I am a callous and bitter man, the kind of man who hates springtime and kittens, but that link has melted my icy heart.
That is some dangerously cute stuff. It could put a smile on even Jack Thompson's face.
This is a nice explanation, though I do wish I understood more of the details of that magic number, where perhaps the bulk of the "WTF?" factor lies. It's a much better explanation, at any rate, then the (complete lack of an) explanation in the original code, which is appallingly comment-free. That flies sometimes, but given how many people are scratching their heads trying to reverse-engineer the thinking here, this is clearly not one of those times.
You can be closed source without having any copying restrictions. I could give out binaries willy-nilly for free, but not tell anyone the source I used to generate them.
Yeah, the only way I can tell someone is blind or seeing is by asking him. There are no other tests possible, none other even conceivable, by which I might tell if someone were blind or not. This is why babies are never diagnosed as blind until they have first acquired speaking skills, the distinction being necessarily unobservable previously.
In my view, the answer to your first question is "No, not directly; the paradox in the grandparent post isn't formulated in terms of sets, and so the von Neumann universe/ZFC approach of rejecting the sethood of certain collections wouldn't be of any use, though perhaps something analogous could be stretched to fit.". But for this to make sense, I suppose I need to reply to your second question:
The idea of the von Neumann universe and ZFC is that not all collections of objects are sets; rather, the only collections that are sets are those that can be built up from previously generated sets, eventually bottoming out at the empty set. Specifically, the von Neumann universe generates sets in a hierarchy of stages like this:
Stage 0: The empty set [also called {} or 0]
Stage 1: 0 and the set containing 0 [also called {0} or 1]
Stage 2: 0, 1, {1}, and {0, 1} [also called 2; in general, the set {0,..., n-1} is called n]
---
Stage n: Collections whose elements were all around in stage n-1
---
Stage omega: Collections whose elements were all generated in one of stages 0, 1, 2, etc. For example, the infinite set {0, {0}, {{0}}, {{{0}}},...} is generated at stage omega, as is the infinite set {0, 1, 2,...}.
Stage omega+1: Collections whose elements were all around in stage omega. For example, the set {{0, 1, 2,...}} is generated here.
---And so on, through omega+2, omega+3, omega+omega, omega*3, omega^2, omega^omega, omega^(omega^omega), and all sorts of crazy higher up stages. At each stage, the associated collections are those which only have elements from previous stages. Certain collections never get generated in this process (for example, the collection of all sets never gets generated, because there is no stage that comes after all stages; similarly for Russell's paradoxical set).
Of course, the question remains "What exactly are the stages? How long do they go on for?". Well, this is tricky. The stages are called the ordinal numbers, and the von Neumann hierarchy doesn't in itself really clarify how long they go on for. The intuition is that every sequence of ordinals (even infinite ones) should be followed by an even greater ordinal. However, using this idea naively results in a contradiction known as the Burali-Forti paradox, which isn't any nicer than Russell's paradox (specifically, the sequence of all ordinals would have to be followed by an ordinal which would be greater than itself!). So, the usual fix is to once again reject the use of arbitrary collections/sequences; rather than saying that EVERY collection of ordinals has a follower, we only state that certain collections have followers (for example, every finite collection of ordinals has a follower; every collection of ordinals that can be put in one-to-one correspondence with a set generated in one of their stages has a follower; etc.).
The axioms of ZFC capture ideas from the von Neumann hierarchy along with a particular bunch of useful closure conditions on the ordinals (which map into useful closure conditions on sets), while leaving out all the dangerously powerful closure conditions that are known to lead to contradictions. It's strong enough to do all the mathematics that core mathematicians normally do (indeed, it's actually rather ridiculously overpowered for this) while at the same time being weak enough that no one knows how to prove a contradiction from it.
To summarize, the von Neumann hierarchy and ZFC avoid Russell's paradox by not adopting the principle that arbitrary collections of sets are themselves sets; rather, they only adopt the weaker idea that sets constructed iteratively from previous sets exist. This prevents the various known paradoxical sets from being constructed. As for whether this "solves" Russell's paradox, or merely highlights it by sidestepping it, can be debated. (Why shouldn't we be able to consider arbitrary collections
If you look back at the post that started this whole mess, you'll see the sentence "Maybe you're part of the one or two percent that will go on to do something interesting." Hope this helps.
Yeah, that "duh" tag really grates on me; it seems like half the articles here get tagged with it, often in situations just as dubious as this one.
It's interesting the way the tagging system causes people to rally around certain words, though; I wouldn't have predicted, before it was implemented, that "duh" in particular would grow to be a particular annoyance for me, but it's become entrenched now.
Actually, I recall that Wargasm's title had little to do with the game (beyond the "War" part, anyway).
To quote IGN: "Apparently someone at DID's marketing department thought that the 'kids' might dig it if they took their latest action title, put a woman in a bulging flak jacket on the cover (maybe she's carrying a bunch of grenades) and named it after something sexual. The PR staff picked up on the Beavis and Butthead vibe and began sending us sheaves of mail with jokes like 'It's time for Wargasm' and 'Multiple Wargasm.' After this ridiculous blitzkrieg of banality (which must have humiliated the actual design team beyond measure) Tal, Jason and I began furiously scrapping over who was going to have to review a game that was sure to be as embarrassing on the inside as it was on the out. I lost (for those of you who keep track of such things, let me warn you that Jason carries brass knuckles) and sadly loaded the game only to find that there was nothing within the actual software that had anything to do with the title, the chick on the cover or anything else we received in the mail. What I did find was a solid action game that is surprisingly hard to put down."
So the name clearly turned people off to the game who might otherwise have enjoyed it. That's about as bad as a name can be.
Nonsense! Without the rules given in school, how would we know how to speak properly? It's not like we learn these skills anywhere else. People like you are simply hastening the degrade of our ability to communicate. As for the grandparent, he would've been correct to reword his sentence as "I'm as much of a gamer as most people are, but honestly, towards whom the hell are they marketing this?"
Doesn't exactly roll of the tongue, but that's the price one pays for the smug sense of self-satisfaction which comes with being "correct".
[Disclaimer/Translation: I agree; it's bizarre that so many people buy into such a contrived "rule".]
Yeah, he's willing to pay up to 500 bucks, and so getting it at 475 or 150 is good to him. It is. He prefers it to not getting the item at all.
But don't ignore the fact that getting it at 150 remains better than getting it at 475, and getting it at 100 remains better still. A strategy that will probably get the item at 100, then, is better than one which will probably get it at 475, even if both strategies are better than not entering the auction at all.
"Winning" an auction may be about getting the item for an amount one is willing to pay. But there remain degrees of winning, some better than others.
I don't think he was implying that those were Latin words. Rather, his point was that people get all worked up over using "correct" plurals for Latin words, and yet, oddly, don't bother doing anything similar for words from other languages. People will say "virii" (which isn't correct in any language) and "penes" and so forth, but will never bother chiding others for saying "saunas" rather than "saunat", or what have you (in the English-speaking world, of course).
I've never understood all the hate for Super Mario 2. It may not have been a "real" Mario game, but Miyamoto was actually quite involved with it, and it was a damn good game. What aspect of it causes you to call it "teh suck"?
The number of mappings from N to N is \aleph_0^{\aleph_0}, not \aleph_0^2 = \aleph_0 * \aleph_0.
After all, the number of sets of natural numbers is uncountable, right? Well, every set can be represented by a function from natural numbers to either 0 or 1 depending on membership in the set. So, clearly, the number of mappings from N to N must be uncountable as well.
Well, I disagree with your assertions about "meaning" and what counts as math, but as far as your core idea goes, in a certain sense, you're correct: Even though you can prove the existence of uncountable sets in ZFC, there, counterintuitively, exist countable models of ZFC. Look up the Lowenheim-Skolem theorem (or "Skolem's Paradox") for more information.
I don't know much about Slate, or Smalltalk, or anything, for that matter, but a quick look at the article shows that both of your "confusing" examples are not that confusing...
As someone else already pointed out, "3 + 4 * 5" ==> 35 (not 23) because Slate has no differences in binary operator precedence and therefore always associates to the left.
And as for "(3/4) == (3/4)" ==> false, this is because "==" tests for _object_ equality, and the two different instances of "(3/4)", while representing the same numerical value, are distinct objects. It appears that "=" tests for numerical equality.
In the paper they wrote, they claim that Grover's algorithm provides an exponential speed up over classical search algorithms. If I'm not mistaken, Grover's algorithm takes time O(N^(1/2)) while classical search algorithms take time O(N), which is only a quadratic speedup, not an exponential one...
I'm sure the rest of their paper is well done, but this bothers me anyway.
I thought that might have been it, but I wasn't sure if perpetual motion itself was outlawed, or just devices which output usable energy. Thanks for setting me straight.
Slashdot Mirror
Well, if he thought they had the same meaning, he wouldn't have proposed the switch, would he?
I am a callous and bitter man, the kind of man who hates springtime and kittens, but that link has melted my icy heart. That is some dangerously cute stuff. It could put a smile on even Jack Thompson's face.
Um, by splitting 0 dollars among 10 people, what you've demonstrated is that 0/10 = 0, which is thoroughly uncontroversial.
This is a nice explanation, though I do wish I understood more of the details of that magic number, where perhaps the bulk of the "WTF?" factor lies. It's a much better explanation, at any rate, then the (complete lack of an) explanation in the original code, which is appallingly comment-free. That flies sometimes, but given how many people are scratching their heads trying to reverse-engineer the thinking here, this is clearly not one of those times.
You can be closed source without having any copying restrictions. I could give out binaries willy-nilly for free, but not tell anyone the source I used to generate them.
Yeah, the only way I can tell someone is blind or seeing is by asking him. There are no other tests possible, none other even conceivable, by which I might tell if someone were blind or not. This is why babies are never diagnosed as blind until they have first acquired speaking skills, the distinction being necessarily unobservable previously.
What exactly is an "official" source of information?
Hm. I have some time to kill...
..., n-1} is called n]
...} is generated at stage omega, as is the infinite set {0, 1, 2, ...}.
...}} is generated here.
In my view, the answer to your first question is "No, not directly; the paradox in the grandparent post isn't formulated in terms of sets, and so the von Neumann universe/ZFC approach of rejecting the sethood of certain collections wouldn't be of any use, though perhaps something analogous could be stretched to fit.". But for this to make sense, I suppose I need to reply to your second question:
The idea of the von Neumann universe and ZFC is that not all collections of objects are sets; rather, the only collections that are sets are those that can be built up from previously generated sets, eventually bottoming out at the empty set. Specifically, the von Neumann universe generates sets in a hierarchy of stages like this:
Stage 0: The empty set [also called {} or 0]
Stage 1: 0 and the set containing 0 [also called {0} or 1]
Stage 2: 0, 1, {1}, and {0, 1} [also called 2; in general, the set {0,
---
Stage n: Collections whose elements were all around in stage n-1
---
Stage omega: Collections whose elements were all generated in one of stages 0, 1, 2, etc. For example, the infinite set {0, {0}, {{0}}, {{{0}}},
Stage omega+1: Collections whose elements were all around in stage omega. For example, the set {{0, 1, 2,
---And so on, through omega+2, omega+3, omega+omega, omega*3, omega^2, omega^omega, omega^(omega^omega), and all sorts of crazy higher up stages. At each stage, the associated collections are those which only have elements from previous stages. Certain collections never get generated in this process (for example, the collection of all sets never gets generated, because there is no stage that comes after all stages; similarly for Russell's paradoxical set).
Of course, the question remains "What exactly are the stages? How long do they go on for?". Well, this is tricky. The stages are called the ordinal numbers, and the von Neumann hierarchy doesn't in itself really clarify how long they go on for. The intuition is that every sequence of ordinals (even infinite ones) should be followed by an even greater ordinal. However, using this idea naively results in a contradiction known as the Burali-Forti paradox, which isn't any nicer than Russell's paradox (specifically, the sequence of all ordinals would have to be followed by an ordinal which would be greater than itself!). So, the usual fix is to once again reject the use of arbitrary collections/sequences; rather than saying that EVERY collection of ordinals has a follower, we only state that certain collections have followers (for example, every finite collection of ordinals has a follower; every collection of ordinals that can be put in one-to-one correspondence with a set generated in one of their stages has a follower; etc.).
The axioms of ZFC capture ideas from the von Neumann hierarchy along with a particular bunch of useful closure conditions on the ordinals (which map into useful closure conditions on sets), while leaving out all the dangerously powerful closure conditions that are known to lead to contradictions. It's strong enough to do all the mathematics that core mathematicians normally do (indeed, it's actually rather ridiculously overpowered for this) while at the same time being weak enough that no one knows how to prove a contradiction from it.
To summarize, the von Neumann hierarchy and ZFC avoid Russell's paradox by not adopting the principle that arbitrary collections of sets are themselves sets; rather, they only adopt the weaker idea that sets constructed iteratively from previous sets exist. This prevents the various known paradoxical sets from being constructed. As for whether this "solves" Russell's paradox, or merely highlights it by sidestepping it, can be debated. (Why shouldn't we be able to consider arbitrary collections
If you look back at the post that started this whole mess, you'll see the sentence "Maybe you're part of the one or two percent that will go on to do something interesting." Hope this helps.
Yeah, that "duh" tag really grates on me; it seems like half the articles here get tagged with it, often in situations just as dubious as this one. It's interesting the way the tagging system causes people to rally around certain words, though; I wouldn't have predicted, before it was implemented, that "duh" in particular would grow to be a particular annoyance for me, but it's become entrenched now.
Actually, I recall that Wargasm's title had little to do with the game (beyond the "War" part, anyway). To quote IGN: "Apparently someone at DID's marketing department thought that the 'kids' might dig it if they took their latest action title, put a woman in a bulging flak jacket on the cover (maybe she's carrying a bunch of grenades) and named it after something sexual. The PR staff picked up on the Beavis and Butthead vibe and began sending us sheaves of mail with jokes like 'It's time for Wargasm' and 'Multiple Wargasm.' After this ridiculous blitzkrieg of banality (which must have humiliated the actual design team beyond measure) Tal, Jason and I began furiously scrapping over who was going to have to review a game that was sure to be as embarrassing on the inside as it was on the out. I lost (for those of you who keep track of such things, let me warn you that Jason carries brass knuckles) and sadly loaded the game only to find that there was nothing within the actual software that had anything to do with the title, the chick on the cover or anything else we received in the mail. What I did find was a solid action game that is surprisingly hard to put down." So the name clearly turned people off to the game who might otherwise have enjoyed it. That's about as bad as a name can be.
Nonsense! Without the rules given in school, how would we know how to speak properly? It's not like we learn these skills anywhere else. People like you are simply hastening the degrade of our ability to communicate. As for the grandparent, he would've been correct to reword his sentence as "I'm as much of a gamer as most people are, but honestly, towards whom the hell are they marketing this?" Doesn't exactly roll of the tongue, but that's the price one pays for the smug sense of self-satisfaction which comes with being "correct". [Disclaimer/Translation: I agree; it's bizarre that so many people buy into such a contrived "rule".]
Yeah, he's willing to pay up to 500 bucks, and so getting it at 475 or 150 is good to him. It is. He prefers it to not getting the item at all. But don't ignore the fact that getting it at 150 remains better than getting it at 475, and getting it at 100 remains better still. A strategy that will probably get the item at 100, then, is better than one which will probably get it at 475, even if both strategies are better than not entering the auction at all. "Winning" an auction may be about getting the item for an amount one is willing to pay. But there remain degrees of winning, some better than others.
I don't think he was implying that those were Latin words. Rather, his point was that people get all worked up over using "correct" plurals for Latin words, and yet, oddly, don't bother doing anything similar for words from other languages. People will say "virii" (which isn't correct in any language) and "penes" and so forth, but will never bother chiding others for saying "saunas" rather than "saunat", or what have you (in the English-speaking world, of course).
I've never understood all the hate for Super Mario 2. It may not have been a "real" Mario game, but Miyamoto was actually quite involved with it, and it was a damn good game. What aspect of it causes you to call it "teh suck"?
Oh, and \aleph_0^{\aleph_0} > \aleph_0, which is the important thing...
The number of mappings from N to N is \aleph_0^{\aleph_0}, not \aleph_0^2 = \aleph_0 * \aleph_0. After all, the number of sets of natural numbers is uncountable, right? Well, every set can be represented by a function from natural numbers to either 0 or 1 depending on membership in the set. So, clearly, the number of mappings from N to N must be uncountable as well.
Well, I disagree with your assertions about "meaning" and what counts as math, but as far as your core idea goes, in a certain sense, you're correct: Even though you can prove the existence of uncountable sets in ZFC, there, counterintuitively, exist countable models of ZFC. Look up the Lowenheim-Skolem theorem (or "Skolem's Paradox") for more information.
Well, that second 'i' isn't put there at all in the U.S. Not that I'm taking sides on this...
I don't know much about Slate, or Smalltalk, or anything, for that matter, but a quick look at the article shows that both of your "confusing" examples are not that confusing...
As someone else already pointed out, "3 + 4 * 5" ==> 35 (not 23) because Slate has no differences in binary operator precedence and therefore always associates to the left.
And as for "(3/4) == (3/4)" ==> false, this is because "==" tests for _object_ equality, and the two different instances of "(3/4)", while representing the same numerical value, are distinct objects. It appears that "=" tests for numerical equality.
In the paper they wrote, they claim that Grover's algorithm provides an exponential speed up over classical search algorithms. If I'm not mistaken, Grover's algorithm takes time O(N^(1/2)) while classical search algorithms take time O(N), which is only a quadratic speedup, not an exponential one... I'm sure the rest of their paper is well done, but this bothers me anyway.
I thought that might have been it, but I wasn't sure if perpetual motion itself was outlawed, or just devices which output usable energy. Thanks for setting me straight.
Why isn't the Earth's revolution around the sun an example of a perpetual motion machine? I'm sure it isn't, but I can't see why not.