The list of all possible real numbers between 0 and 1 would consist of a zero followed by a decimal point, followed by all possible combinations of the digits 0 through 9. I maintain it is not possible to then 'twiddle' one of the digits and arrive at a combination of the digits 0 through 9 that is not already in the set of "all possible combinations of digits 0 through 9."
Cantor's argument says it is possible to find a combination of digits consisting of the numbers 0 through 9 that is not in "all possible combinations of the digits 0 through 9." I have a logical problem with that.
The thing is that you haven't come up with "all possible combinations of digits 0 through 9." after a decimal point. That *is* the interval [0,1] What you've come up with is a map from the natural numbers 1,2,3.... into [0,1]. That is what enumerating means. Enumerating doesn't necessarilly mean that you have listed all of them.
The trick to the diagonal proof relies completely on the fact that everything you deal with in the proof is countably infinite i.e. it's the "littlest infinity", which, granted, you still haven't agreed that such a thing even makes sense.
But the fact that you have an enumerated list is what allows you to find a real number not on that list. While there are an infinite number of numbers in your list, it's trivial to point out the 10th, the millionth, the 250 to the googol to the googol to the googolth number you have listed since that is what an enumeration is.
So if the initial assumption makes sense then every real number is in your list in a specific place. There are infinitely many, but every one exists at some specific *finite* place in the list. So when you create your new number by changing one digit from each number in the list, it truly does not exist in the list. If it did exist in the list, then it would exist in one and only one specific location in that list. So ask yourself which position in the list could it possibly occupy? Not the first, because it's first digit is different. Not the hundredth, not the trillionth, and so on and so on and so on.
As I have posted before, though, Cantor's method using the diagonal does not prove the uncountability to me, as a paradox is created when you claim you have listed all the numbers then claim you find a new number that is not listed. Both can not be true, so either not all the numbers were listed in the beginning or the number 'found' IS on the original list.
This is almost true, but you don't just claim that you have listed all the numbers, you claim something even stronger. You claim that you have listed them *by number* i.e. the first, the second etc. This is what makes it different. So it's true that you claim to have listed them all in order and then demonstrate that you have found one that isn't in the list. This isn't a parodox though, it's a contradiction. Reaching this contradiction shows you that your initial assumption, that the reals are countable, was false.
To really get this, I think there are 2 things that you really need to understand and accept. One is the idea of proof by contradiction. i.e. assume what you want to prove is not true. If this leads you to a contradiction, then what you assumed is false, and what you originally wanted to prove is therefore true. The second is why the new number that you created can not possibly be in the list. If it were, then there would exist a finite number, n, that it's mapped to by. i.e. it's in bucket # n. But by the very nature of the way the number was constructed, it can't be in bucket n because it's a different number in the nth decimal place. The fact that n could be any natural number at all and it isn't a specific one proves it for all natural numbers, hence there isn't a natural number that maps to our new one.
I guess one more thing that might help is to realise that while any statement proven mathematically *is* absolute truth, it's only absolute truth within a very specific un
Sure you can. Give me any real number between 1 and 2. Write it down on one of these infinite number of pieces of paper. Put that piece of paper in one of these infinite number of buckets. Repeat for all the real number between 1 and 2 without duplicating any.
The bolded sentence is the problem. You need to be able to specify a way to determine the n'th real number you are going to choose. There isn't a way to do this.
For the integers, it's easy
f(1)=0 for n > 1, n even, f(n)=> -1^n(n/2) for n > 1, n odd, f(n)=> -1^n((n-1)/2)
so essentially, you are ordering them in a different way.
1=>0 2=>1 3=>-1 4=>2 5=>-2 etc. this function from the natural numbers to the integers is both one to one (no integer is mapped to by more than one natural) and onto (each integer is mapped to).
Then *by the definition of 2 sets having the same "size"* the set of natural numbers and the set of integers is the same size.
You can do something similar to produce a function from the naturals to the rational numbers that meets those requirements.
For the reals it is different. Not only can't you find a neat and tidy function that you can write down, but you can prove that it isn't possible for one to exist.
More than an infinite number? Interesting concept, but I see a logical problem here.
There is no logical problem, I think the issue is that you don't know the exact way that these words are defined.
It's not like I am saying that there are 5 integers and that there are 10 real numbers. When you say that 2 sets have the same size, you are saying that there exists a 1 to one function from one that maps onto the other. That is the definition.
If you think about, it makes a lot of sense to define it this way. It's your bucket example exactly. If you can put one and only one thing in each bucket, and everything is in a bucket, then intuitively there are the same "number" of things as buckets.
You are mapping between MEMBERS of infinite sets and saying they are different, and I am saying the NUMBER of members in two infinite sets (i.e., mapping both sets to the infinite set of positive integers) is the same - both are exactly equal to the number of positive integers. Two things equal to a third thing are equal to each other.
Being able to map between the members in a one to one correspondence is how you say that 2 sets have the same number of members.
The size of the integers is known, so testing for 2 different sets to have the same size by comparing each to the integers seperately works just fine. The problem is that when you try to do this with the reals, you fail every time.
Sounds like you have information or education I don't have. Can you point me toward this information? reductio ad absurdum is a classical technique, I have noy seen it used in this context, though. Links?
This isn't reductio ad absurdum, it is a proof by contradiction. This works like: Assume that A is true This implies B. This implies C. C is known to be false, i.e. not C is true
So if you can start out assuming that something is true and it leads you to a contradiction, in this case C and not C then you know that your initial assumption is wrong.
I read this to say that we assume that all the digits are either a 2 or a 4, so creating a number whose only digits are 2 or 4, however accomplished, only proves our assumption, and no contradiction that I can see despite your statement that there is one.
The core assumption isn't that the digits are either 2 or 4, it's that you have created a map from the natural numbers to the reals that is one to one and includes every real.
The trick of making a number out of this list by taking one digit from each of the other numbers and changing it gives you a new number which is not in your list. You know it is not in your list
Non-intuitively, if you defined your set as (two rocks per each bin, each rock put into a separate bin) [2*infinity] they would still fit into the existing bins! [2*infinity=infinity]
This is true to a point. How it's actually defined is that 2 sets are the "same size" i.e. your rocks and your buckets if each rock can go into a bucket and no buckets are left empty.
There are an infinite number of decimals between the integer counting numbers (1.1, 1.01, 1.001,...) BUT THEY CAN STILL BE ASSIGNED, one to a bin, TO MY INFINITE NUMBER OF BINS . [infinity * infinity = infinity]
This is where it breaks down. The problem is that when you try and figure out a way to determine how to assign the real numbers to your buckets (heck even try it with just the numbers between zero and one) you realise that you can not do it. Even in theory, it is absolutely impossible to determine a way to do this. There are far too many real numbers.
It kind of (i.e. the pedants will roust me for it but it is true enough for purposes of trying to understand it without getting into the details) works like this:
infinity + infinity=infinity infinity * infinity = infinity infinity ^ infinity (or even just 2^infinity) = biggerinfinity.
The typical way of proving that there does not exist any way to assign all the real number rocks to the natural number buckets is to assume that you can.
So assume that you have a mapping from the natural numbers onto the reals between 0 and 1. Onto means that every real between zero and 1 is mapped to.
each real number will be a decimal number in the form 0.abcdxyz......
Heck, even assume that all of the digits are either 2 or 4. This leaves you intuitively with far less numbers. Certainly no more.
So now, if we can create a real number between zero and one whose only digits are 2 or 4, then we have proven that our list is not complete.
So take the first digit of the number 1 maps to. If it is 2, change it to 4. If it is 4, change it to 2. Go to the second digit of the number mapped to by 2. Do the same thing.
Continue all the way down. Of course it's an infinitely long list, so doing it manually won't work. You can say for every n, if the nth digit of the number mapped to by n is 2 make it 4 else make it 2.
So you have constructed a real number which is between zero and 1 and all of whose digits are either 2 or 4 yet this number is not in your list.
The only assumption that we made is that it is possible to construct a mapping between the natural numbers and this subset of the reals. This assumption alone led to a contradiction, therefore it must be false.
The only problem with your "map" is it is skewed by state (Red/Blue) It you look at a more detailed level you will see that the areas of most population, and quite frankly the poorest areas, would BE THE ONES ON welfare and voted Democrat!. The fact is the greater population of the rest of the state canceled the blue votes out. So you argument is silly.
Actually, your link has nothing to do with economics.
However, it is true that the most populous areas do vote Democratic. The simple fact is that the most populous areas are the richest areas rather than the poorest ones.
Are you seriously trying to say that Las Vegas is the poorest area of Nevada? It is the richest, most populous, and most Democratic area of Nevada. More or less the same thing holds across the entire country. The most populous areas are where the money is made, and they are overwhelmingly Democratic.
To paraphrase words of Michael Jordan, to his mother. "Once you start making money real money, you become a Republican."
Certtainly, the very richest are Republican more often than not, but most of the poorest are as well. The rich ones are "smart" to vote Republican if smart means caring only about themselves, but the majority of the Repbulicans are so poor that they have to rely on the Democrats to pay their bills. The fact that Republican policies help to maintain this shows the lack of understanding required for these people to continue voting against their own interests.
I suppose if I hunted her down and killed her, the resulting story would be "Reporter Murdered by Enraged IT Worker"...
Maybe not. If you were free on bail, were interviewed and made it quite clear to the reporter interviewing you why you did it and your total lack of resistance to doing it again, you might actually get the proper headline.
You're in for a long miserable life if you think that way. "A young conservative has no heart; an old liberal has no brain."
Which is an absolutely false statement.
The plain, simple, undeniable fact is that liberal policies work far better than conservative policies.
Look at the facts. almost all "red" states with the exception of Texas, Colorado, and Nevada are welfare states. That is their economies function solely through the charity of the rest of the country. Specifically, the Liberal states. If you break it down by county, then even within the red states, the republican areas are poorer and the liberal areas are much better off financially.
Now given that conservative policies clearly do not work at all, it doesn't make sense to say that somebody with no brain would vote liberal. Now, the converse of that statement is pretty clearly true. This is, of course, if you actually argue based on the facts rather that merely repeating ignorant quotes from long ago.
Here is a link showing this. If you don't like that one, there are many more around. Alternatively, you could just look at the figures yourself. It doesn't really matter though, since you won't find anything to contradict this.
I don't know if evil is the right word, but there is clearly something very seriously wrong with the way these people think for a number of reasons:
First, they claim to be for individual responsibility, small government and against welfare, yet they survive solely through welfare provided by the blue states.
This is either the ultimate hypocricy or an ignorance so severe that it is mind blowing.
They have outdated oppressive social policies which help keep themselves back, yet they voted in this election primarily based on forcing their ignorant ideas on the part of the nation whose mature responsible attitudes are a large part of what allows them to provide the charity upon which their lives are completely dependent.
So it is clear that they lack the most basic common sense since they maintain contradictory ideas (welfare bad--give us welfare) as if they were even rational, let alone good. They also are trying to drag down the people who are paying their bills.
Once we can no longer afford to pay their bills, who do they think will? That is clearly voting directly against their own best interests, which is evidence of severe flaws in their ability to reason.
Now, when I provide the means for somebody else to survive, I don't expect them to grovel at my feet, but I damn well do expect them not to undermine the very things which allow me to provide them that charity.
So, you might want to update your quote to reflect reality as it does make you look extremely ignorant of the simple basic facts of life.
I, for one, will be doing everything I can to encourage my representatives to end welfare payments to these states. I think it will be very interesting to see how long they claim to believe what they do when they actually have to live up to it as people with integrity do.
I mean think about it. If we made a requirement that states could not receive more in federal money than they put in, then all of the liberal states would be better off and almost all of the conservative states would be bankrupt. And they have the audacity to try and tell us how to run an economy or a society.
I am convinced that the margin of error on this election will be the lowest recorded in American history.
So are you saying that the Dems and Repubs did a more or less equal job of rigging things?
The real issue isn't what happened with this particular election, rather it's that these machines are *known* to be faulty, they are *known* to be easily hackable, and they are *known* to leave no audit trail by design.
* Why do you think Clinton and Bush are for privatizing Social Security? To keep the stock bubble booming!
Partly true -- however, that would still be artificial, and would fail. I think we need to strike a balance between an entirely government based social security and a privatized one.
There are certainly arguments either way, but it has a seemingly insurmountable problem. Curiously, it's essentially the same problem that makes banning guns unworkable IMHO.
How would it be possible to handle the transition? This is something I haven't ever really heard anybody address satisfactorily.
If you switch it all to private right now, what do you do? Divide the total balance by the number of people and mail them a check? Sending everybody back the amount they put in wouldn't be workable even if the scumbags in Washington (yes, all of them) hadn't already looted it because there isn't that much avalable. Has anybody proposed a realistic method?
The gun issue seems really similar to me. Some other countries seem to do ok without guns (save the statistics, I said ok, not better), but that doesn't mean we could outlaw them and magically they'd all disappear. Assume every non criminal citizen went down and turned theirs in (yeah, right). That would be a recipe for disaster.
Similarly with privatizing Social Security. It seems like there are two different more or less equally valid ways of doing something, but choosing one eliminates the possibility of later choosing the other without a major disaster.
You might think we would have learned from prohibition that outlawing substances has a net result of creating guys like Al Capone or Pablo Escobar, as the case may be.
Don't forget Joe Kennedy who owned the trucks that transported a lot of the hooch which is where they made their fortune.
Going from there straight to gay marraige is just to big of a shock for middle America and the reaction will be to do their utmost to instill judges that will reverse the previous decisions.
My real problem is that "middle America" is full of backwards ass people whose own backwards assness is holding them back. They're held back to the point that they need my charity to survive (subsidies, tax imbalance etc.) yet they want to drag the part of the country that actually pulls their own weight down to their level.
I don't give a crap what they want to ban in their backwards ass states. It's when these ignorant fools try and tell the rest of the country what they can do, even though they have proven that they don't really have a clue about what works that I have a problem.
There really isn't a point in trying to convince these people. They claim to be Chriatians, but Jesus himself told them directly not to be such fuckheads about things, yet they reject that in favor of a single passage in the old testament. Where's the lobby for an amendment to ban pork? It has exactly the same status in the bible as the gay thing. Oh yeah, it isn't a moral issue at all.. This vote came down to who hates the same people as me and who will actively work to oppress them. And these fucking cowards have the audacity to wave the flag around.
And while I know you didn't mean it this way, I found your comment about "then they would both have better lives" ironic given that the dead child didn't have a chance to have any life, let alone a better one.
Sure I did. The woman and her child. There wasn't another child in the first place. That is the point.
Now can you guarantee a great life for every person you force into the world at gunpoint? Can you even guarantee them a decent life?
Now here's the real test though. Do you support sex education in schools including the teaching of the proper use of birth control?
Heaven forbid you should prevent people from learning basic facts of biology and how to avoid pregnancy yet punish them *and* their child if it happens.
Even when he was being impeached (a perfect time to attack him, you must admit), the media spent its time saying rude things abut the impeachers, not about Clinton.
I amjust amazed at how you people can just lie out your ass with a straight face. In the first place, this isn't what happened. This was treated as if it were a legitimate grounds for impeachment. It was a fucking blowjob. The whole process destroyed the integrity of our congress and showed the Republicans for exactly what they are: Partisan hacks who care about their party far more than they do about their country. They spent his entire administration trying to nail him for anything and this was al they could come up with.
Now that isn't how the press handlled it. They demonized Clinton as if he had lied to commit mass murder.
Oh, speaking of that, where were the questions before the war? None. The press didn't do a god damned thing and just sold the war.
Now that we all know that it was based completely of falsified evidendce and lies, where is the call for impeachment? Had that been Clinton, he would have been fried over that. As it's Bush, nothing has happened, and due in large part to the way the media has coddled him, the traitor actually got eleted legitimately.
The really fucking sad thing is that according to the polls, most Americans knew he was a lying sack of shit, they knew he is a mass murderer, and they know he's crap for the country. So why did they vote for him? Because he shares their ignorant hatreds and promised to actively work to discriminate against a large segment of our society because he hates who they were born.
That is where our country is right now. Sharing of ignorant hatred is our most important "moral value"
Or do you really believe that your beliefs about Bush were formed entirely from your own morla uprightness and superior reasoning, and had nothing to do with the inputs from any of the media?
I knew before he was elected that he had been a constant failure at everything he had ever done, and that he most likely made his first million threough illegal insider trading. I haven't seen that on Fox or CBS or CNN. I knew before the first election that he had jumped ahead of more deserving people into the Texas Air National Guard and then deserted that post. That was only reported recently, and when it turned out that all the facts were correct, but the exact words were on a different peice of paer, it got dropped and treated as if that were the story rather than the cowardice and treason of the President.
So given that, your assertions are without merit and not even remotely rational.
Look up the numbers. How much do they pay in tax dollars, and how much do they take in from the Fed.
They take a lot more than they give. Simple basic fact.
Farmers are subsidized so food prices can be kept down.
Farmers are subsidized, and additionally food prices are kept artificially high.
How the hell can you even make an insane statement like that? A subsidy would be part of the price of food anyway, hence driving it higher so your statement isn't even remotely possible.
And please the last thing a state like TN needs is a bunch of liberals they can get along just fine without you.
Right. The state is poor as shit and wioth a horrible educational system. Look into where tho state's income comes fom and you will see that a lot of comes in subsidies from the states that actually pull their own weight rather than living off of handouts. If you don't like it, start pulling your own weight, don't make nonsensical statements which a quick search would prove wrong.
Attitudes like this are what makes it all the harder for people who want to help you obtain that goal. You expect understanding of your lifestyle while criticizing that of others.
I happily married to a woman, so it's not my lifestyle I'm talking about.
Regardless, I'm not sure what the problem with my attitude is. I believe that all people are created equal, at least in the sense that the government has no right whatsoever to treat them differently just because of who they were born.
That person claimed that he had the right to shove his religious beliefs down people's throats at the point of a gun (as all laws are). That is extremely ignorant, extremely hateful, and totally unAmerican.
Are you trying to say that abortion is moral and anti-abortion is immoral?
I am certainly not saying that abortion *is* moral, however I am most certainly saying that forcing somebody at the point of a gun to have a child that they do not want and can't afford is extremely immoral. There is only one person who is fit to decide whether or not they are ready to have a child and that is that person. It is not my job nor yours to tell them that they have to. Doing so could well condemn them and that child to poverty whereas if they waited until they were ready then they would both have better lives. That is a decision for them to make.
It's really pathetic to hear you talk about killing innocents when you suupport Bush's lies that have killed tens of thousands. You have no moral leg to stand on. You are hypocritical to even pretend that you have morals given that.
When did you get access to Langley computers? YOU MUST BE GOD, YOU KNOW WHAT IS TRUE AND WHAT IS LIE, HUH? You know better than anyone, it's a fact, you have evidence, empiracle. Go ahead and post it to the CNN site if what you say is true, it's all lies. You sound so sure of yourself, you must have proof.
The CIA flat out told Bush not to give that state of the union speech because the info was bad. He did it anyway telling us it was solid evidence. This is undisputed fact. You really should have been paying attention.
SADDAM KILLED 400,000 PEOPLE BECAUSE HE COULD!! 400,000!!!! WHOS THE FUCKING CRIMINAL HERE? DO YOU HATE IRAQI'S? YOU'D RATHER THAT THEY STILL HAVE A MANAICAL DICK WITH TEETH RULING THEM?
That is completely irrelevant, and yelling won't change that. That was not the justification we were given for going to war. That is the only relevant issue to this discussion. We were told that we were under imminent threat of attack when that was blatantly false. Ditto for WMDs.
Now, given that you're so worked up about the poor innocent Iraqis (well except for the ones we murdered. Your "morals" are so relative it's scary) you, of course, know that the US was complicit in their deaths? You know that we encouraged the Kurds to revolt and told them that we'd back them and then betrayed them allowing Saddam to gas them with the gas that Donald Rumsfeld sold them? Sure. Real moral people you are supporting there.
Really? Then the rural states are not largely subsidized by the rest? So phones, cell phones, electric cables and all of that would actually be affordable even though the cost per person is orders of magnitude higher than in densely populated areas? We don't pay far above market value for food and then pay again to subsidize you and pay farmers not to grow food?
Sure.
It's quite clear that you are clueless.
I am guessing you wouldn't know a hard days work if it slapped you in the face.
Well guess what Sparky? You guessed wrong. I've done construction, dug ditches, worked on a freaking tulip farm, worked on a fishing boat, done carpentry and many other jobs.
That doesn't change the simple fact that the Liberal states largely subsidize the existence of people in the conservative states. If it's hard to deal with, then maybe you should move somewhere where you wouldn't be a burden on the parts of the country that actually pull their own weight. If not, the least you could do is thank us.
I believe Hitler never did anything to people here in America, so maybe we should have not gone after him either.
What a ridiculous comparison. You completely ignore the most relevant part as well. Our entire justification was based on lies. That's bearing false witness and is completely immoral by most decent people's standards.
Somebody was forced at gunpoint to have an a baby?
What else would you call illegalization of abortion? According to the polls that is the primary reason people had for voting for Bush. His supposed "morals". They conveniently ignore the lies and murder. Typical of the hypocritical scum who would actually vote for that traitor.
You don't think you have freedom of religion?
When the religious wackos are trying to put their religion into our lawsand they are actually being treated as if that was even remotely consistent with the basis of this country then of course not. What kind of a ridiculous question is that?
Also, when did Jon Stewart suddenly become so intelligent.
It's partly a joke and partly just showing some respect for his stand on Crossfire which any person who has any interest in real discourse coming back to this country should admire.
Well golly gee whiz mister we don't got the technical stuff you all got but if you don't shut yer trap we'll stop growin yer FOOD you stupid fuck!
We pay far more than market value for the food, and we pay for it twice. Once when we buy it and once when we subsidize you. Additionally, I've lived all over California and in Illinois. Both states grow far more food than they need.
If we didn't pay inflated pices for the food you grow, you would not even have an economy.
You very existence is dependent upon our charity you ignorant dipshit.
Personally I don't give a fuck if you want to live your life according to a hate based faith, but keep that shit that drags your pathetically poor state down and keeps it there. If you drag us down with you, then whose charity will you live off of?
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The list of all possible real numbers between 0 and 1 would consist of a zero followed by a decimal point, followed by all possible combinations of the digits 0 through 9. I maintain it is not possible to then 'twiddle' one of the digits and arrive at a combination of the digits 0 through 9 that is not already in the set of "all possible combinations of digits 0 through 9."
Cantor's argument says it is possible to find a combination of digits consisting of the numbers 0 through 9 that is not in "all possible combinations of the digits 0 through 9." I have a logical problem with that.
The thing is that you haven't come up with "all possible combinations of digits 0 through 9." after a decimal point. That *is* the interval [0,1] What you've come up with is a map from the natural numbers 1,2,3.... into [0,1]. That is what enumerating means. Enumerating doesn't necessarilly mean that you have listed all of them.
The trick to the diagonal proof relies completely on the fact that everything you deal with in the proof is countably infinite i.e. it's the "littlest infinity", which, granted, you still haven't agreed that such a thing even makes sense.
But the fact that you have an enumerated list is what allows you to find a real number not on that list. While there are an infinite number of numbers in your list, it's trivial to point out the 10th, the millionth, the 250 to the googol to the googol to the googolth number you have listed since that is what an enumeration is.
So if the initial assumption makes sense then every real number is in your list in a specific place. There are infinitely many, but every one exists at some specific *finite* place in the list. So when you create your new number by changing one digit from each number in the list, it truly does not exist in the list.
If it did exist in the list, then it would exist in one and only one specific location in that list. So ask yourself which position in the list could it possibly occupy?
Not the first, because it's first digit is different. Not the hundredth, not the trillionth, and so on and so on and so on.
As I have posted before, though, Cantor's method using the diagonal does not prove the uncountability to me, as a paradox is created when you claim you have listed all the numbers then claim you find a new number that is not listed. Both can not be true, so either not all the numbers were listed in the beginning or the number 'found' IS on the original list.
This is almost true, but you don't just claim that you have listed all the numbers, you claim something even stronger. You claim that you have listed them *by number* i.e. the first, the second etc. This is what makes it different.
So it's true that you claim to have listed them all in order and then demonstrate that you have found one that isn't in the list. This isn't a parodox though, it's a contradiction. Reaching this contradiction shows you that your initial assumption, that the reals are countable, was false.
To really get this, I think there are 2 things that you really need to understand and accept.
One is the idea of proof by contradiction. i.e. assume what you want to prove is not true. If this leads you to a contradiction, then what you assumed is false, and what you originally wanted to prove is therefore true.
The second is why the new number that you created can not possibly be in the list. If it were, then there would exist a finite number, n, that it's mapped to by. i.e. it's in bucket # n.
But by the very nature of the way the number was constructed, it can't be in bucket n because it's a different number in the nth decimal place. The fact that n could be any natural number at all and it isn't a specific one proves it for all natural numbers, hence there isn't a natural number that maps to our new one.
I guess one more thing that might help is to realise that while any statement proven mathematically *is* absolute truth, it's only absolute truth within a very specific un
Those new buildings on Waukegan Road? Some of those buildings are full of corporate documents...
Hmmmm..... Can you say Abbott Labs?
Sure you can. Give me any real number between 1 and 2. Write it down on one of these infinite number of pieces of paper. Put that piece of paper in one of these infinite number of buckets. Repeat for all the real number between 1 and 2 without duplicating any.
The bolded sentence is the problem.
You need to be able to specify a way to determine the n'th real number you are going to choose.
There isn't a way to do this.
For the integers, it's easy
f(1)=0
for n > 1, n even, f(n)=> -1^n(n/2)
for n > 1, n odd, f(n)=> -1^n((n-1)/2)
so essentially, you are ordering them in a different way.
1=>0
2=>1
3=>-1
4=>2
5=>-2
etc.
this function from the natural numbers to the integers is both one to one (no integer is mapped to by more than one natural) and onto (each integer is mapped to).
Then *by the definition of 2 sets having the same "size"* the set of natural numbers and the set of integers is the same size.
You can do something similar to produce a function from the naturals to the rational numbers that meets those requirements.
For the reals it is different.
Not only can't you find a neat and tidy function that you can write down, but you can prove that it isn't possible for one to exist.
More than an infinite number? Interesting concept, but I see a logical problem here.
There is no logical problem, I think the issue is that you don't know the exact way that these words are defined.
It's not like I am saying that there are 5 integers and that there are 10 real numbers.
When you say that 2 sets have the same size, you are saying that there exists a 1 to one function from one that maps onto the other.
That is the definition.
If you think about, it makes a lot of sense to define it this way. It's your bucket example exactly.
If you can put one and only one thing in each bucket, and everything is in a bucket, then intuitively there are the same "number" of things as buckets.
You are mapping between MEMBERS of infinite sets and saying they are different, and I am saying the NUMBER of members in two infinite sets (i.e., mapping both sets to the infinite set of positive integers) is the same - both are exactly equal to the number of positive integers. Two things equal to a third thing are equal to each other.
Being able to map between the members in a one to one correspondence is how you say that 2 sets have the same number of members.
The size of the integers is known, so testing for 2 different sets to have the same size by comparing each to the integers seperately works just fine.
The problem is that when you try to do this with the reals, you fail every time.
Sounds like you have information or education I don't have. Can you point me toward this information? reductio ad absurdum is a classical technique, I have noy seen it used in this context, though. Links?
This isn't reductio ad absurdum, it is a proof by contradiction.
This works like:
Assume that A is true
This implies B.
This implies C.
C is known to be false, i.e. not C is true
So if you can start out assuming that something is true and it leads you to a contradiction, in this case C and not C then you know that your initial assumption is wrong.
I read this to say that we assume that all the digits are either a 2 or a 4, so creating a number whose only digits are 2 or 4, however accomplished, only proves our assumption, and no contradiction that I can see despite your statement that there is one.
The core assumption isn't that the digits are either 2 or 4, it's that you have created a map from the natural numbers to the reals that is one to one and includes every real.
The trick of making a number out of this list by taking one digit from each of the other numbers and changing it gives you a new number which is not in your list. You know it is not in your list
Non-intuitively, if you defined your set as (two rocks per each bin, each rock put into a separate bin) [2*infinity] they would still fit into the existing bins! [2*infinity=infinity]
This is true to a point. How it's actually defined is that 2 sets are the "same size" i.e. your rocks and your buckets if each rock can go into a bucket and no buckets are left empty.
There are an infinite number of decimals between the integer counting numbers (1.1, 1.01, 1.001,...) BUT THEY CAN STILL BE ASSIGNED, one to a bin, TO MY INFINITE NUMBER OF BINS . [infinity * infinity = infinity]
This is where it breaks down.
The problem is that when you try and figure out a way to determine how to assign the real numbers to your buckets (heck even try it with just the numbers between zero and one) you realise that you can not do it.
Even in theory, it is absolutely impossible to determine a way to do this. There are far too many real numbers.
It kind of (i.e. the pedants will roust me for it but it is true enough for purposes of trying to understand it without getting into the details) works like this:
infinity + infinity=infinity
infinity * infinity = infinity
infinity ^ infinity (or even just 2^infinity) = biggerinfinity.
The typical way of proving that there does not exist any way to assign all the real number rocks to the natural number buckets is to assume that you can.
So assume that you have a mapping from the natural numbers onto the reals between 0 and 1.
Onto means that every real between zero and 1 is mapped to.
each real number will be a decimal number in the form 0.abcdxyz......
Heck, even assume that all of the digits are either 2 or 4. This leaves you intuitively with far less numbers. Certainly no more.
So now, if we can create a real number between zero and one whose only digits are 2 or 4, then we have proven that our list is not complete.
So take the first digit of the number 1 maps to. If it is 2, change it to 4. If it is 4, change it to 2.
Go to the second digit of the number mapped to by 2. Do the same thing.
Continue all the way down. Of course it's an infinitely long list, so doing it manually won't work. You can say for every n, if the nth digit of the number mapped to by n is 2 make it 4 else make it 2.
So you have constructed a real number which is between zero and 1 and all of whose digits are either 2 or 4 yet this number is not in your list.
The only assumption that we made is that it is possible to construct a mapping between the natural numbers and this subset of the reals. This assumption alone led to a contradiction, therefore it must be false.
there is about 2.27x10^613 primes in that space
"there are".
No need to be so formal:
"there're"
The only problem with your "map" is it is skewed by state (Red/Blue) It you look at a more detailed level you will see that the areas of most population, and quite frankly the poorest areas, would BE THE ONES ON welfare and voted Democrat!. The fact is the greater population of the rest of the state canceled the blue votes out. So you argument is silly.
Actually, your link has nothing to do with economics.
However, it is true that the most populous areas do vote Democratic. The simple fact is that the most populous areas are the richest areas rather than the poorest ones.
Are you seriously trying to say that Las Vegas is the poorest area of Nevada?
It is the richest, most populous, and most Democratic area of Nevada.
More or less the same thing holds across the entire country.
The most populous areas are where the money is made, and they are overwhelmingly Democratic.
To paraphrase words of Michael Jordan, to his mother. "Once you start making money real money, you become a Republican."
Certtainly, the very richest are Republican more often than not, but most of the poorest are as well.
The rich ones are "smart" to vote Republican if smart means caring only about themselves, but the majority of the Repbulicans are so poor that they have to rely on the Democrats to pay their bills. The fact that Republican policies help to maintain this shows the lack of understanding required for these people to continue voting against their own interests.
I suppose if I hunted her down and killed her, the resulting story would be "Reporter Murdered by Enraged IT Worker"...
Maybe not.
If you were free on bail, were interviewed and made it quite clear to the reporter interviewing you why you did it and your total lack of resistance to doing it again, you might actually get the proper headline.
You're in for a long miserable life if you think that way. "A young conservative has no heart; an old liberal has no brain."
Which is an absolutely false statement.
The plain, simple, undeniable fact is that liberal policies work far better than conservative policies.
Look at the facts. almost all "red" states with the exception of Texas, Colorado, and Nevada are welfare states. That is their economies function solely through the charity of the rest of the country. Specifically, the Liberal states. If you break it down by county, then even within the red states, the republican areas are poorer and the liberal areas are much better off financially.
Now given that conservative policies clearly do not work at all, it doesn't make sense to say that somebody with no brain would vote liberal.
Now, the converse of that statement is pretty clearly true. This is, of course, if you actually argue based on the facts rather that merely repeating ignorant quotes from long ago.
Here is a link showing this. If you don't like that one, there are many more around. Alternatively, you could just look at the figures yourself.
It doesn't really matter though, since you won't find anything to contradict this.
I don't know if evil is the right word, but there is clearly something very seriously wrong with the way these people think for a number of reasons:
First, they claim to be for individual responsibility, small government and against welfare, yet they survive solely through welfare provided by the blue states.
This is either the ultimate hypocricy or an ignorance so severe that it is mind blowing.
They have outdated oppressive social policies which help keep themselves back, yet they voted in this election primarily based on forcing their ignorant ideas on the part of the nation whose mature responsible attitudes are a large part of what allows them to provide the charity upon which their lives are completely dependent.
So it is clear that they lack the most basic common sense since they maintain contradictory ideas (welfare bad--give us welfare) as if they were even rational, let alone good.
They also are trying to drag down the people who are paying their bills.
Once we can no longer afford to pay their bills, who do they think will? That is clearly voting directly against their own best interests, which is evidence of severe flaws in their ability to reason.
Now, when I provide the means for somebody else to survive, I don't expect them to grovel at my feet, but I damn well do expect them not to undermine the very things which allow me to provide them that charity.
So, you might want to update your quote to reflect reality as it does make you look extremely ignorant of the simple basic facts of life.
I, for one, will be doing everything
I can to encourage my representatives to end welfare payments to these states. I think it will be very interesting to see how long they claim to believe what they do when they actually have to live up to it as people with integrity do.
I mean think about it. If we made a requirement that states could not receive more in federal money than they put in, then all of the liberal states would be better off and almost all of the conservative states would be bankrupt.
And they have the audacity to try and tell us how to run an economy or a society.
I am convinced that the margin of error on this election will be the lowest recorded in American history.
So are you saying that the Dems and Repubs did a more or less equal job of rigging things?
The real issue isn't what happened with this particular election, rather it's that these machines are *known* to be faulty, they are *known* to be easily hackable, and they are *known* to leave no audit trail by design.
Come on, Larry. Why are you posting AC?
No. Much like I have faith that the sun will rise tomorrow I have faith in my fellow /. posters to take this discussion off target.
Ahhh... but your faith that the sun will rise tomorrow is only backed up by one piece of evidence a day.
* Why do you think Clinton and Bush are for privatizing Social Security? To keep the stock bubble booming!
Partly true -- however, that would still be artificial, and would fail. I think we need to strike a balance between an entirely government based social security and a privatized one.
There are certainly arguments either way, but it has a seemingly insurmountable problem. Curiously, it's essentially the same problem that makes banning guns unworkable IMHO.
How would it be possible to handle the transition?
This is something I haven't ever really heard anybody address satisfactorily.
If you switch it all to private right now, what do you do? Divide the total balance by the number of people and mail them a check? Sending everybody back the amount they put in wouldn't be workable even if the scumbags in Washington (yes, all of them) hadn't already looted it because there isn't that much avalable. Has anybody proposed a realistic method?
The gun issue seems really similar to me. Some other countries seem to do ok without guns (save the statistics, I said ok, not better), but that doesn't mean we could outlaw them and magically they'd all disappear. Assume every non criminal citizen went down and turned theirs in (yeah, right). That would be a recipe for disaster.
Similarly with privatizing Social Security.
It seems like there are two different more or less equally valid ways of doing something, but choosing one eliminates the possibility of later choosing the other without a major disaster.
Indeed, very informative, thanks.
Can you imagine the repercussions of dropping napalm in the wrong place?
Ummmm..... Bush would get a third term?
We cant eliminate rats, crabgrass, fleas. or ragweed; we're not gonna eliminate coca, hemp, poppies, or datura either.
What in the heck is datura?
You might think we would have learned from prohibition that outlawing substances has a net result of creating guys like Al Capone or Pablo Escobar, as the case may be.
Don't forget Joe Kennedy who owned the trucks that transported a lot of the hooch which is where they made their fortune.
Going from there straight to gay marraige is just to big of a shock for middle America and the reaction will be to do their utmost to instill judges that will reverse the previous decisions.
My real problem is that "middle America" is full of backwards ass people whose own backwards assness is holding them back. They're held back to the point that they need my charity to survive (subsidies, tax imbalance etc.) yet they want to drag the part of the country that actually pulls their own weight down to their level.
I don't give a crap what they want to ban in their backwards ass states. It's when these ignorant fools try and tell the rest of the country what they can do, even though they have proven that they don't really have a clue about what works that I have a problem.
There really isn't a point in trying to convince these people. They claim to be Chriatians, but Jesus himself told them directly not to be such fuckheads about things, yet they reject that in favor of a single passage in the old testament.
Where's the lobby for an amendment to ban pork?
It has exactly the same status in the bible as the gay thing.
Oh yeah, it isn't a moral issue at all.. This vote came down to who hates the same people as me and who will actively work to oppress them.
And these fucking cowards have the audacity to wave the flag around.
And while I know you didn't mean it this way, I found your comment about "then they would both have better lives" ironic given that the dead child didn't have a chance to have any life, let alone a better one.
Sure I did. The woman and her child. There wasn't another child in the first place. That is the point.
Now can you guarantee a great life for every person you force into the world at gunpoint?
Can you even guarantee them a decent life?
Now here's the real test though.
Do you support sex education in schools including the teaching of the proper use of birth control?
Heaven forbid you should prevent people from learning basic facts of biology and how to avoid pregnancy yet punish them *and* their child if it happens.
Even when he was being impeached (a perfect time to attack him, you must admit), the media spent its time saying rude things abut the impeachers, not about Clinton.
I amjust amazed at how you people can just lie out your ass with a straight face.
In the first place, this isn't what happened. This was treated as if it were a legitimate grounds for impeachment. It was a fucking blowjob. The whole process destroyed the integrity of our congress and showed the Republicans for exactly what they are:
Partisan hacks who care about their party far more than they do about their country. They spent his entire administration trying to nail him for anything and this was al they could come up with.
Now that isn't how the press handlled it. They demonized Clinton as if he had lied to commit mass murder.
Oh, speaking of that, where were the questions before the war?
None.
The press didn't do a god damned thing and just sold the war.
Now that we all know that it was based completely of falsified evidendce and lies, where is the call for impeachment?
Had that been Clinton, he would have been fried over that. As it's Bush, nothing has happened, and due in large part to the way the media has coddled him, the traitor actually got eleted legitimately.
The really fucking sad thing is that according to the polls, most Americans knew he was a lying sack of shit, they knew he is a mass murderer, and they know he's crap for the country.
So why did they vote for him?
Because he shares their ignorant hatreds and promised to actively work to discriminate against a large segment of our society because he hates who they were born.
That is where our country is right now. Sharing of ignorant hatred is our most important "moral value"
Or do you really believe that your beliefs about Bush were formed entirely from your own morla uprightness and superior reasoning, and had nothing to do with the inputs from any of the media?
I knew before he was elected that he had been a constant failure at everything he had ever done, and that he most likely made his first million threough illegal insider trading.
I haven't seen that on Fox or CBS or CNN.
I knew before the first election that he had jumped ahead of more deserving people into the Texas Air National Guard and then deserted that post. That was only reported recently, and when it turned out that all the facts were correct, but the exact words were on a different peice of paer, it got dropped and treated as if that were the story rather than the cowardice and treason of the President.
So given that, your assertions are without merit and not even remotely rational.
Look up the numbers.
How much do they pay in tax dollars, and how much do they take in from the Fed.
They take a lot more than they give. Simple basic fact.
Farmers are subsidized so food prices can be kept down.
Farmers are subsidized, and additionally food prices are kept artificially high.
How the hell can you even make an insane statement like that?
A subsidy would be part of the price of food anyway, hence driving it higher so your statement isn't even remotely possible.
And please the last thing a state like TN needs is a bunch of liberals they can get along just fine without you.
Right. The state is poor as shit and wioth a horrible educational system.
Look into where tho state's income comes fom and you will see that a lot of comes in subsidies from the states that actually pull their own weight rather than living off of handouts.
If you don't like it, start pulling your own weight, don't make nonsensical statements which a quick search would prove wrong.
Attitudes like this are what makes it all the harder for people who want to help you obtain that goal. You expect understanding of your lifestyle while criticizing that of others.
I happily married to a woman, so it's not my lifestyle I'm talking about.
Regardless, I'm not sure what the problem with my attitude is.
I believe that all people are created equal, at least in the sense that the government has no right whatsoever to treat them differently just because of who they were born.
That person claimed that he had the right to shove his religious beliefs down people's throats at the point of a gun (as all laws are).
That is extremely ignorant, extremely hateful, and totally unAmerican.
Was anything I said in the least inaccurate?
Are you trying to say that abortion is moral and anti-abortion is immoral?
I am certainly not saying that abortion *is* moral, however I am most certainly saying that forcing somebody at the point of a gun to have a child that they do not want and can't afford is extremely immoral.
There is only one person who is fit to decide whether or not they are ready to have a child and that is that person. It is not my job nor yours to tell them that they have to.
Doing so could well condemn them and that child to poverty whereas if they waited until they were ready then they would both have better lives.
That is a decision for them to make.
It's really pathetic to hear you talk about killing innocents when you suupport Bush's lies that have killed tens of thousands.
You have no moral leg to stand on. You are hypocritical to even pretend that you have morals given that.
When did you get access to Langley computers? YOU MUST BE GOD, YOU KNOW WHAT IS TRUE AND WHAT IS LIE, HUH? You know better than anyone, it's a fact, you have evidence, empiracle. Go ahead and post it to the CNN site if what you say is true, it's all lies. You sound so sure of yourself, you must have proof.
The CIA flat out told Bush not to give that state of the union speech because the info was bad.
He did it anyway telling us it was solid evidence.
This is undisputed fact. You really should have been paying attention.
SADDAM KILLED 400,000 PEOPLE BECAUSE HE COULD!! 400,000!!!! WHOS THE FUCKING CRIMINAL HERE? DO YOU HATE IRAQI'S? YOU'D RATHER THAT THEY STILL HAVE A MANAICAL DICK WITH TEETH RULING THEM?
That is completely irrelevant, and yelling won't change that. That was not the justification we were given for going to war. That is the only relevant issue to this discussion. We were told that we were under imminent threat of attack when that was blatantly false. Ditto for WMDs.
Now, given that you're so worked up about the poor innocent Iraqis (well except for the ones we murdered. Your "morals" are so relative it's scary) you, of course, know that the US was complicit in their deaths? You know that we encouraged the Kurds to revolt and told them that we'd back them and then betrayed them allowing Saddam to gas them with the gas that Donald Rumsfeld sold them? Sure. Real moral people you are supporting there.
You are clueless.
Really? Then the rural states are not largely subsidized by the rest? So phones, cell phones, electric cables and all of that would actually be affordable even though the cost per person is orders of magnitude higher than in densely populated areas?
We don't pay far above market value for food and then pay again to subsidize you and pay farmers not to grow food?
Sure.
It's quite clear that you are clueless.
I am guessing you wouldn't know a hard days work if it slapped you in the face.
Well guess what Sparky? You guessed wrong.
I've done construction, dug ditches, worked on a freaking tulip farm, worked on a fishing boat, done carpentry and many other jobs.
That doesn't change the simple fact that the Liberal states largely subsidize the existence of people in the conservative states.
If it's hard to deal with, then maybe you should move somewhere where you wouldn't be a burden on the parts of the country that actually pull their own weight.
If not, the least you could do is thank us.
I believe Hitler never did anything to people here in America, so maybe we should have not gone after him either.
What a ridiculous comparison. You completely ignore the most relevant part as well.
Our entire justification was based on lies.
That's bearing false witness and is completely immoral by most decent people's standards.
Somebody was forced at gunpoint to have an a baby?
What else would you call illegalization of abortion?
According to the polls that is the primary reason people had for voting for Bush. His supposed "morals". They conveniently ignore the lies and murder. Typical of the hypocritical scum who would actually vote for that traitor.
You don't think you have freedom of religion?
When the religious wackos are trying to put their religion into our lawsand they are actually being treated as if that was even remotely consistent with the basis of this country then of course not. What kind of a ridiculous question is that?
Also, when did Jon Stewart suddenly become so intelligent.
It's partly a joke and partly just showing some respect for his stand on Crossfire which any person who has any interest in real discourse coming back to this country should admire.
Well golly gee whiz mister we don't got the technical stuff you all got but if you don't shut yer trap we'll stop growin yer FOOD you stupid fuck!
We pay far more than market value for the food, and we pay for it twice. Once when we buy it and once when we subsidize you.
Additionally, I've lived all over California and in Illinois.
Both states grow far more food than they need.
If we didn't pay inflated pices for the food you grow, you would not even have an economy.
You very existence is dependent upon our charity you ignorant dipshit.
Personally I don't give a fuck if you want to live your life according to a hate based faith, but keep that shit that drags your pathetically poor state down and keeps it there. If you drag us down with you, then whose charity will you live off of?