Calm down. In this context, "in anger" is used to denote something that is a deliberate act of war, to distinguish it from tests, accidents, and the like. I'm sure you'll agree that the atomic bombings at the end of WWII were acts of war.
The only two times a nuclear bomb was used in anger, they were both used to prepare the way for the surrender and occupation of the target. Until and unless some evidence presents itself to the contrary, I will have to say that you are wrong.
Likewise, every time we have a science story on slashdot, there's a hundred Monday-morning quarterbacks who know nothing about the field but feel free to criticize the scientists' approach anyway.
As far as I know, no Office bug has ever killed or threatened to kill anybody, so this would not apply. Non-lethal bugfixes are not obligatory in any way, and they may be provided at the publisher's discretion.
X-Plane can be made to send pretty much any data in the simulator through UDP packets, and it can accept pretty much any commands the user can give (and then some) via UDP. For whatever bits the UDP interface misses, plugins can take over, but I don't even think you'd need to go that far for something like a motion platform or custom hardware controls. I dunno how you get easier to hack than that.
Wow, that Netherlands case seems to be going too far in the other direction. Certainly, multimillion dollar rewards are going way too far, but the equivalent of a couple of months' wages for extreme negligence resulting in permanent physical handicap is just ridiculous.
When you buy a product, that's all you buy. You're not buying the product plus a lifetime right to patches. The software company provides those patches at its convenience, as a service to its customers. If it wanted to restrict the availability of patches to people whose last names when converted to ASCII sum to an odd number, they're entirely within their rights. You have no right to download any patches they don't want you to.
If you're referring to the Mac version of Office, that's a different product and would have a different EULA. If you're referring to Virtual PC, that's running a real copy of Windows, and it's made by Microsoft anyway.
feel free to correct me here, but last time i knew, water wasnt the greatest gamma rad stopper
What are you trying to shield from gammas for? Are you trying to survive a gamma-ray burster or something?
Also, what makes you think that the extremely thin walls on the Shuttle or ISS do any better of a job at blocking anything than this will do? The current state of the art in space radiation protection is basically, "we don't stay up too long and we're inside the magnetosphere, so we'll just grin and bear it".
Thanks for the info. It certainly speaks well for your cause that you're willing to check this out after some bozo like me suggests a random possibility.
Are birth defects linked to the quality of prenatal care and nutrition? I ask because that graph seems to start climbing right around the time when the UN sanctions really started to hit the country hard. I don't know if this would be true, but it's a possibility; correlation does not imply causation, after all.
Interesting you should mention that. Bungie used to be a big fan of this technique, which they used in all of their Marathon and Myth games. The game basically just recorded the player's actions, and saved this record into the file when you hit save. Restore was a matter of loading the level and then replaying the save file until the end. How they managed not to break save files every time they released a new version, I will never know.
One big advantage of this approach is that you can use the exact same code to create game films. Create a file the exact same way, but display the results when you play it back, and presto! Bungie did this for Marathon and Myth as well.
The big disadvantage is that you have to be absolutely sure that the game will play back exactly the same way every time. As far as I know, the Aleph One open-source version of the Marathon engine has never, and probably will never, read normal Marathon saved games or films because of subtle differences in how the engines operate.
This is going to sound funny, but... Goody! That means that I can define a new set, the squissh set!;) It is the set of natural numbers plus aleph null. There is such a set already. And it has the properties that I have been describing. Although, it would be far more practical to simply include the statement 'except where are equal to aleph null' in each of those cases you gave.
Yes, it would. In fact, you can do a lot of useful things in graph theory by defining a semiring with the "usual" operations over the the reals union plus and minus infinity. However, you have to be very careful about what operations you allow and how they work.
"Hmm, I thought 1/infinity wasn't zero, but now you say it is." "If you can't come up with a consistent answer, then your theory is flawed." Yes, I see my error here. I have been inconsistent somewhere, and I will now correct it. I have been using the term 'equals' (=) where I should have been using the term 'limit'.
Goodness, I hope you're not this imprecise in your engineering designs.
The limit of this: R + N*(1/I) , as I -> infinity, is R, where R=a real number, N=a finite integer.
If you patch my previous statements with this, they are now consistent.
Except they aren't. You said that the largest rational in (0,1) was 1 - 1/infinity, but by your definition this is equal to 1 and is outside the range.
There are lots of examples in math and science where ignoring some small problem makes many things much easier, but including them vastly complicates things, even to the point of making them intractable with the current tools/understanding. But if you try to put those small problems back in and deal with them, while it complicates things greatly, you can gain new insight that could not be reached any other way. And sometimes you have to leave those problems out until you have developed the proper tools to deal with them, in this case, tools to deal with infinity, and infinitesimals. Like the limit. (which stupid me, I forgot to use! It has been years since I touched math...)
As I stated above, adding infinity to one of these sets can produce useful results. However, by definition as generally accepted by the mathematical world, infinity is not part of these sets.
"Back to the original question: do you accept that we can have a type which does not have a largest possible value?"
Sure, I just don't see how the set of natural numbers is one of these. Since we have 'numbers' like infinity that we can use, at any rate. Take away that concept, and then yea, there is no largest natural number.
For the love of God... didn't we just spend a day or two showing that "natural numbers" does not include infinity? If you want a set that is "natural numbers plus infinity", by all means go ahead, but call it something else!
In some way, the equations infinity + 1 = infinity, and 1/infinity = 0 are wrong. It is just that we can usually ignore the rounding error.
This is not computation, this is math, there is no rounding error. If you allow me to add 1 to infinity, then the result must either equal infinity or not make any sense. Here's a brief outline of why:
infinity + 1 > infinity -- This makes no sense, because infinity is defined as being something which is greater than all the other members of the set, but now we just found one that's even greater.
infinity + 1 < infinity -- If you add 1 and get a smaller number, something is seriously wrong.
infinity + 1 = infinity -- This is at least somewhat self-consistent.
The proper answer is to disallow the operations that let you prove things like 1 = 0. For example, saying that infinity + 1 = infinity is fine, and even intuitive. However, we should not be allowed to say that infinity - infinity = 0. Similarly, saying that 1/infinity = 0 is ok, but infinity/infinity = 1 is not. Or you can simp
"Yours is flawed because it starts with the assumption that it is."
Yes, I thought I made this clear in my post. Whatever. Both proofs are nothing more than mathematical tautologies. All that they do tell us is that, as far as this proof is conscerned, aleph null can be either a natural number or not, both are consistent.
I thought you were claiming that this is how things are, when in fact this is not how things are.
Aleph null is not a natural number, this is in the definition.
If we change the definitions of vegtable, fruit, and mineral so that the tomato is now a vegtable, or a mineral, does this change the nature of the tomato?
That's not what you're doing. You're changing the definition of "tomato" to include all fruit currently accepted as "tomato", plus one particular type of pumpkin.
Using this definition of 'natural number' only changes the names that I have to use to describe the result. Can I use the term 'the squisshes' again? What, if any practical differences would there be to defining natural numbers as a_n = a_n-1 + 1 with a_1 = 1 AND b_n+1 = b_n - 1 with b_1 = (aleph null) ? (In general, would this break anything?)
Yes, it would break things. There are a lot of properties that are true for the natural numbers as they are known and defined by mathematicians everywhere that would not be true for the natural numbers plus aleph null. Minor things like associativity, negation, etc. would be completely destroyed.
(a + b) + c = a + (b + c) -- let a = infinity, b > 0 (inf + b) + c = inf + (b + c) -- by your def. of infinity, inf + b = inf inf + c = inf + (b + c) -- subtract a from both sides c = b + c -- false
These operations are all perfectly legitimate on the naturals, but adding "infinity" to the mix lets me come up with totally bogus statements. Here's another one:
1 = 1 -- add a to both sides a + 1 = a + 1 -- let a = infinity inf + 1 = inf + 1 -- inf + 1 = inf inf = inf + 1 -- put a back a = a + 1 -- subtract a 0 = 1
Do you really want to use a mathematical theory which allows me to prove that 0 = 1?
"1/(aleph null) = 0" Not always true. Choose a random real number on the interval (0,1) inclusive or exclusive, it doesn't matter. What is the probability of choosing any particular number in this interval? 1/infinity. If this equals exactly zero, then the sum of all the probabilities is also zero, which means that it is impossible to choose any number. But you can, so 1/infinity cannot equal exactly zero.
I thought you said you were an engineer. Didn't they make you take calculus? Do you remember any of it? This argument hasn't made any sense in mathematics since the 1600s.
R + N*(1/infinity) = R
Hmm, I thought 1/infinity wasn't zero, but now you say it is. This kind of inconsistency is what happens when you toss around these terms without thinking them through the way you're doing, and it's what mathematicians strive to avoid.
R + N*(1/infinity) = R -- subtract R N*(1/infinity) = 0 -- divide by N 1/infinity = 0
If you can't come up with a consistent answer, then your theory is flawed.
Back to the original question: do you accept that we can have a type which does not have a largest possible value?
I wasn't paying attention and copied a raw , which slashdot decided was part of an HTML tag and so destroyed a bit of the message.
It only killed some quoted material which wasn't extremely necessary anyway, and caused some of my text to be italicized. I hope it's still understandable.
Your simple proof for the naturals is flawed, because it starts with the assumptin that aleph null is not a natural number.
Yours is flawed because it starts with the assumption that it is.
It should come as no surprise that you conclude that it is not a natural number. Simple proof of this: infinity + 1 = infinity. Aleph null = infinity. (one of them anyway) So, if we start with the assumption that aleph null is a natural number:
"First, we observe that for every natural number x, the following property is always true: x
Maybe you won't believe me, but hopefully you'll believe Wolfram Research:
[The positive integers] are the solution to the simple linear recurrance equation a_n = a_n-1 + 1 with a_1 = 1.
There you have it. The property above isn't even an observation, it's part of the definition of "positive integer", which is the same as "natural number".
If we assume that aleph null is a natural number, the largest rational on the interval (0,1) exclusive is 1 - 1/(aleph null)
This doesn't work even if we allow you to use aleph null in the denominator and do arithmetic on it, because 1/(aleph null) = 0, and so 1 - 1/(aleph null) = 1, and it is not inside the desired interval.
Any rational number in (0,1) must be less than 1 by a finite amount.
"The number of natural numbers is called aleph null. It is not a natural number."
A better proof would be welcome.
Hopefully proof by definition is acceptable to you.
"In other words, you have some of the vocabulary, but none of the knowledge."
Yup. Would I be calling aleph null squissh if that wasn't the case? knowledge != inteligence, just facts.
In this case, you're missing some really fundamental things, like "it's possible to have an infinite set which has no largest member", and we're spending ages just trying to convince you that it's possible and true for the naturals, when this has been established mathematical fact for hundreds of years.
I don't see where you get off saying that I'm "wrong" and that I'm making "bogus legal arguments" when I state very clearly in my post that I'm just talking about how I think things should work, and not how they actually do work. It's good to both read and comprehend the entirety of a post before you respond to it with crap.
"You are clearly a wannabe math guy who didn't actually pay attention to his courses."
No. I am an engineer who had a math major for a roommate. I don't care much for theory if it does not apply to the problem at hand. Most of what you have said, while it may be correct from a mathematician's point of view, is simply irrelevant.
Ok, so I was wrong. You simply can talk the talk, but you can't walk the walk. In other words, you have some of the vocabulary, but none of the knowledge.
The problem at hand is to find some way to find a number larger than the median in an array of numbers in less time than it takes to look at all of the numbers. The answer is simple, the largest possible number.
Such a thing does not exist in the naturals, integers, rationals, or reals. Here's a simple proof for the naturals:
First, we observe that for every natural number x, the following property is always true: x < x+1.
Let x be the largest natural number. This means that for all natural numbers y where y != x, y < x. Now, let z = x + 1. By the above property, x < z. Contradiction. Thus x is not the largest natural number, and such a number does not exist.
This is an extremely basic proof, and anybody who is even vaguely mathematician-like will verify its truth for you.
Infinity is large enough to satisfy the problem in all cases.
This is true. However, infinity is not a natural number.
While I may be wrong in calling the number of natural numbers 'countably intinite', the name is irrelevant. Since I do not know it's proper name, I'll call it squissh:^) What is it? well, I'll start counting them. Starting with 1.
1.. #1
2.. #2
3.. #3
Hmmm. see a pattern? I do.
I see a pattern you do not:
x.. #x x+1.. #x+1
This pattern is true for every x. There is always a larger number, and the list never ends.
The number of natural numbers is the same as the largest natural number, squissh. Which means that squissh is also a natural number, and it is larger than all other natural numbers.
False. The number of natural numbers is called aleph null. It is not a natural number.
So if I have an array of arbitrarily large natural numbers, and can represent those numbers, then squissh is equal to or greater than their median.
Squissh is strictly greater than their median, and every member of the array, because squissh is greater than every natural number.
And since I can represent any natural number, I can also represent squissh. Done.
False, as demonstrated above.
"What is _MAX for the naturals?" Squissh. If the computer has a variable type that can represent the naturals, then it can represent this number, whatever it is. If this variable type cannot represent this number, then it cannot represent all naturals, only a subset of them. And this means that there is a finite largest number that can be in the array, and that number can be represented. Always.
Demonstrably false.
"You can get any natural number by starting from 0 and then adding 1 a finite number of times." " If I allow you to write a string of arbitrary, finite length consisting only of the digits 0-9, you can write any natural number."
This implies that all natural numbers are finite numbers. But the set of natural numbers is infinite, there is an infinite number of them. But I just showed that the number of nautral numbers is also a natural number. Which means that there is at least one infinite natural number.
This simply does not make sense. Just because the quantity is infinite doesn't mean the size is infinite.
For a simple demonstration, take the number of rational numbers betwee
And you can do this. However, the dealer may not agree, in which case your only recourse is to suck it up or leave the game. This is exactly what Orbitz's choices should be: either let people deep link despite their non-binding request not to, or actively prevent people from doing so in a technological (not legal) manner.
I doubt you'd be able to patent it, there's too much prior art.
Just one example, poke around the Apple Store for a bit. Find something cool. Try to send the link to a friend. It won't work for them.
This kind of thing is trivial for a database-backed web site, which both the Apple Store and Orbitz undoubtedly are. They're either being lazy or stupid, or the web monkeys are revolting against management by implementing their ideas in a stupid way.
Certainly! Most MMORPGs do this already, just to name one example. Of course, they will promptly lose 95% of their customers because nobody wants to install some unweildy custom application just to book airline tickets, but they'd be well within their rights to require such a monstrosity.
It's not very hard to block external links even without the Referer header, this header just makes it slightly easier. Removing it won't do very much except force lazy web developers to expend a little more effort to stop customers from visiting their sites.
First, Orbitz should be allowed to do whatever they want. This includes prohibiting deep linking.
Second, Orbitz should not be allowed to legally enforce anything that doesn't have a signed contract behind it.
These two combine to form what I think is the correct set of possibilities. Orbitz can yell and scream as much as they like. If they really want to, they can even implement technical restrictions. It's almost trivial to prevent deep linking with a technical solution; embed a unique, expiring identifier into every URL. If you want to get really draconian, tie it to the incoming IP address, or a cookie, etc. Problem solved. What they cannot do is allow deep linking on a technical level but then sue people for doing it.
I believe that disallowing people from sending a correct request to your web server and retrieving a correct response is utterly bogus and the law should come down on the side of the requester.
However, I don't know how the law actually is, I just think that's how it should be.
Personally, my objections to a national ID card have nothing to do with identity theft. Having a national ID system makes it likely that the national ID will be required for all sorts of inappropriate things. Will I have to show my national ID to mail a package, buy sports tickets with a credit card, book a train ride, vote, etc.? Once that happens, having a national ID card will be mandatory, whether explicitly or implicitly. Once having and showing an ID card is mandatory, anonymity goes out the window. I believe (and entities like the Supreme Court agree) that anonymity is essential to practicing my fundamental rights.
Slashdot Mirror
Calm down. In this context, "in anger" is used to denote something that is a deliberate act of war, to distinguish it from tests, accidents, and the like. I'm sure you'll agree that the atomic bombings at the end of WWII were acts of war.
The only two times a nuclear bomb was used in anger, they were both used to prepare the way for the surrender and occupation of the target. Until and unless some evidence presents itself to the contrary, I will have to say that you are wrong.
Likewise, every time we have a science story on slashdot, there's a hundred Monday-morning quarterbacks who know nothing about the field but feel free to criticize the scientists' approach anyway.
As far as I know, no Office bug has ever killed or threatened to kill anybody, so this would not apply. Non-lethal bugfixes are not obligatory in any way, and they may be provided at the publisher's discretion.
X-Plane can be made to send pretty much any data in the simulator through UDP packets, and it can accept pretty much any commands the user can give (and then some) via UDP. For whatever bits the UDP interface misses, plugins can take over, but I don't even think you'd need to go that far for something like a motion platform or custom hardware controls. I dunno how you get easier to hack than that.
The HTTP request is never going to get made if the browser never gets a DNS response.
Wow, that Netherlands case seems to be going too far in the other direction. Certainly, multimillion dollar rewards are going way too far, but the equivalent of a couple of months' wages for extreme negligence resulting in permanent physical handicap is just ridiculous.
You have this almost exactly backwards.
When you buy a product, that's all you buy. You're not buying the product plus a lifetime right to patches. The software company provides those patches at its convenience, as a service to its customers. If it wanted to restrict the availability of patches to people whose last names when converted to ASCII sum to an odd number, they're entirely within their rights. You have no right to download any patches they don't want you to.
If you're referring to the Mac version of Office, that's a different product and would have a different EULA. If you're referring to Virtual PC, that's running a real copy of Windows, and it's made by Microsoft anyway.
feel free to correct me here, but last time i knew, water wasnt the greatest gamma rad stopper
What are you trying to shield from gammas for? Are you trying to survive a gamma-ray burster or something?
Also, what makes you think that the extremely thin walls on the Shuttle or ISS do any better of a job at blocking anything than this will do? The current state of the art in space radiation protection is basically, "we don't stay up too long and we're inside the magnetosphere, so we'll just grin and bear it".
Thanks for the info. It certainly speaks well for your cause that you're willing to check this out after some bozo like me suggests a random possibility.
Are birth defects linked to the quality of prenatal care and nutrition? I ask because that graph seems to start climbing right around the time when the UN sanctions really started to hit the country hard. I don't know if this would be true, but it's a possibility; correlation does not imply causation, after all.
Interesting you should mention that. Bungie used to be a big fan of this technique, which they used in all of their Marathon and Myth games. The game basically just recorded the player's actions, and saved this record into the file when you hit save. Restore was a matter of loading the level and then replaying the save file until the end. How they managed not to break save files every time they released a new version, I will never know.
One big advantage of this approach is that you can use the exact same code to create game films. Create a file the exact same way, but display the results when you play it back, and presto! Bungie did this for Marathon and Myth as well.
The big disadvantage is that you have to be absolutely sure that the game will play back exactly the same way every time. As far as I know, the Aleph One open-source version of the Marathon engine has never, and probably will never, read normal Marathon saved games or films because of subtle differences in how the engines operate.
"Yes, it would break things."
;) It is the set of natural numbers plus aleph null. There is such a set already. And it has the properties that I have been describing. Although, it would be far more practical to simply include the statement 'except where are equal to aleph null' in each of those cases you gave.
This is going to sound funny, but... Goody! That means that I can define a new set, the squissh set!
Yes, it would. In fact, you can do a lot of useful things in graph theory by defining a semiring with the "usual" operations over the the reals union plus and minus infinity. However, you have to be very careful about what operations you allow and how they work.
"Hmm, I thought 1/infinity wasn't zero, but now you say it is." "If you can't come up with a consistent answer, then your theory is flawed." Yes, I see my error here. I have been inconsistent somewhere, and I will now correct it. I have been using the term 'equals' (=) where I should have been using the term 'limit'.
Goodness, I hope you're not this imprecise in your engineering designs.
The limit of this: R + N*(1/I) , as I -> infinity, is R, where R=a real number, N=a finite integer.
If you patch my previous statements with this, they are now consistent.
Except they aren't. You said that the largest rational in (0,1) was 1 - 1/infinity, but by your definition this is equal to 1 and is outside the range.
There are lots of examples in math and science where ignoring some small problem makes many things much easier, but including them vastly complicates things, even to the point of making them intractable with the current tools/understanding. But if you try to put those small problems back in and deal with them, while it complicates things greatly, you can gain new insight that could not be reached any other way. And sometimes you have to leave those problems out until you have developed the proper tools to deal with them, in this case, tools to deal with infinity, and infinitesimals. Like the limit. (which stupid me, I forgot to use! It has been years since I touched math...)
As I stated above, adding infinity to one of these sets can produce useful results. However, by definition as generally accepted by the mathematical world, infinity is not part of these sets.
"Back to the original question: do you accept that we can have a type which does not have a largest possible value?"
Sure, I just don't see how the set of natural numbers is one of these. Since we have 'numbers' like infinity that we can use, at any rate. Take away that concept, and then yea, there is no largest natural number.
For the love of God... didn't we just spend a day or two showing that "natural numbers" does not include infinity? If you want a set that is "natural numbers plus infinity", by all means go ahead, but call it something else!
In some way, the equations infinity + 1 = infinity, and 1/infinity = 0 are wrong. It is just that we can usually ignore the rounding error.
This is not computation, this is math, there is no rounding error. If you allow me to add 1 to infinity, then the result must either equal infinity or not make any sense. Here's a brief outline of why:
infinity + 1 > infinity -- This makes no sense, because infinity is defined as being something which is greater than all the other members of the set, but now we just found one that's even greater.
infinity + 1 < infinity -- If you add 1 and get a smaller number, something is seriously wrong.
infinity + 1 = infinity -- This is at least somewhat self-consistent.
The proper answer is to disallow the operations that let you prove things like 1 = 0. For example, saying that infinity + 1 = infinity is fine, and even intuitive. However, we should not be allowed to say that infinity - infinity = 0. Similarly, saying that 1/infinity = 0 is ok, but infinity/infinity = 1 is not. Or you can simp
"Yours is flawed because it starts with the assumption that it is."
Yes, I thought I made this clear in my post. Whatever. Both proofs are nothing more than mathematical tautologies. All that they do tell us is that, as far as this proof is conscerned, aleph null can be either a natural number or not, both are consistent.
I thought you were claiming that this is how things are, when in fact this is not how things are.
Aleph null is not a natural number, this is in the definition.
If we change the definitions of vegtable, fruit, and mineral so that the tomato is now a vegtable, or a mineral, does this change the nature of the tomato?
That's not what you're doing. You're changing the definition of "tomato" to include all fruit currently accepted as "tomato", plus one particular type of pumpkin.
Using this definition of 'natural number' only changes the names that I have to use to describe the result. Can I use the term 'the squisshes' again? What, if any practical differences would there be to defining natural numbers as a_n = a_n-1 + 1 with a_1 = 1 AND b_n+1 = b_n - 1 with b_1 = (aleph null) ? (In general, would this break anything?)
Yes, it would break things. There are a lot of properties that are true for the natural numbers as they are known and defined by mathematicians everywhere that would not be true for the natural numbers plus aleph null. Minor things like associativity, negation, etc. would be completely destroyed.
(a + b) + c = a + (b + c) -- let a = infinity, b > 0
(inf + b) + c = inf + (b + c) -- by your def. of infinity, inf + b = inf
inf + c = inf + (b + c) -- subtract a from both sides
c = b + c -- false
These operations are all perfectly legitimate on the naturals, but adding "infinity" to the mix lets me come up with totally bogus statements. Here's another one:
1 = 1 -- add a to both sides
a + 1 = a + 1 -- let a = infinity
inf + 1 = inf + 1 -- inf + 1 = inf
inf = inf + 1 -- put a back
a = a + 1 -- subtract a
0 = 1
Do you really want to use a mathematical theory which allows me to prove that 0 = 1?
"1/(aleph null) = 0" Not always true. Choose a random real number on the interval (0,1) inclusive or exclusive, it doesn't matter. What is the probability of choosing any particular number in this interval? 1/infinity. If this equals exactly zero, then the sum of all the probabilities is also zero, which means that it is impossible to choose any number. But you can, so 1/infinity cannot equal exactly zero.
I thought you said you were an engineer. Didn't they make you take calculus? Do you remember any of it? This argument hasn't made any sense in mathematics since the 1600s.
R + N*(1/infinity) = R
Hmm, I thought 1/infinity wasn't zero, but now you say it is. This kind of inconsistency is what happens when you toss around these terms without thinking them through the way you're doing, and it's what mathematicians strive to avoid.
R + N*(1/infinity) = R -- subtract R
N*(1/infinity) = 0 -- divide by N
1/infinity = 0
If you can't come up with a consistent answer, then your theory is flawed.
Back to the original question: do you accept that we can have a type which does not have a largest possible value?
I wasn't paying attention and copied a raw , which slashdot decided was part of an HTML tag and so destroyed a bit of the message.
It only killed some quoted material which wasn't extremely necessary anyway, and caused some of my text to be italicized. I hope it's still understandable.
Yours is flawed because it starts with the assumption that it is.
It should come as no surprise that you conclude that it is not a natural number. Simple proof of this: infinity + 1 = infinity. Aleph null = infinity. (one of them anyway) So, if we start with the assumption that aleph null is a natural number:
"First, we observe that for every natural number x, the following property is always true: x
Maybe you won't believe me, but hopefully you'll believe Wolfram Research: There you have it. The property above isn't even an observation, it's part of the definition of "positive integer", which is the same as "natural number".
Princeton has a very similar definition in WordNet.
Ditto for some site I've never heard of.
If we assume that aleph null is a natural number, the largest rational on the interval (0,1) exclusive is 1 - 1/(aleph null)
This doesn't work even if we allow you to use aleph null in the denominator and do arithmetic on it, because 1/(aleph null) = 0, and so 1 - 1/(aleph null) = 1, and it is not inside the desired interval.
Any rational number in (0,1) must be less than 1 by a finite amount.
"The number of natural numbers is called aleph null. It is not a natural number."
A better proof would be welcome.
Hopefully proof by definition is acceptable to you.
"In other words, you have some of the vocabulary, but none of the knowledge."
Yup. Would I be calling aleph null squissh if that wasn't the case? knowledge != inteligence, just facts.
In this case, you're missing some really fundamental things, like "it's possible to have an infinite set which has no largest member", and we're spending ages just trying to convince you that it's possible and true for the naturals, when this has been established mathematical fact for hundreds of years.
I don't see where you get off saying that I'm "wrong" and that I'm making "bogus legal arguments" when I state very clearly in my post that I'm just talking about how I think things should work, and not how they actually do work. It's good to both read and comprehend the entirety of a post before you respond to it with crap.
No. I am an engineer who had a math major for a roommate. I don't care much for theory if it does not apply to the problem at hand. Most of what you have said, while it may be correct from a mathematician's point of view, is simply irrelevant.
Ok, so I was wrong. You simply can talk the talk, but you can't walk the walk. In other words, you have some of the vocabulary, but none of the knowledge.
The problem at hand is to find some way to find a number larger than the median in an array of numbers in less time than it takes to look at all of the numbers. The answer is simple, the largest possible number.
Such a thing does not exist in the naturals, integers, rationals, or reals. Here's a simple proof for the naturals:
This is an extremely basic proof, and anybody who is even vaguely mathematician-like will verify its truth for you.
:^) What is it? well, I'll start counting them. Starting with 1.
.. #1
.. #2
.. #3
.. #x .. #x+1
Infinity is large enough to satisfy the problem in all cases.
This is true. However, infinity is not a natural number.
While I may be wrong in calling the number of natural numbers 'countably intinite', the name is irrelevant. Since I do not know it's proper name, I'll call it squissh
1
2
3
Hmmm. see a pattern? I do.
I see a pattern you do not:
x
x+1
This pattern is true for every x. There is always a larger number, and the list never ends.
The number of natural numbers is the same as the largest natural number, squissh. Which means that squissh is also a natural number, and it is larger than all other natural numbers.
False. The number of natural numbers is called aleph null. It is not a natural number.
So if I have an array of arbitrarily large natural numbers, and can represent those numbers, then squissh is equal to or greater than their median.
Squissh is strictly greater than their median, and every member of the array, because squissh is greater than every natural number.
And since I can represent any natural number, I can also represent squissh. Done.
False, as demonstrated above.
"What is _MAX for the naturals?" Squissh. If the computer has a variable type that can represent the naturals, then it can represent this number, whatever it is. If this variable type cannot represent this number, then it cannot represent all naturals, only a subset of them. And this means that there is a finite largest number that can be in the array, and that number can be represented. Always.
Demonstrably false.
"You can get any natural number by starting from 0 and then adding 1 a finite number of times."
" If I allow you to write a string of arbitrary, finite length consisting only of the digits 0-9, you can write any natural number."
This implies that all natural numbers are finite numbers. But the set of natural numbers is infinite, there is an infinite number of them. But I just showed that the number of nautral numbers is also a natural number. Which means that there is at least one infinite natural number.
This simply does not make sense. Just because the quantity is infinite doesn't mean the size is infinite.
For a simple demonstration, take the number of rational numbers betwee
And you can do this. However, the dealer may not agree, in which case your only recourse is to suck it up or leave the game. This is exactly what Orbitz's choices should be: either let people deep link despite their non-binding request not to, or actively prevent people from doing so in a technological (not legal) manner.
I doubt you'd be able to patent it, there's too much prior art.
Just one example, poke around the Apple Store for a bit. Find something cool. Try to send the link to a friend. It won't work for them.
This kind of thing is trivial for a database-backed web site, which both the Apple Store and Orbitz undoubtedly are. They're either being lazy or stupid, or the web monkeys are revolting against management by implementing their ideas in a stupid way.
Certainly! Most MMORPGs do this already, just to name one example. Of course, they will promptly lose 95% of their customers because nobody wants to install some unweildy custom application just to book airline tickets, but they'd be well within their rights to require such a monstrosity.
It's not very hard to block external links even without the Referer header, this header just makes it slightly easier. Removing it won't do very much except force lazy web developers to expend a little more effort to stop customers from visiting their sites.
Here's my opinion:
First, Orbitz should be allowed to do whatever they want. This includes prohibiting deep linking.
Second, Orbitz should not be allowed to legally enforce anything that doesn't have a signed contract behind it.
These two combine to form what I think is the correct set of possibilities. Orbitz can yell and scream as much as they like. If they really want to, they can even implement technical restrictions. It's almost trivial to prevent deep linking with a technical solution; embed a unique, expiring identifier into every URL. If you want to get really draconian, tie it to the incoming IP address, or a cookie, etc. Problem solved. What they cannot do is allow deep linking on a technical level but then sue people for doing it.
I believe that disallowing people from sending a correct request to your web server and retrieving a correct response is utterly bogus and the law should come down on the side of the requester.
However, I don't know how the law actually is, I just think that's how it should be.
This is wandering off-topic, but what the hey....
Personally, my objections to a national ID card have nothing to do with identity theft. Having a national ID system makes it likely that the national ID will be required for all sorts of inappropriate things. Will I have to show my national ID to mail a package, buy sports tickets with a credit card, book a train ride, vote, etc.? Once that happens, having a national ID card will be mandatory, whether explicitly or implicitly. Once having and showing an ID card is mandatory, anonymity goes out the window. I believe (and entities like the Supreme Court agree) that anonymity is essential to practicing my fundamental rights.