Well no, if it's shown to be undecidable then either it's true or there exists some counterexample.
You don't know what "undecidable" means is all that you've said. If it's undecidable, then it can not be shown to be true or false. from within the system. Providing a counterexample would prove it false within the system. If it's true or there exists a counter example, then it is decidable.
And since it's impossible to search all of the integers looking for one, you can never say that one doesn't exist,
That is totally irrelevant. All you said there is that providing examples for a statement does not constitute a proof which is certainly true but has nothing to do with the point. Undecidability *proves* that there can not possibly exist a counter example. It's just not done from within arithmetic.
No worries, your sort is well known for spouting meaningless bullshit and babbling nonsense like that when called on your ignorance and lies. Too bad you're incapable of actually defending your nonsense position, but common decency does demand that you stop spouting such idiocy given that. I certainly don't expect anything like that to stop you though. You wouldn't be making such idiotic statements in the first place if you were concerned about decency or anything of the sort.
It fails utterly. Any "thought experiment" you can possibly come up with will likewise fail just as utterly. That's the difference between intuition and proof. Your intuition is wrong. Dead wrong. Absolutely, positively wrong.
Here's one reason why:
[This thought experiment should be able to be expanded to a formal inductive proof if that is what you want.]
No, it can't. Not to prove what you're trying to prove. Induction can only prove a statement true for all *finite* values of 'n'. So, yes, obviously once you pick a specific fixed number, n, if you take the set of all natural numbers up to and including n and compare that with the set of all primes up to and including n (assuming n is prime of course, otherwise it's not included) then your first set will be larger.
That's all well and good for what it is, but it has no bearing whatsoever on what you're attempting to demonstrate.
Let me also state, that I completely understand the OP's proof. I just think that it isn't the whole story.
Then you do *not* understand the OP's proof *at all*. You don't even understand what a proof is.
And, BTW, the first assumption from my proof is only false when talking about infinite sets.
But that's all we're talking about. Both the primes and the naturals are infinite sets, so in the context of the discussion your assumption is absolutely false. In fact the truth or falsehood of your assumption depends entirely upon whether the set in question is finite or infinite. The negation of your statement can, in fact, be taken as the *definition* of an infinite set. It's absolutely true that a set meeting the negation of your statement is always infinite. It's also true that any set meeting that criteria must be infinite.
I cannot say that I know all of the math behind why infinite sets are allowed to break the intuitive definition of cardinality, but if it is like alot of things in math, it could be an arbitrary decision.
They are not "allowed" to do anything. They do. They did so long before Cantor developed the theory of the infinite and proved it absolutely. It's an inherent property of infinity.
There isn't an intuitive "definition' of cardinality. There is intuition. There is a definition of sets having the same cardinality. These are 2 different things. Very rarely if ever are decisions about definitions or axioms "arbitrary". It might well be the case that it's possible to choose one of 2 mutually contradictory options in some given situation without "breaking" anything, but which one to go with is generally chosen based on which one creates more interesting mathematics. Physicists have different ideas about such things. This is why, IIRC, mathematicians generally like the Axiom of Choice as it leads to many interesting consequences while physicists generally don't like it because it leads to pathological cases like unmeasurable sets. The decisions aren't arbitrary in general.
Math is dependent on the axioms (or at least the logic portion of it). I can choose the axioms I want whether or not they fit with your view of reality. Granted, it may not mean much, but in this case, I think that an alternative axiom set could produce interesting results.
You're certainly welcome to make up whatever you like, but you'd be best served by actually doing so rather than claiming that somehow your unstated (and most likely not very well thought out) new axioms should be treated as if they have any value. This is especially true given that you haven't proposed any problem with the existing mathematics, or anything interesting you hope to gain from the exercise. The fact that you don't find something intuitively obvious is completely meaningless as to the value of the mathematics, it only demonstrates a lack of imagination and a lack of effort to understand on your part.
I think it's rather silly to split hairs, but that's just me. Anyway, since you seem to be in the business of splitting hairs, I will indulge you.
It's not "splitting hairs", it's called being precise which is critical.
Let me give you a quick proof that there are not as many things in the set of primes as there are in the set of natural numbers:
1. A proper subset contains fewer things than its superset.
OK, your first assumption is demonstrably false, therefore your entire argument is unsound. The OP (and a few other people as well) have absolutely positively and beyond the possibility of a doubt demonstrated that to be the case. The problem is that you are using ambiguous terms in an attempt to prove something. You must have well defined terms or you are not going to be capable of constructing a logical argument. What does it mean for a set to "contain fewer things" than another set? For finite sets, you can lay them out side by side and you are done. For infinite sets you'll never even get close to finishing so you need a more precise way of stating that relationship.
I suspect the only reason we have this disagreement is because you start from a different set of axioms and/or take a different route to arrive at the knowledge you so want to prove me wrong with. [And, btw, you're the one who brought the word cardinality into this. I was speaking of degrees of infinities which may be thought of as aleph numbers and thus cardinalities --or-- as a more proper degree which respects some other (and more important to some) things that we know about sets].
The only disagreement is between your intuition and the way things actually are. "more proper" "things we know about sets" These statements do not make any sense. You are talking about how finite sets work and demanding that infinite sets must conform exactly to finite sets.
But, once again, this sort of collision of models is entirely expected given Godel's theorem. The 'problem' here can be looked at in several ways but since you did not understand my geometry illustration and since you do not seem to want to learn anything from me (or to even think that is possible for you to do so, I will not be wasting my time going in to these additional insights).
No, it has nothing at all to do with Godel's theorems. There is no "collision of models". You are not conveying "insights", you are only conveying your lack of understanding of the subject.
Well, if you realize that, then you should be well on your way to understanding the proof I gave above.
That wasn't a proof. It was an unsound argument based on known false assumptions.
You've got to be kidding me. Of course it is a convenience, that's why I used it. However, it becomes much less convenient when people start splitting hairs like this.
He wasn't splitting hairs. He's absolutely right, and you are absolutely wrong.
I was speaking of the density of the set. When you do your neat folding trick where you take all the negatives and you put them in with the positives with zero at the beginning, are you not doubling the number of numbers that come after zero in that set? The geometric analogy I provided would've shown you this if you'd understood it. [And, if you don't believe this statement, I can prove it.]
Do you even know what density is? Don't worry, you don't. It is a well defined term and it does not apply to sets. Sets have no order, they have no concept of distance between numbers or anything of the sort. No, you can not prove it. The OP already proved it to be false. You just do not understand the ideas under discussion. It's ok, they're not simple concepts, but they are very well understood.
Ok, now do this. Take the first n positive numbers where n=5: {1, 2, 3, 4, 5}. Now, of those, which are primes? 2, 3, 5. Hmmm, only 3. There are more naturals than primes.
Cardinality is simply an artificial construct and does not entirely capture the real nature of things.
No, No, No.
It does capture the real nature of things. Their real nature might be unintuitive, but it absolutely, brilliantly does capture the real nature of *infinite* things.
Say I have 5 pebbles in one hand and 5 coins in the other. Would you agree that the "5ness" contained in each of my hands is a real thing. You can take one pebble and one coin, then another and then another until my hands are empty. Each coin will be matched up with one *and only one* pebble.
The problem is not at all that it doesn't capture the real nature of things, it's that your intuition is not geared to deal with infinite quantities. That's what makes mathematics so freaking powerful. It can conclusively demonstrate things which you, quite reasonably mind you, would automatically assume to be obviously false.
Making a statement that the cardinality of the primes is exactly equal to the cardinality of the rationals doesn't lose any information about either. Primes are still indivisible, and rationals are still all the ratios of all the integers (barring zero denominators, of course) every other true statement about either is still true. You only gain *more* information about them,which can be incredibly useful.
This is the interesting thing about Godel, to me. Models work in isolation; but when they reach a certain critical mass, they start to break down.
Again, no. The only thing that is broken down by this is your intuition. The "models" work better than ever with this than they ever could without it. You have no idea how much mathematics would not exist without this.
In this particular case, maybe we need another definition of cardinality that doesn't lose the intuitive nature of the relationships between the sizes of these sets).
No, that's exactly what you don't need. Reality need not conform to your preconceptions.
You are stuck in a rut based on your long experience with finite quantities. Cardinal numbers (well, the infinite ones anyhow) are not part of the system of numbers which you are familiar with. They are characteristics of sets. Just like coins and pebbles are entirely different things, but when I hold 5 in each hand those collections share a certain property. The property of "5ness".
Infinities work differently. This is a proven fact (not a well verified theory, but an absolute and unassailable fact.) which is extremely important and incredibly useful. Cardinality does maintain the intuitive relationship you like as long as you are talking about finite sets. Once you hit the first infinity, things change. But the concept of cardinality and the definition of equal cardinality doesn't change at all by crossing that boundary. The fact that we don't need to change it to meet our finite intuition gives us new information.
Is it just that you don't like the particular example the OP gave? Are you comfortable with the fact that infinity+1=infinity, 2* infinity = infinity ( taking "infinity" to mean the smallest infinity here)? If so, then you already know and accept that the naturals have the same number of numbers as the naturals plus zero. You already know and accept that there are *exactly* as many positive integers as there are integers. Infinity * infinity = infinity expresses the fact that the naturals and the rationals have the same exact "number" of numbers.
This might be unintuitive, but it's not at all arbitrary and you can't define cardinality in such a way as to make this go away without making it a completely meaningless or inconsistent idea. It is an inherent property of the way things are....OK, "things" being the sets of numbers in question anyhow.
The continuum hypothesis says there is a set bigger than the integers but smaller than the reals.
Well, close....in the sense that most of the words are accurate. The hypothesis was that there is *not* such a set.
Also, you can't necessarily say that you'll get consistent mathematics by adding an undecidable statement to existing axioms. You also have to know that the existing axioms are already consistent. ZF set theory has not been shown to be consistent, and in fact, it has been proven that it can't be proven to be consistent from within the theory itself...well, that is...unless it actually is inconsistent.
Yeesh, I think I liked functional analysis better than formal set theory and that's saying something.
How can a mathematical truth be 'known" if it cannot be proven?
Well, *assume* for the moment that...say... the Goldbach conjecture is shown to be undecidable. This means that it can not be proven or disproven (from within the system). Therefore, it has to be true. Why you ask? Well, if it were to be false, then you would be able to come up with integers which violate it. But then you would have proven it false and hence contradicted the undecidability.
So it cannot be proven yet nonetheless, it's now a known truth.
Maybe, maybe not. Had that fascist fuck gotten the traitor's death he well earned, then it just would have meant an even longer reign of the Bush crime family. Talk about a lose lose situation.
I DO believe that the resentment is misplaced, as it's merely "jokers" like you who give the rest of your countrymen (and countrywomen) a bad rap, not the general populace.
Then, quite frankly, you are a fucking idiot. He's one of the few people here who actually realizes that it's a joke.
"You deserve/have the right to [item] regardless of your ability to pay." This is not a capitalist catchphrase. This is socialism at work.
No, that is not *necessarily* socialism at work. It depends entirely upon who the "deserving" party is. If it's some poor entity, then it's socialism. If it's a rich entity then it's fascism.
That's the difference between the left and the right. Whose benefit they want to use the state to fuck everyone else over in favor of.
So, when you see somebody claiming some big government bullshit is necessarily "socialism" without paying attention to the details which make all the difference, you can easily recognize them as a big government ninny state right wing extremist fascist shitbag.
the difference would be if your piss poor decisions was as a captain of a ship and we are heading for an iceberg. bitching about your past paychecks would not change the fact that we need to do something to not hit the iceberg
Hitting the iceberg is inevitable at this point. Our whole economy is greatly inflated and absolutely needs to correct itself. All that can be accomplished by increasing the artificial propping up of the economy is changing the date it occurs and *increase* the damage done by it.
So it's not at all like your iceberg analogy. Had the captain of the Titanic heeded the warnings and allowed the ship to be slowed down, then the iceberg could have been avoided. We're way past that point.
Read more history. The racists stayed with the Democratic Party for several decades, until they just couldn't stomach some of the things the left was doing. Then they left, reluctantly.
By "the things", you mean the civil rights act signed by Johnson? Heaven forbid those damn niggers be granted equal rights like the fucking constitution guarantees, right? Lost the south for a generation etc. etc. Longer actually as it turns out, but dirt poor dipshits tend to want to blame others for their problems.
Also, note that, traditionally, the "welfare leeches" were, and are, staunch Democrats to this day. Remember, Welfare was also an invention of the Democrats.
Wrong. Welfare, obviously is an invention of the left, but Republicans are the primary recipients. Agricultural subsidies, electricity, internet, phones etc etc etc. all of these are heavily subsidized and the primary recipients are rural states and regions i.e. Republicans by and large. Add in the military which is primarily a welfare scam and it's overwhelming. Then, of course, there's corporate welfare which goes primarily to the very few people who actually are sane (if evil) for voting Republican.
Almost all "red states" are welfare states. That means they receive more in taxes than they pay. Every resident of these states are on welfare. Just because they don't receive GA checks doesn't mean they aren't welfare leeches
Of course, even the "blue' welfare recipients are getting paid primarily by the blues states as they're the ones paying the country's bills, unlike the welfare states.
Wow, that's the first time I've ever been called a lefty! I'm not sure whether to applaud you, or suggest that you're not bright enough to pour sand from a boot with instructions printed on the heel, since I've been posting here for years and years, mostly to the right.
You're the one saying that people unwilling to pay for luxury services should receive them out of my pocket. That's loony lefty. I care sweet fuck all, what other nonsense you've spewed over the years. If you think poor people who refuse to pull their own weight should be paid for their failure out of my pocket, that makes you far to the loony side of left on that topic anyhow. Let me guess, you're a poor religious loon so your left when it means you get my money, but you still want to inject your fucked up hate based religious bullshit into my life, which is extremist right wing nonsense.
Though perhaps, if you're as far right as you sound like (just a tad right of David Duke), I look like the left from there.
Nope, I'm a Liberal as in what the word actually means. not how it's commonly used in America these days. You know, Liberalism, the basis of America? That's the center. I don't like extremist lefties, because I don't like the power of the state being used against me to promote some bullshit interpretation of "equality" and I fucking despise the right, or fascists, because I don't like the power of the state being used to keep the rich and powerful rich and powerful at my expense because they despise equality as in "equal under the law". So I'm in the middle and you seem to be an extremist to the left and right depending on the issue and more importantly the recipient.
Oh, you should also read enough history to know what you're talking about when you discuss religion in this country. From your rant above, you haven't a clue in the world about the subject.
"the United States is in no way founded on the Christian religion" --- Treaty of Tripoli. Passed unanimously and widely celebrated by the Americans of the time. We have fallen far driven primarily by the religious loons coming out of the woodwork with the threat of the "godless communists" They added their god bullshit to the Pledge of allegiance in the 50s dipshit. You obviously don't know a god damned thing about the history of keeping religion out of our government and the critical importance of that to America becoming what it was before the fucking loons
I was really looking forward to your reply on this topic, but if this is what you're going to give me, I'm wasting my time looking for a decent opposing viewpoint.
Opposing what? You made a non sequitor and haven't expressed anything approaching a point on the topic.
My, you have a distorted picture of the universe. HINT: the Rural Electrification Act was passed by the Democrats, not the Republicans.
Which has what to do with anything? I'm well aware of that fact. It has nothing to do with the fact that the leeches, as a whole, are the Republican voters.
Another hint: most people in the country aren't terribly interested in being part of a group that favours gun control (they pretty much all hunt), gay marriages (traditional is a way of life in the country, for good or ill), etc.
I don't favor gun control, yet I don't favor voting for fascists either. Gay marriage is a non issue except to religious extremists. This country was founded primarily on the idea that religious delusion has no place in the government in a civil society. That is the primary defining characteristic of America, so catering to those who despise this country is completely unconstitutional. Their thoughts on that issue have no legitimacy. They have no right to try and shove their evil hate based faith into our laws.
Note further that until the Democrats decided to swing to the left in the 60's, most people in the country were staunch Democrats. Hell, my grandfather would have died rather than vote Republican. (personally, I've always believed he did - he died the year that Dukakis was the Democratic candidate for President)
Yes, the racists were Democrats. They left en masse for the Republican party leading to its current make up of the bottom of the barrel of our society. Welfare leeches, religious extremists, racists and corporate welfare leeches. That's a complete breakdown of the Republican party membership. Well, ok, and deeply stupid fools who haven't paid any attention to what the Republican party has become over the last several decades.
Note, by the way, that from the point of view of most everyone else "promoting ignorant and damaging behavior at my expense is complete crap" works nicely. Too bad they think that most of the things YOU think are important are part of "ignorant and damaging behavior at my expense" to THEM.
Except this isn't symmetrical. America isn't a democracy specifically to keep shitbags like them from being able to shove their delusional religious shit into the government of a free society. Even if it was symmetrical, it simply isn't true. I'm not asking them for shit except to stay the fuck out of other people's business. They are insisting that I be robbed to make their welfare payments and to push their religious delusions. Huge fucking difference.
An example of "ignorant and damaging behaviour at my expense" is the notion that electricity is a privilege only for urban dwellers, or the very wealthy. Sorry, I think it would be nice if every farmhouse could have lights at night. Likewise phone service. I remember having to drive 20 miles to the store with my grandfather to make a phone call. I think that the idea that he be wealthy, or move to a city, to get phone service in his home is "ignorant and damaging".
Well, then., he can damn well pay for it himself. You're a typical loony lefty who thinks government cheese is free. I pay my fucking bills. I can't afford to pay yours too. Anything of the sort is only a privilege for those who can afford it. How about you buy me a Ferrari? No, then fuck off, put your grubby little mitt away keep it out of my pocket and pull you own god damned weight.
And ditto for broadband. Five years ago, I considered it a luxury. Now it's a basic necessity, like phone service or electricity. or indoor plumbing, for that matter. And so it should be provided to everyone, not just to the self-serving idiots who can't imagine that some people don't live the way they do.
If you want to leech off of me without pulling your own weight, then not a chance in fucking hell at the very least as long as your trying to grant legitimacy to religious extremists trying to destroy America.
If you think they should be provided all these neat things I have to pay for myself, then *you* pay for it. Otherwise shut the fuck up leech.
I said nothing except the fact that we need to have a military or we will eventually get run over by another country.
No, in response to somebody else saying that the military will more likely be used for evil than good, which is obviously true, you went off about how other countries have militaries which is a complete non sequitor.
A wise man once told me, "When in Rome, good fences keep good neighbors." You seem to have agreed with this, but the rest of your rant is completely unwarranted
Knocking down other people's fences to steal their shit and to rob your own populace makes bad neighbors. That's the difference between what you're saying and reality.
While having *a* military might be a necessary evil, having *our* military is a completely unnecessary evil.
So, no, that isn't what you said at all. If that's what you meant, you should actually say that. It would still be a non sequitor though.
We have 2 choices, give them the money and hope we can smooth out a crash/depression (by no means certain) or don't and accept the 'correction' (cringely has a great piece last week likening this to a forest fire). The correction would work, it would hurt a lot, but at the end of the day it would work.
If at a bare minimum, every board member and every corporate officer are stripped of all of their assets and sent to prison for life with no possibility for parole, then and only then, is it even worth discussing. Only then would it be possible for it to have any positive effect.
Sorry, but as long as the people *responsible* for causing this problem have one penny to their names, don't you fucking dare ask me to sacrifice my future so that they can continue their failed policies living in extreme luxury. It's a completely asinine idea.
Say I made a bunch of piss poor decisions and came begging you for a handout. Would you eagerly pay up massive amounts of cash if I wasn't willing to touch my savings to cover it? I doubt it.
Don't ask me to do the same thing on a far more massive scale for people who have far more than either of us which they made through fraud.
Now, if the information gathered under this Bill results in a broadband equivalent of the Rural Electrification Act, it'll be a good thing. Annoying, to have my broadband rates raised to pay to provide broadband to the areas it isn't provided, but worth it, in the big picture.
Not worth it in the big picture. In the big picture it leads to welfare leeches claiming that they hate socialism while remaining the largest recipients of it while not contributing at all. This sort of delusional belief is a primary contributor to the hostility of the rural people for those who pull their own weight. It's why they keep voting for fascist Republicans against their own best interests and against everybody else's.
Sorry, but promoting ignorant and damaging behavior at my expense is complete crap if you actually look at the big picture instead of the narrow view you are taking while claiming it's the big picture.
The idea is to close the loophole that ISPs and telecom operators have used for decades to keep their licenses while not having to serve those areas that are financially unrewarding to serve with the same service.
How about we don't close the loophole and quit making me pay for the rural welfare leeches? Maybe if they actually paid their own way for once they'd pull their heads out of their asses and quit pretending that they are free market capitalists when they're just the recipients of welfare. Maybe they'd quit voting against their and everybody else's best interests?
Sorry, but I am sick to death of being robbed to pay welfare to a bunch of whiny bitches who refuse to deal with reality. People always end up hating those they are forced to beg for charity so let's just quit making the people who pull their own weight the targets of the hatred of ignorant extremists.
Tell the rest of the world to get rid of their militaries and all animosity towards the rest of their fellow man, and maybe the US will get rid of theirs, too. Until then, I don't see a lot happening in this area.
But that's a totally insane response that doesn't deal with the reality. We spend more than the next 10 countries put together on our military and we use it primarily for evil purposes. We have not used it for the defense of America in 2 generations, so your "point" is complete nonsense based completely on the entirely religious belief that our leaders aren't capable of doing wrong which is, again, completely insane.
Try looking around the real world once in a while and putting down the crack pipe. You'll be quite surprised to realize that most people don't want to kill us, they just want us to stop robbing, raping and murdering them.
It won't get better for some time, so consider what I did back in 1981. Join the military in a non-bullet-catcher specialty. The new Webb G.I. Bill is a FULL four-year ride to college with a monthly stipend, so you can wait out the failing economy while adding something employers respect to your resume, then get paid to go to school for another four years.
Oh, so you mean be a welfare leech and add to the problem by putting an even greater strain on the resources of those who are doing worthwhile work? You're encouraging increasing the problem rather than being part of the solution. How about go find a way to make yourself useful to people and add value rather than be just another leech?
How about it's tough enough these days without another douchebag signing up for the dole?
What a bunch of entitlement crybaby thieves you fuckers are.
Grow up and realize that money doesn't grow on trees and you are not entitled to rob me blind while signing up to be a tool which is primarily used as a weapon against this country. In short, quit holding out your hand begging for welfare and get to work doing something useful for which people might be willing to pay you.
We should cut the military budget. This will make it easier for red states to resist Chairman Obama's socialist mandates. He can't really have a socialist state without a big army to keep pieces of it from breaking away, can he?
You do know that the red states are pretty much the same as the welfare states with only a couple exceptions, right? You're pegging the wrong party as socialists. The republicans don't like paying for their welfare and expect the rest of us to do it for them, but they would howl like banshees if we ever tried to cut off their welfare payments. This includes the military, given its extremely excessive budget, but, again, the Republican welfare queens are the ones who keep increasing it to put more of their buddies on the dole.
Seriously, you might consider pulling your head out of your ass and looking around once in a while. You'll look less stupid if you don't keep repeating the same old blatantly false propaganda that your Dear Leaders keep shoveling down your throat, "comrade".
If you only learn one thing in your life, it should be "actions speak louder than words". I learned that when I was a little child, so I never fell for any idiotic cults or scams as you obviously have.
No, it's only the fundamentalist Christians who would think that, in the rest of the world you can be a Christian without believing every single word in the Bible is the absolute truth.
No, you really can't be a Christian without believing every single word, since at that point you've either decided that you know better than god, or that god wasn't involved, hence it's not really a religion. You can call yourself a Christian, but so can anybody, and you can find backing for any sick shit you want to do in the Bible, making it totally worthless for any "moral" guidance.
BTW, I'm not a Christian, I'm just saying you shouldn't judge a faith or set of beliefs by the crazy extremists, otherwise you end up with the "because most suicide bombers are Muslims, most Muslims are suicide bombers" type of argument.
All Muslims are suicide bombers or similar. The Koran is quite explicit about it. All Christians and Jews are evil vile brutal murdering scum as well.
The thing you're missing is that there really aren't very many of any of the above in the world these days. If you go read the bible and the Koran, that will be obvious. The ones who believe it are absolutely required by their gos to be savage murderers. Most who claim to be of a given faith don't really believe it though. We'd still have stoning for a whole host of petty crimes if they did. Well, I guess there are a fair amount of Muslims who actually believe their idiot book, but Christians of faith mostly died out in the Enlightenment, and that is an absolutely great thing. Look how far the world has come since.
I sometimes wonder if Martin Luther, had he realized what would be the ultimate fate of his cause, would have snuck back up to the cathedral door and pulled the nails out.
No, the holocaust alone would have made it all worthwhile to him...well if it had succeeded anyhow. It was his plan in most of the details even, don't you know?
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Well no, if it's shown to be undecidable then either it's true or there exists some counterexample.
You don't know what "undecidable" means is all that you've said. If it's undecidable, then it can not be shown to be true or false. from within the system. Providing a counterexample would prove it false within the system. If it's true or there exists a counter example, then it is decidable.
And since it's impossible to search all of the integers looking for one, you can never say that one doesn't exist,
That is totally irrelevant. All you said there is that providing examples for a statement does not constitute a proof which is certainly true but has nothing to do with the point. Undecidability *proves* that there can not possibly exist a counter example. It's just not done from within arithmetic.
No worries, your sort is well known for spouting meaningless bullshit and babbling nonsense like that when called on your ignorance and lies.
Too bad you're incapable of actually defending your nonsense position, but common decency does demand that you stop spouting such idiocy given that. I certainly don't expect anything like that to stop you though. You wouldn't be making such idiotic statements in the first place if you were concerned about decency or anything of the sort.
Ok, how about this thought experiment.
It fails utterly. Any "thought experiment" you can possibly come up with will likewise fail just as utterly. That's the difference between intuition and proof. Your intuition is wrong. Dead wrong. Absolutely, positively wrong.
Here's one reason why:
[This thought experiment should be able to be expanded to a formal inductive proof if that is what you want.]
No, it can't. Not to prove what you're trying to prove. Induction can only prove a statement true for all *finite* values of 'n'. So, yes, obviously once you pick a specific fixed number, n, if you take the set of all natural numbers up to and including n and compare that with the set of all primes up to and including n (assuming n is prime of course, otherwise it's not included) then your first set will be larger.
That's all well and good for what it is, but it has no bearing whatsoever on what you're attempting to demonstrate.
Let me also state, that I completely understand the OP's proof. I just think that it isn't the whole story.
Then you do *not* understand the OP's proof *at all*. You don't even understand what a proof is.
And, BTW, the first assumption from my proof is only false when talking about infinite sets.
But that's all we're talking about. Both the primes and the naturals are infinite sets, so in the context of the discussion your assumption is absolutely false. In fact the truth or falsehood of your assumption depends entirely upon whether the set in question is finite or infinite. The negation of your statement can, in fact, be taken as the *definition* of an infinite set. It's absolutely true that a set meeting the negation of your statement is always infinite. It's also true that any set meeting that criteria must be infinite.
I cannot say that I know all of the math behind why infinite sets are allowed to break the intuitive definition of cardinality, but if it is like alot of things in math, it could be an arbitrary decision.
They are not "allowed" to do anything. They do. They did so long before Cantor developed the theory of the infinite and proved it absolutely. It's an inherent property of infinity.
There isn't an intuitive "definition' of cardinality. There is intuition. There is a definition of sets having the same cardinality. These are 2 different things. Very rarely if ever are decisions about definitions or axioms "arbitrary". It might well be the case that it's possible to choose one of 2 mutually contradictory options in some given situation without "breaking" anything, but which one to go with is generally chosen based on which one creates more interesting mathematics. Physicists have different ideas about such things. This is why, IIRC, mathematicians generally like the Axiom of Choice as it leads to many interesting consequences while physicists generally don't like it because it leads to pathological cases like unmeasurable sets. The decisions aren't arbitrary in general.
Math is dependent on the axioms (or at least the logic portion of it). I can choose the axioms I want whether or not they fit with your view of reality. Granted, it may not mean much, but in this case, I think that an alternative axiom set could produce interesting results.
You're certainly welcome to make up whatever you like, but you'd be best served by actually doing so rather than claiming that somehow your unstated (and most likely not very well thought out) new axioms should be treated as if they have any value. This is especially true given that you haven't proposed any problem with the existing mathematics, or anything interesting you hope to gain from the exercise. The fact that you don't find something intuitively obvious is completely meaningless as to the value of the mathematics, it only demonstrates a lack of imagination and a lack of effort to understand on your part.
I think it's rather silly to split hairs, but that's just me. Anyway, since you seem to be in the business of splitting hairs, I will indulge you.
It's not "splitting hairs", it's called being precise which is critical.
Let me give you a quick proof that there are not as many things in the set of primes as there are in the set of natural numbers:
1. A proper subset contains fewer things than its superset.
OK, your first assumption is demonstrably false, therefore your entire argument is unsound. The OP (and a few other people as well) have absolutely positively and beyond the possibility of a doubt demonstrated that to be the case. The problem is that you are using ambiguous terms in an attempt to prove something. You must have well defined terms or you are not going to be capable of constructing a logical argument.
What does it mean for a set to "contain fewer things" than another set? For finite sets, you can lay them out side by side and you are done. For infinite sets you'll never even get close to finishing so you need a more precise way of stating that relationship.
I suspect the only reason we have this disagreement is because you start from a different set of axioms and/or take a different route to arrive at the knowledge you so want to prove me wrong with. [And, btw, you're the one who brought the word cardinality into this. I was speaking of degrees of infinities which may be thought of as aleph numbers and thus cardinalities --or-- as a more proper degree which respects some other (and more important to some) things that we know about sets].
The only disagreement is between your intuition and the way things actually are. "more proper" "things we know about sets" These statements do not make any sense. You are talking about how finite sets work and demanding that infinite sets must conform exactly to finite sets.
But, once again, this sort of collision of models is entirely expected given Godel's theorem. The 'problem' here can be looked at in several ways but since you did not understand my geometry illustration and since you do not seem to want to learn anything from me (or to even think that is possible for you to do so, I will not be wasting my time going in to these additional insights).
No, it has nothing at all to do with Godel's theorems. There is no "collision of models". You are not conveying "insights", you are only conveying your lack of understanding of the subject.
Well, if you realize that, then you should be well on your way to understanding the proof I gave above.
That wasn't a proof. It was an unsound argument based on known false assumptions.
You've got to be kidding me. Of course it is a convenience, that's why I used it. However, it becomes much less convenient when people start splitting hairs like this.
He wasn't splitting hairs. He's absolutely right, and you are absolutely wrong.
I was speaking of the density of the set. When you do your neat folding trick where you take all the negatives and you put them in with the positives with zero at the beginning, are you not doubling the number of numbers that come after zero in that set? The geometric analogy I provided would've shown you this if you'd understood it. [And, if you don't believe this statement, I can prove it.]
Do you even know what density is? Don't worry, you don't. It is a well defined term and it does not apply to sets. Sets have no order, they have no concept of distance between numbers or anything of the sort. No, you can not prove it. The OP already proved it to be false. You just do not understand the ideas under discussion. It's ok, they're not simple concepts, but they are very well understood.
Ok, now do this. Take the first n positive numbers where n=5: {1, 2, 3, 4, 5}. Now, of those, which are primes? 2, 3, 5. Hmmm, only 3. There are more naturals than primes.
Here is a nice analogy of what you just did:
Take the set {3,5}. Wow, they're
Cardinality is simply an artificial construct and does not entirely capture the real nature of things.
No, No, No.
It does capture the real nature of things. Their real nature might be unintuitive, but it absolutely, brilliantly does capture the real nature of *infinite* things.
Say I have 5 pebbles in one hand and 5 coins in the other. Would you agree that the "5ness" contained in each of my hands is a real thing. You can take one pebble and one coin, then another and then another until my hands are empty. Each coin will be matched up with one *and only one* pebble.
The problem is not at all that it doesn't capture the real nature of things, it's that your intuition is not geared to deal with infinite quantities. That's what makes mathematics so freaking powerful. It can conclusively demonstrate things which you, quite reasonably mind you, would automatically assume to be obviously false.
Making a statement that the cardinality of the primes is exactly equal to the cardinality of the rationals doesn't lose any information about either. Primes are still indivisible, and rationals are still all the ratios of all the integers (barring zero denominators, of course) every other true statement about either is still true. You only gain *more* information about them,which can be incredibly useful.
This is the interesting thing about Godel, to me. Models work in isolation; but when they reach a certain critical mass, they start to break down.
Again, no. The only thing that is broken down by this is your intuition. The "models" work better than ever with this than they ever could without it. You have no idea how much mathematics would not exist without this.
In this particular case, maybe we need another definition of cardinality that doesn't lose the intuitive nature of the relationships between the sizes of these sets).
No, that's exactly what you don't need. Reality need not conform to your preconceptions.
You are stuck in a rut based on your long experience with finite quantities. Cardinal numbers (well, the infinite ones anyhow) are not part of the system of numbers which you are familiar with. They are characteristics of sets. Just like coins and pebbles are entirely different things, but when I hold 5 in each hand those collections share a certain property. The property of "5ness".
Infinities work differently. This is a proven fact (not a well verified theory, but an absolute and unassailable fact.) which is extremely important and incredibly useful. Cardinality does maintain the intuitive relationship you like as long as you are talking about finite sets. Once you hit the first infinity, things change. But the concept of cardinality and the definition of equal cardinality doesn't change at all by crossing that boundary. The fact that we don't need to change it to meet our finite intuition gives us new information.
Is it just that you don't like the particular example the OP gave? Are you comfortable with the fact that infinity+1=infinity, 2* infinity = infinity ( taking "infinity" to mean the smallest infinity here)? If so, then you already know and accept that the naturals have the same number of numbers as the naturals plus zero. You already know and accept that there are *exactly* as many positive integers as there are integers. Infinity * infinity = infinity expresses the fact that the naturals and the rationals have the same exact "number" of numbers.
This might be unintuitive, but it's not at all arbitrary and you can't define cardinality in such a way as to make this go away without making it a completely meaningless or inconsistent idea. It is an inherent property of the way things are....OK, "things" being the sets of numbers in question anyhow.
The continuum hypothesis says there is a set bigger than the integers but smaller than the reals.
Well, close....in the sense that most of the words are accurate. The hypothesis was that there is *not* such a set.
Also, you can't necessarily say that you'll get consistent mathematics by adding an undecidable statement to existing axioms. You also have to know that the existing axioms are already consistent. ZF set theory has not been shown to be consistent, and in fact, it has been proven that it can't be proven to be consistent from within the theory itself...well, that is...unless it actually is inconsistent.
Yeesh, I think I liked functional analysis better than formal set theory and that's saying something.
How can a mathematical truth be 'known" if it cannot be proven?
Well, *assume* for the moment that...say... the Goldbach conjecture is shown to be undecidable. This means that it can not be proven or disproven (from within the system). Therefore, it has to be true.
Why you ask? Well, if it were to be false, then you would be able to come up with integers which violate it. But then you would have proven it false and hence contradicted the undecidability.
So it cannot be proven yet nonetheless, it's now a known truth.
But not well enough!
Maybe, maybe not. Had that fascist fuck gotten the traitor's death he well earned, then it just would have meant an even longer reign of the Bush crime family.
Talk about a lose lose situation.
I DO believe that the resentment is misplaced, as it's merely "jokers" like you who give the rest of your countrymen (and countrywomen) a bad rap, not the general populace.
Then, quite frankly, you are a fucking idiot.
He's one of the few people here who actually realizes that it's a joke.
"You deserve/have the right to [item] regardless of your ability to pay." This is not a capitalist catchphrase. This is socialism at work.
No, that is not *necessarily* socialism at work. It depends entirely upon who the "deserving" party is.
If it's some poor entity, then it's socialism.
If it's a rich entity then it's fascism.
That's the difference between the left and the right. Whose benefit they want to use the state to fuck everyone else over in favor of.
So, when you see somebody claiming some big government bullshit is necessarily "socialism" without paying attention to the details which make all the difference, you can easily recognize them as a big government ninny state right wing extremist fascist shitbag.
Oh, and that's you, by the way.
the difference would be if your piss poor decisions was as a captain of a ship and we are heading for an iceberg. bitching about your past paychecks would not change the fact that we need to do something to not hit the iceberg
Hitting the iceberg is inevitable at this point. Our whole economy is greatly inflated and absolutely needs to correct itself. All that can be accomplished by increasing the artificial propping up of the economy is changing the date it occurs and *increase* the damage done by it.
So it's not at all like your iceberg analogy. Had the captain of the Titanic heeded the warnings and allowed the ship to be slowed down, then the iceberg could have been avoided. We're way past that point.
Read more history. The racists stayed with the Democratic Party for several decades, until they just couldn't stomach some of the things the left was doing. Then they left, reluctantly.
By "the things", you mean the civil rights act signed by Johnson? Heaven forbid those damn niggers be granted equal rights like the fucking constitution guarantees, right? Lost the south for a generation etc. etc. Longer actually as it turns out, but dirt poor dipshits tend to want to blame others for their problems.
Also, note that, traditionally, the "welfare leeches" were, and are, staunch Democrats to this day. Remember, Welfare was also an invention of the Democrats.
Wrong. Welfare, obviously is an invention of the left, but Republicans are the primary recipients. Agricultural subsidies, electricity, internet, phones etc etc etc. all of these are heavily subsidized and the primary recipients are rural states and regions i.e. Republicans by and large. Add in the military which is primarily a welfare scam and it's overwhelming. Then, of course, there's corporate welfare which goes primarily to the very few people who actually are sane (if evil) for voting Republican.
Almost all "red states" are welfare states. That means they receive more in taxes than they pay. Every resident of these states are on welfare. Just because they don't receive GA checks doesn't mean they aren't welfare leeches
Of course, even the "blue' welfare recipients are getting paid primarily by the blues states as they're the ones paying the country's bills, unlike the welfare states.
Wow, that's the first time I've ever been called a lefty! I'm not sure whether to applaud you, or suggest that you're not bright enough to pour sand from a boot with instructions printed on the heel, since I've been posting here for years and years, mostly to the right.
You're the one saying that people unwilling to pay for luxury services should receive them out of my pocket. That's loony lefty. I care sweet fuck all, what other nonsense you've spewed over the years. If you think poor people who refuse to pull their own weight should be paid for their failure out of my pocket, that makes you far to the loony side of left on that topic anyhow. Let me guess, you're a poor religious loon so your left when it means you get my money, but you still want to inject your fucked up hate based religious bullshit into my life, which is extremist right wing nonsense.
Though perhaps, if you're as far right as you sound like (just a tad right of David Duke), I look like the left from there.
Nope, I'm a Liberal as in what the word actually means. not how it's commonly used in America these days. You know, Liberalism, the basis of America? That's the center. I don't like extremist lefties, because I don't like the power of the state being used against me to promote some bullshit interpretation of "equality" and I fucking despise the right, or fascists, because I don't like the power of the state being used to keep the rich and powerful rich and powerful at my expense because they despise equality as in "equal under the law". So I'm in the middle and you seem to be an extremist to the left and right depending on the issue and more importantly the recipient.
Oh, you should also read enough history to know what you're talking about when you discuss religion in this country. From your rant above, you haven't a clue in the world about the subject.
"the United States is in no way founded on the Christian religion" --- Treaty of Tripoli. Passed unanimously and widely celebrated by the Americans of the time. We have fallen far driven primarily by the religious loons coming out of the woodwork with the threat of the "godless communists" They added their god bullshit to the Pledge of allegiance in the 50s dipshit. You obviously don't know a god damned thing about the history of keeping religion out of our government and the critical importance of that to America becoming what it was before the fucking loons
I was really looking forward to your reply on this topic, but if this is what you're going to give me, I'm wasting my time looking for a decent opposing viewpoint.
Opposing what? You made a non sequitor and haven't expressed anything approaching a point on the topic.
My, you have a distorted picture of the universe. HINT: the Rural Electrification Act was passed by the Democrats, not the Republicans.
Which has what to do with anything? I'm well aware of that fact. It has nothing to do with the fact that the leeches, as a whole, are the Republican voters.
Another hint: most people in the country aren't terribly interested in being part of a group that favours gun control (they pretty much all hunt), gay marriages (traditional is a way of life in the country, for good or ill), etc.
I don't favor gun control, yet I don't favor voting for fascists either. Gay marriage is a non issue except to religious extremists. This country was founded primarily on the idea that religious delusion has no place in the government in a civil society. That is the primary defining characteristic of America, so catering to those who despise this country is completely unconstitutional. Their thoughts on that issue have no legitimacy. They have no right to try and shove their evil hate based faith into our laws.
Note further that until the Democrats decided to swing to the left in the 60's, most people in the country were staunch Democrats. Hell, my grandfather would have died rather than vote Republican. (personally, I've always believed he did - he died the year that Dukakis was the Democratic candidate for President)
Yes, the racists were Democrats. They left en masse for the Republican party leading to its current make up of the bottom of the barrel of our society. Welfare leeches, religious extremists, racists and corporate welfare leeches. That's a complete breakdown of the Republican party membership. Well, ok, and deeply stupid fools who haven't paid any attention to what the Republican party has become over the last several decades.
Note, by the way, that from the point of view of most everyone else "promoting ignorant and damaging behavior at my expense is complete crap" works nicely. Too bad they think that most of the things YOU think are important are part of "ignorant and damaging behavior at my expense" to THEM.
Except this isn't symmetrical. America isn't a democracy specifically to keep shitbags like them from being able to shove their delusional religious shit into the government of a free society.
Even if it was symmetrical, it simply isn't true. I'm not asking them for shit except to stay the fuck out of other people's business. They are insisting that I be robbed to make their welfare payments and to push their religious delusions. Huge fucking difference.
An example of "ignorant and damaging behaviour at my expense" is the notion that electricity is a privilege only for urban dwellers, or the very wealthy. Sorry, I think it would be nice if every farmhouse could have lights at night. Likewise phone service. I remember having to drive 20 miles to the store with my grandfather to make a phone call. I think that the idea that he be wealthy, or move to a city, to get phone service in his home is "ignorant and damaging".
Well, then., he can damn well pay for it himself. You're a typical loony lefty who thinks government cheese is free. I pay my fucking bills. I can't afford to pay yours too. Anything of the sort is only a privilege for those who can afford it. How about you buy me a Ferrari? No, then fuck off, put your grubby little mitt away keep it out of my pocket and pull you own god damned weight.
And ditto for broadband. Five years ago, I considered it a luxury. Now it's a basic necessity, like phone service or electricity. or indoor plumbing, for that matter. And so it should be provided to everyone, not just to the self-serving idiots who can't imagine that some people don't live the way they do.
If you want to leech off of me without pulling your own weight, then not a chance in fucking hell at the very least as long as your trying to grant legitimacy to religious extremists trying to destroy America.
If you think they should be provided all these neat things I have to pay for myself, then *you* pay for it. Otherwise shut the fuck up leech.
I said nothing except the fact that we need to have a military or we will eventually get run over by another country.
No, in response to somebody else saying that the military will more likely be used for evil than good, which is obviously true, you went off about how other countries have militaries which is a complete non sequitor.
A wise man once told me, "When in Rome, good fences keep good neighbors." You seem to have agreed with this, but the rest of your rant is completely unwarranted
Knocking down other people's fences to steal their shit and to rob your own populace makes bad neighbors. That's the difference between what you're saying and reality.
While having *a* military might be a necessary evil, having *our* military is a completely unnecessary evil.
So, no, that isn't what you said at all. If that's what you meant, you should actually say that. It would still be a non sequitor though.
We have 2 choices, give them the money and hope we can smooth out a crash/depression (by no means certain) or don't and accept the 'correction' (cringely has a great piece last week likening this to a forest fire). The correction would work, it would hurt a lot, but at the end of the day it would work.
If at a bare minimum, every board member and every corporate officer are stripped of all of their assets and sent to prison for life with no possibility for parole, then and only then, is it even worth discussing. Only then would it be possible for it to have any positive effect.
Sorry, but as long as the people *responsible* for causing this problem have one penny to their names, don't you fucking dare ask me to sacrifice my future so that they can continue their failed policies living in extreme luxury. It's a completely asinine idea.
Say I made a bunch of piss poor decisions and came begging you for a handout. Would you eagerly pay up massive amounts of cash if I wasn't willing to touch my savings to cover it? I doubt it.
Don't ask me to do the same thing on a far more massive scale for people who have far more than either of us which they made through fraud.
Now, if the information gathered under this Bill results in a broadband equivalent of the Rural Electrification Act, it'll be a good thing. Annoying, to have my broadband rates raised to pay to provide broadband to the areas it isn't provided, but worth it, in the big picture.
Not worth it in the big picture. In the big picture it leads to welfare leeches claiming that they hate socialism while remaining the largest recipients of it while not contributing at all.
This sort of delusional belief is a primary contributor to the hostility of the rural people for those who pull their own weight. It's why they keep voting for fascist Republicans against their own best interests and against everybody else's.
Sorry, but promoting ignorant and damaging behavior at my expense is complete crap if you actually look at the big picture instead of the narrow view you are taking while claiming it's the big picture.
The idea is to close the loophole that ISPs and telecom operators have used for decades to keep their licenses while not having to serve those areas that are financially unrewarding to serve with the same service.
How about we don't close the loophole and quit making me pay for the rural welfare leeches? Maybe if they actually paid their own way for once they'd pull their heads out of their asses and quit pretending that they are free market capitalists when they're just the recipients of welfare. Maybe they'd quit voting against their and everybody else's best interests?
Sorry, but I am sick to death of being robbed to pay welfare to a bunch of whiny bitches who refuse to deal with reality. People always end up hating those they are forced to beg for charity so let's just quit making the people who pull their own weight the targets of the hatred of ignorant extremists.
How long will it be?
How long is forever?
Add a week.
Tell the rest of the world to get rid of their militaries and all animosity towards the rest of their fellow man, and maybe the US will get rid of theirs, too. Until then, I don't see a lot happening in this area.
But that's a totally insane response that doesn't deal with the reality. We spend more than the next 10 countries put together on our military and we use it primarily for evil purposes. We have not used it for the defense of America in 2 generations, so your "point" is complete nonsense based completely on the entirely religious belief that our leaders aren't capable of doing wrong which is, again, completely insane.
Try looking around the real world once in a while and putting down the crack pipe. You'll be quite surprised to realize that most people don't want to kill us, they just want us to stop robbing, raping and murdering them.
Looks like someone fell for it. ;)
Personally, I'd avoid burying any survivors...
I don't think he fell for it. He did specify that they themselves could specify where they were to be buried. I'm guessing he got the joke.
It won't get better for some time, so consider what I did back in 1981. Join the military in a non-bullet-catcher specialty. The new Webb G.I. Bill is a FULL four-year ride to college with a monthly stipend, so you can wait out the failing economy while adding something employers respect to your resume, then get paid to go to school for another four years.
Oh, so you mean be a welfare leech and add to the problem by putting an even greater strain on the resources of those who are doing worthwhile work?
You're encouraging increasing the problem rather than being part of the solution. How about go find a way to make yourself useful to people and add value rather than be just another leech?
How about it's tough enough these days without another douchebag signing up for the dole?
What a bunch of entitlement crybaby thieves you fuckers are.
Grow up and realize that money doesn't grow on trees and you are not entitled to rob me blind while signing up to be a tool which is primarily used as a weapon against this country. In short, quit holding out your hand begging for welfare and get to work doing something useful for which people might be willing to pay you.
We should cut the military budget. This will make it easier for red states to resist Chairman Obama's socialist mandates. He can't really have a socialist state without a big army to keep pieces of it from breaking away, can he?
You do know that the red states are pretty much the same as the welfare states with only a couple exceptions, right? You're pegging the wrong party as socialists. The republicans don't like paying for their welfare and expect the rest of us to do it for them, but they would howl like banshees if we ever tried to cut off their welfare payments. This includes the military, given its extremely excessive budget, but, again, the Republican welfare queens are the ones who keep increasing it to put more of their buddies on the dole.
Seriously, you might consider pulling your head out of your ass and looking around once in a while. You'll look less stupid if you don't keep repeating the same old blatantly false propaganda that your Dear Leaders keep shoveling down your throat, "comrade".
If you only learn one thing in your life, it should be "actions speak louder than words". I learned that when I was a little child, so I never fell for any idiotic cults or scams as you obviously have.
No, it's only the fundamentalist Christians who would think that, in the rest of the world you can be a Christian without believing every single word in the Bible is the absolute truth.
No, you really can't be a Christian without believing every single word, since at that point you've either decided that you know better than god, or that god wasn't involved, hence it's not really a religion. You can call yourself a Christian, but so can anybody, and you can find backing for any sick shit you want to do in the Bible, making it totally worthless for any "moral" guidance.
BTW, I'm not a Christian, I'm just saying you shouldn't judge a faith or set of beliefs by the crazy extremists, otherwise you end up with the "because most suicide bombers are Muslims, most Muslims are suicide bombers" type of argument.
All Muslims are suicide bombers or similar. The Koran is quite explicit about it. All Christians and Jews are evil vile brutal murdering scum as well.
The thing you're missing is that there really aren't very many of any of the above in the world these days. If you go read the bible and the Koran, that will be obvious. The ones who believe it are absolutely required by their gos to be savage murderers. Most who claim to be of a given faith don't really believe it though. We'd still have stoning for a whole host of petty crimes if they did. Well, I guess there are a fair amount of Muslims who actually believe their idiot book, but Christians of faith mostly died out in the Enlightenment, and that is an absolutely great thing. Look how far the world has come since.
I sometimes wonder if Martin Luther, had he realized what would be the ultimate fate of his cause, would have snuck back up to the cathedral door and pulled the nails out.
No, the holocaust alone would have made it all worthwhile to him...well if it had succeeded anyhow. It was his plan in most of the details even, don't you know?