Interesting point. Actually both you and your parent post are correct, the system composed of a PC and Windows was the most open system, not because either the PC or windows was, but because Microsoft didn't make any PC's, and IBM made both PC's and OS/2. This made the combination of OS/2 and a PC (even if IBM didn't make it) the more closed system.
"Most Mormons will put their religion before their profession."
I agree. But the Republican party leadership here in Utah is not 'Most Mormons'. The way that they act in their profession (politics) tells me that they do put their profession first.
Second, point, 'Utah Mormons' tend to act a bit differently from those outside the state. Perhaps you are unfamiliar with the utah variety?
You misunderstood what I meant about scientific rigor. I can pray to God, and get an answer. This answer is irrefutable proof that someone answered. However, this proof is something that is only valid for me. I can't show it to you. You have to take my word on it. Plus, not getting an answer does not prove the non-existance of God, he simply did not want to reply. This means that you have no reliable way to test my results. This breaks the scientific method.
"Utah, and the US in general, are not examples of democracy."
You are mostly correct here. There are sometimes laws put on the ballot for a direct vote here in Utah, but most are done by the legislature. However this does not change what I said. I was talking about the group-think, not this bill in particular. Democracy forgot to "Never underestimate the power of stupid people in large groups"
"The Mormon church is very much involved in politics. They run the political scene in Utah"
I wish that I could say that this was total nonsense. It is not true, but there is a reason that a person might think that. The Republican Party Leadership in Utah is mostly Mormon. (no surprise, ~70% of Utah is Mormon) It is in the interests of the Republican Leadership in Utah to give this impression. And they do a good, subtle job of it. (It has to be subtle, if it wasn't, church headquarters would do something about it. - as another reply stated.) The truth is that The Republican Party leadership runs politics in Utah, and one of the tools they use to hold power is to cater to a few of the more visible Mormon beliefs. This assures them of most of the mormon vote. It is sad that in one of, if not the most, Republican states, that the republicans still monkey with voting districts etc. to marginalize the Democrats.
It is my opinion that if Church leaders came out and said that being a Republican was evil, most of the Republican leadership would leave the church before leaving the Rebuplican Party. They are Mormon in name, and Republican at heart. Unfortunately, most Utahn's don't see this. Nor did you.
As for the BoM, Most mormons have a simpilistic interpretation of the history in it. (All native Americans decended from BoM people, The Jaredites killed off all Jaredite decendants, Final battle in New York - etc. ) This view is unsupportable from either a logical view or an archaeological one. On the other hand, the BoM does fit several things in history fairly well. It pegs the Olmec civ. timeline within a couple hundred years.[1] In short it is a better guide to Central American history than anything written prior to the early 1900's, and was published 80+ years earlier. It is as good a history book as the Bible is.[2]
"you learn more about a man from his enemies than his friends." And you can learn a lot about someone from the enemies he makes. Most anti-mormons are liars and frauds. The rest are filled with a mild hate that you showed. As for the analogy: Toyota can make a Toyota, The consumer magazine couldn't make a go-cart. I would only trust the magazine for information that was a comparison to other cars, or information that Toyota would want to hide.
People you should not trust for information about a religion are A) Active members of another one (especially ministers etc.) or B) Former members, like yourself.
[1]Both the BoM and archeological evidence have about that margin of error on the subject.
[2] Yes, this means fairly lousy on most points, with a few exceptions.
You information is a bit off. It is just that liquor licences are hard to come by and there is a bit of a loophole for 'private clubs'. So a lot of places that would normally just serve liquor call themselves a private club and charge a membership fee to get around the harder licence. There are normal bars.
Also you did not RTFA, because this is an opt-in thing. Your porn will only be blocked if you ask your ISP to block it, or install the free blocking software from the ISP. (ISP's choice of which you get) For this bill, "Won't somebody think of the childeren?" wasn't just retoric.
On the other hand, I agree with your comment about the red tape. For this bill and the liqour laws. Stupid meddling in other peoples lives.
There are more Mormons outside the US than in. And there are more Mormons in the US that are not in Utah than are in Utah. I think California has almost as many Mormons as Utah.
The problem is when any group is a vast majority of the population, They start to do funny things. Especially when the defining charactoristic is something that you cannot apply scientific rigor to. This is the problem with 'Utah Mormons" - yes, they do tend to act differently than Mormons outside of Utah/Idaho.
This is why Democracy is a lousy form of government. It's only real asset, is that it takes longer to corrupt than most other forms of government.
And yes, IAA Mormon. Utah is not 'backward' (or advanced..) It just suffers from too much group-think. That it is Mormon group-think is less important.
Your grasp of history is sadly lacking. The Dark Ages were not caused by the Catholic Church putting scientists to death. If for no other reason than that the Dark Ages began in the 5th century AD, and the Catholic Church did not have all that much power until 1000 AD or so.
Precisely. You have to explicitly enforce all rules that you want followed. Which means among other things you have to figure out all of the condidtions. And be carefull that you don't impose 'rules' that you are not aware of.
To put it another way, if you want the design to function under 'changed conditions' evolove it under changing conditions. And remember, the design is aware of all of the conditions that it is under, whether you are or not.
It is a tricky process, one that most designers are not prepared to deal with. (obvious statement I guess, if they were, darwinian designs would be commonplace, instead of research curiosities.)
But the 'guns of the government' are already in the way. They are called copyright and patent laws.
I am a free market guy as much or than the next person, but this is not a case of free market or no, this is a debate over what and how much the government is going to intervene.
Cheat? Of course. They don't know your rules. The only rules that they know are the laws of physics (the real ones, not our approximations) and those that you explicitly impose. In this case, the researchers did not impose a 'you can't use EM radiation from something else' Should have put it in a Farady cage.
This property is actually why darwinian produced designs are so attractive. They can make use of anything. Including laws of physics that we haven't discovered yet.
Longer than it would take for the computer(s) that were involved to be blacklisted off the internet and any communications between those computers and any TC computers disabled.
Try learning a little about what TC is. Unlike most security schemes, TC cannot be broken by having a few small cracks in the system. It has to have a huge hole in it.
The only huge hole that I can see is the adoption phase. If it never takes off it is powerless. Otherwise...
" You also have to keep in mind that IBM is a big player in linux development."
Yes, they are. Why? Are they in it for the freedom? the free software ($$)? Freedom from Microsoft? Competitive advantages?
Note, not all of these possibilities are incompatible with Trusted Computing. Only one is, really. IBM can get freedom from microsoft even if TC locks down all computers. They are big enough to get their linux binaries signed.
"What the harddware manufactures care about is selling hardware."
The biggest threat to IBM is for Microsoft to stop letting IBM stuff interoperate with TC. If TC only works with Windows, this is possible. If an IBM compiled linux binary works, and enough people are using linux on IBM hardware, then Microsoft can't lock them out. This just may be insurance that Microsoft can't lock IBM out, not IBM trying to get a wider market, or careing about your 'freedom'.
It's a P4. Remember they slow their clock down if the chip gets too hot. Encoding software fills the pipeline pretty well, that's why the P4 does so well at it. So the chip used a lot of power. My guess is that stock cooling just isn't good enough on high-clockspeed P4's.
Redo this on a slower P4 or an athlon{XP,64} and I don't think that you will see a difference. That said, if they did not do several trials of this test..... 0.5% difference is likely less than the margin of error.
"Expect major disruptions if they ever decide to revamp some kernel component like paging or smp."
If you would recall a recent interview Linus did, he said that there probably wasn't going to be anything like that in the near future, except possibly the tty stuff. Mostly just work on drivers and such. I would not be too surprised if the real reason that there is no 2.7 branch now is because there simply isn't any major system that needs a rewrite.
"You NEED to take the waste of the burning and turn it into something can be burned again."
And plants do precisely that. Using the sun's power no less. The only difficulty is to get enough plants so this happens as fast as we burn, and still grow enough food etc. This may not be a simple task, but I believe it is doable.
"Appel has lined up federal grant money to help build demonstration plants to process chicken offal and manure in Alabama and crop residuals and grease in Nevada."
"Then you have no rights because no god exists to give them"
I think you are missing something here. An unalienable right is one that you can't give or sell away. I have an unalienable right to think what thoughts I choose, because I can't give that away. I can't not choose what I think about, (psycologists may disagree here, but the point is I have more choice about what I think about than anyone else, and I can't change that.) I do not have an unalienable right to my paycheck. I can give or sell away those rights. (and do, I sell my rights to the money when I buy stuff with it)
Now, if god created man, he created man the way that he is. (except the fall, but that obviously did not change all aspects, besides, he let those changes remain...) This means that we have god given unalienable rights. We have those rights because we do not have the ability to give them away. And we do not have that ability because god made us that way.
However, the existance or not of god does not change the way we are. We are the way we are regardless of how we became that way. Hence we still have these unalienable rights even if we are nothing less than evolved animals.
Yes, it would. In fact, you can do a lot of useful things in graph theory by defining a semiring with the "usual" operations over the the reals union plus and minus infinity. However, you have to be very careful about what operations you allow and how they work.
I have wondered about this before. Things like tan(x) would be continuous functions. Thanks.
infinity + 1 > infinity -- This makes no sense, because infinity is defined as being something which is greater than all the other members of the set, but now we just found one that's even greater.
infinity + 1
infinity + 1 = infinity -- This is at least somewhat self-consistent.
The proper answer is to disallow the operations that let you prove things like 1 = 0. For example, saying that infinity + 1 = infinity is fine, and even intuitive. However, we should not be allowed to say that infinity - infinity = 0. Similarly, saying that 1/infinity = 0 is ok, but infinity/infinity = 1 is not. Or you can simply disallow all operations on infinity, but then one wonders why it's polluting our otherwise nicely consistent set."
Mostly agree. I especially like "This is at least somewhat self-consistent." I think you could also say 'most consistent' as well, just not completely consistent. I would like to point out that the reason that infinity - infinity != 0 is because infinity + 1 = infinity. I suspect that infinity is not consistent with addition. For example:
1 + infinity - infinity = ?
The answer depends on the order of operations - do we do the 1 + infinity first or the infinity - infinity? But, addition is not supposed to depend on the order of operations. This is similar to your (a + b) + c = a + (b + c) example from one of your earlier posts. Which BTW depends on infinity - infinity = 0 being true. So, which should it be? Or is there any completely consistent system? It seems that it would require infinity + 1 to be undefined, and instead use the limit of I + 1 , as I -> infinity to be infinity (as it is). Or in other words, in any equation that involves infinity, you must use 'limit of f(x) as I -> infinity' and not 'equals'. And as you said, "The difference between the two concepts [sum and limit] is enormous"
The limit of 1 / I, as I -> 0 is infinity. but 1 / 0 is undefined, not infinity. Just so 1 / I as I -> infinity is 0, does not mean 1 / infinity = 0 too.
To look at an integral as an infinite sum of infinitesimals is to define an unsolveable problem, but looking at it as a limit is a problem that can be solved. I suspect this is the reason mathematicians stopped thinking this way, not because it doesn't make sense, but because it doesn't work. Remember, Newton was not primarily a mathematician, he was a scientist who had to invent his own math tools. It had to work for it to be of any use to him.
"didn't we just spend a day or two showing that "natural numbers" does not include infinity?" No. you waved at it with a tautology and a series that said nothing about it, and then threw in a definition (that has some value at least) What I was trying to point out is that that definition is inconsistent. I'll try again. After that, it may be best to agree to disagree. (although I have enjoyed the conversation)
The value of each of the natural numbers is equal to the number of natural numbers less than or equal to itself. This property is why they are useful as, and called the 'counting numbers' You can count things with them. By definition the number of natural numbers is infinite. Since the value and the number of values are linked, if the number is infinite then one of the natural numbers must have the value of infinity. If you don't have one with the value of infinity, then you don't have an infinite number of them, or at some point before you reach infinity the relationship between value and number must break down. Specifically, for some N in the natural numbers, there must be more natural numbers less than or equal to N than N. If that is true, well mabe I don't want to see the proof, the proof that 1 is greater than 0 that my roommate did was bad enough:)
This is going to sound funny, but... Goody! That means that I can define a new set, the squissh set!;) It is the set of natural numbers plus aleph null. There is such a set already. And it has the properties that I have been describing. Although, it would be far more practical to simply include the statement 'except where <values> are equal to aleph null' in each of those cases you gave.
"Hmm, I thought 1/infinity wasn't zero, but now you say it is." "If you can't come up with a consistent answer, then your theory is flawed." Yes, I see my error here. I have been inconsistent somewhere, and I will now correct it. I have been using the term 'equals' (=) where I should have been using the term 'limit'.
The limit of this: R + N*(1/I) , as I -> infinity, is R, where R=a real number, N=a finite integer.
If you patch my previous statements with this, they are now consistent.
That's not what you're doing. You're changing the definition of "tomato" to include all fruit currently accepted as "tomato", plus one particular type of pumpkin."
Unless, when they came up with the definition of 'tomato' there was this one very difficult tomato that the gardners did not know how to deal with, and so they exlcuded it from the definition, to make it easier to say 'This is how to grow all tomatoes.'?
There are lots of examples in math and science where ignoring some small problem makes many things much easier, but including them vastly complicates things, even to the point of making them intractable with the current tools/understanding. But if you try to put those small problems back in and deal with them, while it complicates things greatly, you can gain new insight that could not be reached any other way. And sometimes you have to leave those problems out until you have developed the proper tools to deal with them, in this case, tools to deal with infinity, and infinitesimals. Like the limit. (which stupid me, I forgot to use! It has been years since I touched math...)
"Back to the original question: do you accept that we can have a type which does not have a largest possible value?"
Sure, I just don't see how the set of natural numbers is one of these. Since we have 'numbers' like infinity that we can use, at any rate. Take away that concept, and then yea, there is no largest natural number.
In some way, the equations infinity + 1 = infinity, and 1/infinity = 0 are wrong. It is just that we can usually ignore the rounding error. Indeed, for the latter, you can't get a smaller rounding error! But this reality is inherant in the concept of the integral. We are 'summing' an infinite number of 1/infinity width, finite height volumes, and coming up with finite, non-zero results. If 1/infinity was truely zero, then the answer to all integral problems would be zero. You said "This argument hasn't made any sense in mathematics since the 1600s" Why? Because mathematicians stopped thinking this way and started thinking about limits instead?
I haven't done the math, but I would bet that the combined land use of buildings + paved areas is a lot greater than that needed by solar power. That and the buildings can usually do both.
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Interesting point. Actually both you and your parent post are correct, the system composed of a PC and Windows was the most open system, not because either the PC or windows was, but because Microsoft didn't make any PC's, and IBM made both PC's and OS/2. This made the combination of OS/2 and a PC (even if IBM didn't make it) the more closed system.
People did talk differently in the past, and this sounds old to me.
(wouldn't be too surprised if it did but... lies and statistics.)
I agree. But the Republican party leadership here in Utah is not 'Most Mormons'. The way that they act in their profession (politics) tells me that they do put their profession first.
Second, point, 'Utah Mormons' tend to act a bit differently from those outside the state. Perhaps you are unfamiliar with the utah variety?
"Utah, and the US in general, are not examples of democracy."
You are mostly correct here. There are sometimes laws put on the ballot for a direct vote here in Utah, but most are done by the legislature. However this does not change what I said. I was talking about the group-think, not this bill in particular. Democracy forgot to "Never underestimate the power of stupid people in large groups"
And here in Utah, some do.
I wish that I could say that this was total nonsense. It is not true, but there is a reason that a person might think that. The Republican Party Leadership in Utah is mostly Mormon. (no surprise, ~70% of Utah is Mormon) It is in the interests of the Republican Leadership in Utah to give this impression. And they do a good, subtle job of it. (It has to be subtle, if it wasn't, church headquarters would do something about it. - as another reply stated.) The truth is that The Republican Party leadership runs politics in Utah, and one of the tools they use to hold power is to cater to a few of the more visible Mormon beliefs. This assures them of most of the mormon vote. It is sad that in one of, if not the most, Republican states, that the republicans still monkey with voting districts etc. to marginalize the Democrats.
It is my opinion that if Church leaders came out and said that being a Republican was evil, most of the Republican leadership would leave the church before leaving the Rebuplican Party. They are Mormon in name, and Republican at heart. Unfortunately, most Utahn's don't see this. Nor did you.
As for the BoM, Most mormons have a simpilistic interpretation of the history in it. (All native Americans decended from BoM people, The Jaredites killed off all Jaredite decendants, Final battle in New York - etc. ) This view is unsupportable from either a logical view or an archaeological one. On the other hand, the BoM does fit several things in history fairly well. It pegs the Olmec civ. timeline within a couple hundred years.[1] In short it is a better guide to Central American history than anything written prior to the early 1900's, and was published 80+ years earlier. It is as good a history book as the Bible is.[2]
"you learn more about a man from his enemies than his friends." And you can learn a lot about someone from the enemies he makes. Most anti-mormons are liars and frauds. The rest are filled with a mild hate that you showed. As for the analogy: Toyota can make a Toyota, The consumer magazine couldn't make a go-cart. I would only trust the magazine for information that was a comparison to other cars, or information that Toyota would want to hide.
People you should not trust for information about a religion are A) Active members of another one (especially ministers etc.) or B) Former members, like yourself.
[1]Both the BoM and archeological evidence have about that margin of error on the subject.
[2] Yes, this means fairly lousy on most points, with a few exceptions.
Also you did not RTFA, because this is an opt-in thing. Your porn will only be blocked if you ask your ISP to block it, or install the free blocking software from the ISP. (ISP's choice of which you get) For this bill, "Won't somebody think of the childeren?" wasn't just retoric.
On the other hand, I agree with your comment about the red tape. For this bill and the liqour laws. Stupid meddling in other peoples lives.
The problem is when any group is a vast majority of the population, They start to do funny things. Especially when the defining charactoristic is something that you cannot apply scientific rigor to. This is the problem with 'Utah Mormons" - yes, they do tend to act differently than Mormons outside of Utah/Idaho.
This is why Democracy is a lousy form of government. It's only real asset, is that it takes longer to corrupt than most other forms of government.
And yes, IAA Mormon. Utah is not 'backward' (or advanced..) It just suffers from too much group-think. That it is Mormon group-think is less important.
Your grasp of history is sadly lacking. The Dark Ages were not caused by the Catholic Church putting scientists to death. If for no other reason than that the Dark Ages began in the 5th century AD, and the Catholic Church did not have all that much power until 1000 AD or so.
To put it another way, if you want the design to function under 'changed conditions' evolove it under changing conditions. And remember, the design is aware of all of the conditions that it is under, whether you are or not.
It is a tricky process, one that most designers are not prepared to deal with. (obvious statement I guess, if they were, darwinian designs would be commonplace, instead of research curiosities.)
I am a free market guy as much or than the next person, but this is not a case of free market or no, this is a debate over what and how much the government is going to intervene.
This property is actually why darwinian produced designs are so attractive. They can make use of anything. Including laws of physics that we haven't discovered yet.
Try learning a little about what TC is. Unlike most security schemes, TC cannot be broken by having a few small cracks in the system. It has to have a huge hole in it.
The only huge hole that I can see is the adoption phase. If it never takes off it is powerless. Otherwise...
Yes, they are. Why? Are they in it for the freedom? the free software ($$)? Freedom from Microsoft? Competitive advantages?
Note, not all of these possibilities are incompatible with Trusted Computing. Only one is, really. IBM can get freedom from microsoft even if TC locks down all computers. They are big enough to get their linux binaries signed.
"What the harddware manufactures care about is selling hardware."
The biggest threat to IBM is for Microsoft to stop letting IBM stuff interoperate with TC. If TC only works with Windows, this is possible. If an IBM compiled linux binary works, and enough people are using linux on IBM hardware, then Microsoft can't lock them out. This just may be insurance that Microsoft can't lock IBM out, not IBM trying to get a wider market, or careing about your 'freedom'.
So, just why is IBM into Linux, anyway?
That said, eventually there will need to be some major core changes. I hope (and think) that Linus will be smart enough to say 'It's time for 2.7.x!'
Redo this on a slower P4 or an athlon{XP,64} and I don't think that you will see a difference. That said, if they did not do several trials of this test ..... 0.5% difference is likely less than the margin of error.
If you would recall a recent interview Linus did, he said that there probably wasn't going to be anything like that in the near future, except possibly the tty stuff. Mostly just work on drivers and such. I would not be too surprised if the real reason that there is no 2.7 branch now is because there simply isn't any major system that needs a rewrite.
They didn't. It is still there. No patches needed.
And plants do precisely that. Using the sun's power no less. The only difficulty is to get enough plants so this happens as fast as we burn, and still grow enough food etc. This may not be a simple task, but I believe it is doable.
"Appel has lined up federal grant money to help build demonstration plants to process chicken offal and manure in Alabama and crop residuals and grease in Nevada."
I think you are missing something here. An unalienable right is one that you can't give or sell away. I have an unalienable right to think what thoughts I choose, because I can't give that away. I can't not choose what I think about, (psycologists may disagree here, but the point is I have more choice about what I think about than anyone else, and I can't change that.) I do not have an unalienable right to my paycheck. I can give or sell away those rights. (and do, I sell my rights to the money when I buy stuff with it)
Now, if god created man, he created man the way that he is. (except the fall, but that obviously did not change all aspects, besides, he let those changes remain...) This means that we have god given unalienable rights. We have those rights because we do not have the ability to give them away. And we do not have that ability because god made us that way.
However, the existance or not of god does not change the way we are. We are the way we are regardless of how we became that way. Hence we still have these unalienable rights even if we are nothing less than evolved animals.
I have wondered about this before. Things like tan(x) would be continuous functions. Thanks.
infinity + 1 > infinity -- This makes no sense, because infinity is defined as being something which is greater than all the other members of the set, but now we just found one that's even greater.
infinity + 1 infinity + 1 = infinity -- This is at least somewhat self-consistent.
The proper answer is to disallow the operations that let you prove things like 1 = 0. For example, saying that infinity + 1 = infinity is fine, and even intuitive. However, we should not be allowed to say that infinity - infinity = 0. Similarly, saying that 1/infinity = 0 is ok, but infinity/infinity = 1 is not. Or you can simply disallow all operations on infinity, but then one wonders why it's polluting our otherwise nicely consistent set."
Mostly agree. I especially like "This is at least somewhat self-consistent." I think you could also say 'most consistent' as well, just not completely consistent. I would like to point out that the reason that infinity - infinity != 0 is because infinity + 1 = infinity. I suspect that infinity is not consistent with addition. For example:
1 + infinity - infinity = ?
The answer depends on the order of operations - do we do the 1 + infinity first or the infinity - infinity? But, addition is not supposed to depend on the order of operations. This is similar to your (a + b) + c = a + (b + c) example from one of your earlier posts. Which BTW depends on infinity - infinity = 0 being true. So, which should it be? Or is there any completely consistent system? It seems that it would require infinity + 1 to be undefined, and instead use the limit of I + 1 , as I -> infinity to be infinity (as it is). Or in other words, in any equation that involves infinity, you must use 'limit of f(x) as I -> infinity' and not 'equals'. And as you said, "The difference between the two concepts [sum and limit] is enormous"
The limit of 1 / I, as I -> 0 is infinity. but 1 / 0 is undefined, not infinity. Just so 1 / I as I -> infinity is 0, does not mean 1 / infinity = 0 too.
To look at an integral as an infinite sum of infinitesimals is to define an unsolveable problem, but looking at it as a limit is a problem that can be solved. I suspect this is the reason mathematicians stopped thinking this way, not because it doesn't make sense, but because it doesn't work. Remember, Newton was not primarily a mathematician, he was a scientist who had to invent his own math tools. It had to work for it to be of any use to him.
"didn't we just spend a day or two showing that "natural numbers" does not include infinity?" No. you waved at it with a tautology and a series that said nothing about it, and then threw in a definition (that has some value at least) What I was trying to point out is that that definition is inconsistent. I'll try again. After that, it may be best to agree to disagree. (although I have enjoyed the conversation)
The value of each of the natural numbers is equal to the number of natural numbers less than or equal to itself. This property is why they are useful as, and called the 'counting numbers' You can count things with them. By definition the number of natural numbers is infinite. Since the value and the number of values are linked, if the number is infinite then one of the natural numbers must have the value of infinity. If you don't have one with the value of infinity, then you don't have an infinite number of them, or at some point before you reach infinity the relationship between value and number must break down. Specifically, for some N in the natural numbers, there must be more natural numbers less than or equal to N than N. If that is true, well mabe I don't want to see the proof, the proof that 1 is greater than 0 that my roommate did was bad enough :)
This is going to sound funny, but... Goody! That means that I can define a new set, the squissh set! ;) It is the set of natural numbers plus aleph null. There is such a set already. And it has the properties that I have been describing. Although, it would be far more practical to simply include the statement 'except where <values> are equal to aleph null' in each of those cases you gave.
"Hmm, I thought 1/infinity wasn't zero, but now you say it is." "If you can't come up with a consistent answer, then your theory is flawed." Yes, I see my error here. I have been inconsistent somewhere, and I will now correct it. I have been using the term 'equals' (=) where I should have been using the term 'limit'.
The limit of this: R + N*(1/I) , as I -> infinity, is R, where R=a real number, N=a finite integer.
If you patch my previous statements with this, they are now consistent.
That's not what you're doing. You're changing the definition of "tomato" to include all fruit currently accepted as "tomato", plus one particular type of pumpkin."
Unless, when they came up with the definition of 'tomato' there was this one very difficult tomato that the gardners did not know how to deal with, and so they exlcuded it from the definition, to make it easier to say 'This is how to grow all tomatoes.'?
There are lots of examples in math and science where ignoring some small problem makes many things much easier, but including them vastly complicates things, even to the point of making them intractable with the current tools/understanding. But if you try to put those small problems back in and deal with them, while it complicates things greatly, you can gain new insight that could not be reached any other way. And sometimes you have to leave those problems out until you have developed the proper tools to deal with them, in this case, tools to deal with infinity, and infinitesimals. Like the limit. (which stupid me, I forgot to use! It has been years since I touched math...)
"Back to the original question: do you accept that we can have a type which does not have a largest possible value?"
Sure, I just don't see how the set of natural numbers is one of these. Since we have 'numbers' like infinity that we can use, at any rate. Take away that concept, and then yea, there is no largest natural number.
In some way, the equations infinity + 1 = infinity, and 1/infinity = 0 are wrong. It is just that we can usually ignore the rounding error. Indeed, for the latter, you can't get a smaller rounding error! But this reality is inherant in the concept of the integral. We are 'summing' an infinite number of 1/infinity width, finite height volumes, and coming up with finite, non-zero results. If 1/infinity was truely zero, then the answer to all integral problems would be zero. You said "This argument hasn't made any sense in mathematics since the 1600s" Why? Because mathematicians stopped thinking this way and started thinking about limits instead?
I haven't done the math, but I would bet that the combined land use of buildings + paved areas is a lot greater than that needed by solar power. That and the buildings can usually do both.