comments to an extreme, it is not as if this is the first site to publicly post negative (and likely false) statements about named individuals. Consider: http://www.myprofessorsucks.com/. Some of the bad reviews of professors are really bad; "he suggested I come over to the grad dorms to talk about raising my grade," for instance. Any employer reading that might think twice about hiring that professor.
Of course, this issue will go away soon enough. We just need to wait until law firms realize that these postings cannot be taken as a reliable source of information about prospective employees. I suspect universities already feel that way about MyProfessorSucks.com.
But we are not talking about giving libraries and schools the right to keep us from waisting time. Instead, this bill would take away their right to let us. And in so doing, take away our rights to do the same. That's a huge difference.
The problem is, the iPod is TOO random. True randomness guarentees the possibility that a single song can be played 10 times back to back to back to...to back. And nobody wants that when listening to music! The solution? Random for music should mean something different than random. The next track should be different than the one currently playing. Different genre, different artest, different tempo, different key, different . And which different xxxx should not be random either. After a death metal song, I don't want sludge metal (well, I do, but not on random mode). The iPod should pick a song that is truely different than the one I am currently listening to. The point it, in user applications, nobody every wants real randomness. So why do we keep getting it?
By the way, it sounds like you have a beef with the fact some things are illegal in your state (I'm going to hope the beef is with private gambling and NOT child molestation). Your problem shouldn't be with the laws being enforced, but the laws themselves.
This is exactly the point! There are some unjust laws. Always have been, and what's worse, there always will be. So if we allow complete surveillance of our lives by those who make the laws, there is potential for those lawmakers to make any law they desire. And once the law is made, there would be no way of changing it. Even if you don't believe that the government of today is corrupt (and I would agree with you), there is no reason to believe that government will never be so.