Isn't this the kind of thing Free Software was supposed to be against?
No. You have confused trademark with copyright - and possibly with patent.
Anyone can distribute their own flavor of Linux and call it Linux without being threatened by lawsuits over trademarks?
No. You have confused trademark with copyright again.
According to your posting, Microsoft should be able to release an operating system and call it Linux. I am glad they are not allowed to do so under TRADEMARK law.
+5, Insightful? Moderators must be on crack again.
Free (as in speech) software is about being able to reuse others code through PRIOR licensing of copyrighted material. However, I can not make my own DISTRIBUTION based on the Linux kernel and call it RedHat, as that NAME is already trademarked. Nothing in the free software philosophy contradicts that, or says that I should be able to use the RedHat name.
Linux is a word that is used in a fairly generic way by many, but the laws require the trademark holder to defend the mark in business - or lose it to 'public domain' as a generic term. Once lost, people are free to use the word in any way they want - we could have linux soap, Windows applications, Linux air conditioners, soda pop, ANYTHING.
(that's a joke, I say, that's a joke son!)
Wish I had moderator points; you would get a -1, Overrated just for being so wrong.
A quick google searsh shows 4,260,000 web sites found using the search phrase 'world vista', 6,870 sites found using '"World Vista"' and 3,940 sites using 'worldvista'
From their site: WorldVistA is a charitable organization: a nonprofit, 501(c)(3) public-benefit corporation. WorldVistA was incorporated March 18, 2002 to measurably improve health worldwide by making medical software better and more accessible.
Note the specific reference to software. So how is Microsoft able to trademark this combination of words? It is already in use in the same industry, making it ineligable for trademark I thought.
Dirt is also good insulation as well as being great thermal mass. Those two attributes make it not as practical as you would seem to think.
Anything below the frost line is deep enough - some places could require 6-8 feet, I would think 2-3 feet would be more common.
Your efficiency would start out at adequate but, as the temp of the ground immediately surrounding the copper pipe rose the efficiency would drop quickly. At some point temp in would equal temp out, and your 'better idea' no longer works.
I agree with your second point - except that I am not sure that the Federal government has any business regulating it, it should be the state government or the pocket book of 'the people' who are doing the regulating.
On the first point, you are just wrong.
The gasoline in the 1920's was used in very low compression engines which were very wasteful of gasoline - but gasoline was cheap and plentiful, so who cared!
The car manufacturers wanted to sell higher compression engines in heavier cars, but the higher compression caused unpleasant (and damaging) knocking when using the gasoline of the day. Many things were tried to reduce or eliminate knocking - including Bromine, Selenium, Iodine, Anine dyes, and ethol alcohol, as well as benzine (sp?), sulpher, and other aromatics.
Tetraethyl lead was chosen for a variety of reasons, including the patentability of the process for making it. Delco labs was a pioneer in anti-knock fuel additives, and was swallowed by GM as their Research Division. After a time, the Ethyl Lead production was spun off as a separate company, the Ethyl Additive Corp.
From internal memos from GM, the process that produced tetraethly lead cost about one penny per gallon and was sold to the fuel distributers at three cents per gallon - for a net profit of two cents per gallon. The memos estimated that GM would have a profit from the production of tetraethyl lead of $60,000,000 per year BACK IN THE 20's! I have no way of calculating the equivalent in todays dollars, but would assume it would be significant!!
The production facilities were toured by professional chemists who were horrified by the lack of safety precautions. Even when 8 workers died from lead poisoning in the first year - the go-live had to be postponed when the trial runs poisoned the workes to the point they were unable to work on the live production line - the main concern (as stated in internal memos) was the bad PR which could reduce the demand for the lead additive, not the safety of the workers.
Bottom line, it DID NOT cost more to produce gasoline without lead, unless you figure the PROFIT from selling the lead additive reduced the cost of producing the gasoline.
Unless you are talking about lately when regular (with lead) was removed from the market in favor of unleaded gasoline. In that case, I would probably claim that the increase in cost (if any) would be in the refinery hardware needed to increase the octane (the anti-knock component of the gasoline), not in the production of the gasoline itself. In addition, the lead was used as a lubricant in engins with soft valve seats and would have to be replaced with another lubricant - or the valve seats would have to be made of harder materials that did not require the lead buffering.
On a related (but not directly) note, my fiancee buys the higher 89 octane gasoline for a '97 Jeep. I have tried to explain that the higher octane is not needed if there is no knocking, but she seems to think 'a higher price means it is better gas'. My thoughts on the subject are that, if there is no knocking, there is no need for additional anti-knock compounds (higher octane), the vast (or maybe half-vast?) majority buy the lowest price gasoline, so the higher octane - and higher priced - gasoline is more likely to be older with the attending greater possibility of 'stuff' (like water, gum, and other crap and crud), as well as a suspicion of pricing motives when there is, and remains, a $0.10 price difference between the lowest grade (87 octane locally) the mid grade (89 octane) and the highest grade (91 octane locally). As long as she is spending her money on gasoline, I don't care which octane she choses, but will continue to try to educate her.
Octane is added to reduce knocking - or premature detonation/ignition of the air/gas mixture. That means higher octane gasoline DOES NOT BURN AS WELL as lower octane gasoline - but we are charged MORE for the gasoline that burns WORSE. Interesting marketing concept.
It wasn't enlightened self-interest which took the lead out saving millions of kids from brain-damage.
You are right, the producers of the lead based paint and gasoline had better (and lead free) products available, but people would not buy them as they were more expensive than the products containing lead.
So what did the paint producers, the gasoline producers, and the insurance companies do? They lobbied for legislation REQUIRING the elimination of lead in the products. When the laws were passed the profits of the paint companies rose, the profits of the gasoline companies rose, the insurance companies paid out less for care of lead-based brain-damage, and everyone was happy - except the end user who was paying more for what they needed.
Interesting that regular gasoline (with lead) was cheaper than unleaded - and that many cars could not run with unleaded because they had valves that would burn up without the buffering of the lead additives.
Interesting that the lead was BOUGHT and added to the gasoline, but the price of gasoline WITHOUT the additive was higher - and when the additive was banned, only the higher priced version was available. The gasoline producers saved money by not having to buy the additive, and made more profit by selling the same thing at a higher price. SWEEEEET!
Also interesting is that you can buy tetraethyle lead gasoline additives in auto performance stores over the counter for those cars that have to have lead to protect their valves.
Or take the case of auto accidents. For decades Detroit couldn't sell safe cars. Few manufacturers tried and they failed.
Good point. Why did 'Few manufacturers tr[y]'? because the buying public was being led by the nose through advertising (by the auto manufacturing companies) to buy the larger, more powerful models (6000 SUX!! YEAH BABY!!), not the safer, more fuel efficient models.
When there was more profit in selling land barges that got terrible gas milage, that weighed several thousands of pounds, and that were capable of speeds their parts (tires, brakes, etc) could not manage, then there was no point in trying to also make safe models.
Remember the cost/benefit rules the auto manufacturers live by - if it is cheaper to settle a few claims for millions of dollars than to re-tool to FIX a problem, the problem doesn't get fixed no matter the number of people killed.
Remember 'Unsafe at Any Speed' by Ralph Nader?
Remember when seat belts were optional and no one would pay extra for them? Remember when they were required in the front seat but not the back seat? If the back seat is so much safer, why not put the driver in the back seat 'for their own safety'?
Also, when an 'optional' extra cost safety feature is made mandatory, the playing field is leveled in that ALL the manufacturers have to include the 'feature' - and the base price can be jacked up to cover the costs.
The results were immediate and today with many more cars on the road, we have fewer deaths than we did in the '60s.
Interesting. You do realize, don't you, that the government allows MORE depreciation against taxes for large, unsafe, non-fuel efficient vehicles than for smaller, fuel efficient vehicles - which means the government seems to be advocating usage of larger, more dangerous, gas-guzzling trucks over the use of safer, more environmentally friendly vehicles.
[D]on't blame the government just because it can't ensure the safety of every citizen.
It has never been and should not now be the job of the government to ensure the safety of ANY citizen. The Federal government sould be most concerned with the interaction between the states - UNITING the states into the UNITED STATES - and with the interaction of this nation with outher nations and coutries. The states should be most concerend with the providing of infrastructure of needed services at the best cost to their citizens, not to the highest campaign contributer or the company most willing to hire the office holder when they leave office.
Damn, I am long winded! I am sure I had a point in there somewhere, if anyone finds it, could you point it out for me?
Michigan isn't satisfied and is proposing banning all over-the-net wine orders on the flimsy reasoning that kids will be able to buy booze without government control.
Interesting thought process; before the ruling it was OK for Michigan kids to buy Michigan booze over the internet, but it was NOT OK for Michigan kids to buy OUT-OF-STATE booze over the internet. Now either Michigan booze is so waterlike that it is suitable for children and so can be sold over the internet - which should put QUITE a damper on internet sales of Michigan alcoholic beverages to ANYONE, adults as well as children - or the only reason for the previous rules was protectionism and the kids be damned.
Claiming now that the reason for the previous rules was for the protection of children is so obviously false it is pathetic.
I agree with the ruling - it either is OK to buy wine over the internet, or it isn't. The "won't somebody PLEASE think of the children" argument is stupid and reflects badly on those using it as an arguing point.
But it can't have been emitted at both times because it's just a single electron
Whoa!!
I was following up to this point.
According to the article, these are NOT "a single electron" at various points in time, it is DIFFERENT electrons at various point in time.
Take a classic sine wave, varying from +1 to -1 over time.
Take any two slices at varying times and add them together - you will get a value between +2 and -2 - i.e., constructive or destructive reinforcement. This is with ONE electron sampled at varying times.
Point being, if you take ANY two electrons at different points in the sine wave cycle and add their values, you will get the same effect. The two-slit experiment is interesting (in one aspect) because if you use one photon - as you point out - it still seems to pass through both slits AT THE SAME TIME. Part of the 'gee whiz' factor goes away if multiple photons are used.
As I read TFA, they are looking at multiple electrons at multiple times - and getting the results that wuold seem to be expected. I don't see the 'gee whiz' factor in that.
If there were two 'things' with nothing between them then the distance could be stated as being one unit. Now move the objects so that there is a distance of two units between them.
Is there more 'something' between them? Yes, more distance, more 'units' (was one, now two), etc., but if empty space has a physical existance then you would seem to be saying that there is "more nothing" between them as well.
You can describe the space between as having volume, and moving the objects increases the volume between, but it does not increase the contents of that volume - an empty 5 gallon bucket holds as much water as an empty 55 gallon barrel - both hold exactly no water.
I agree the 55 gallon POTENTALLY holds more than the 5 gallon bucket, but potential 'something' does not have a physical existance - that is why it is 'potential'.
I am not saying you are wrong, I am saying I have great difficulty getting my mind around the concept of "more nothing".
I also do not see how this is time rather than space.
The laser is creating a maxima, then a minima, then a maxima. The interaction between the electrons means the time of the maxima causes the electrons to be deflected to the (for example) left and then the minima causes a deflection to the right, then the second maxima causes a deflection to the left again.
Therefore there is a difference in time of arrival at one of the detectors - which is their 'slits in time' - and the defraction seen on the detector is a result (supposedly) of the differences based on time, not on differences in spacial location.
I say supposedly because I am not convinced this is a valid experiment relating to electrons interacting through time.
To my thoughts, if you are using one electron and checking it at varying times then you could get different information/results - I am not set up to perform quantum physics experiments at home!
It would seem to me that you could take a representation of a sine wave, take the value at an arbitrary time, then take another value at another arbitrary time, and analyse the results and get results that should equate to a single electron at varying points in time.
If the results of tfa experiments are not in line with those results, then I would suspect the experiment is not showing what it is purporting to show.
I find it hard to understand how looking at one electron at one time, and another electron at some other time gives useful information - At 10 pm last evening I was at the gas station, where were you at noon today, and what possible conclusions can you derive from those two pieces of information?
Even with large numbers of examples (there must have been thousands of people at service stations sat 10 pm last evening, and I assume there were thousands of people still in bed at noon...;-D ) I don't see that conclusions can be validly drawn from this data.
It's not like their restricting you from running Windows on a competing platform.
What does this have to do with anything, and who said this is what was happening?
I read the article, and unless I missed something, this is NOT the complaint.
I don't use WINE to run Windows(c) OS, I run it to run some (work required) Office apps and some games.
The Office apps were purchased and presumably have rights to be updated the same as any other user of Office apps. Same with the games.
But Microsoft is saying that, because I am using a valid purchased version of their software on an OS other than Windows (by using WINE) they will not allow updates from their servers.
This is the mirror image of their antitrust loss - they were accused of using their market possition (monopoly) in the OS to maintain and grow their market position in other markets, while here they are using their market possition in the other areas to maintain their possition in the OS market.
You say you were a victim of the DR-DOS 'trick', where a competiting product was specifically checked for and then bogus 'error' messages were given, or the applications just didn't work as expected - not because of a problem with DR-DOS, but because the app was PROGRAMMED to work differently when used with DR-DOS. Like is happening here?
You say you worked at WordPerfect. Isn't that the company that worked with Microsoft to be compatable and competitive, then Microsoft changed the APIs and didn't publish them to competitors of their Office (specifically Word(c)) and royally screwed WordPerfect over?
Novell - didn't I hear their networking applications were deliberately 'broken' by Microsoft so that Microsofts' market share of networking would not be threatened? Like here?
They're just saying "don't expect to be able to use our bandwidth and download from us without being a customer first".
No, they are just saying "don't expect to be able to use our bandwidth and download from us without being a Microsoft Windows OS customer first (even if you are a valid Microsoft Office customer)." Very different than what you posted.
Except for the provisions in section 110 (from your linked web site.
(1) performance or display of a work by instructors or pupils in the course of face-to-face teaching activities of a nonprofit educational institution, in a classroom or similar place devoted to instruction, unless, in the case of a motion picture or other audiovisual work, the performance, or the display of individual images, is given by means of a copy that was not lawfully made under this title, and that the person responsible for the performance knew or had reason to believe was not lawfully made;
So assuming you can get a legal copy and you are showing it to students of a specific class in a setting where the general public is not permitted while the showing is going on - as the parent states "in class, as part of the class" it would seem to be covered by Fair Use exclusions.
There are a LOT of ifs, ands and buts included in section 110, including the requirement for maximum SqFt area of estabilshments (2000 not including parking area) for non-food or drinking establishments and using not more than 6 loudspeakers, with not more than 4 in any one room...
Laws are complex, for actual legal advise, consult a shyster! (um, I mean a lawyer)
The "famous proof of Cantor" you refer to, is that the diagonal argument or another proof?
If it is the diagonal argument, can you help me to understand why it does not also prove that the set of positive integers is not uncountably infinite, even though there exists a 1-1 relationship with the members of the set of positive integers and the definition of a countable infinity I found states that a countable infinity is one with a 1-1 correspondence with the positive integers?
The setup is this:
Create a function f such that f(x) = x, i.e., f(1) = 1 and f(f(1)) = 1, f(2) = 2 and f(f(2)) = 2, etc.
Apply this function to each of the positive integers, with the results being a new set WITH EXACTLY THE SAME ELEMENTS as the set of positive integers. Because the set of positive integers is countably infinite, the new set should be also countably infinite unless I am missing something.
List the elements of the result set one to a line and pad the left with zeros to make the lines all the same length.
Apply Cantor's diagonal argument to the listing created above, resulting in a number i. My understanding is that i is claimed to be different in the ith place from any number in the result set which means we have determined a number in the set of f(n) = n where f(i) != i.
Based on Cantor's argument being a proof that the set of real numbers is not countably infinite, haven't we just proven that the countably infinite set of positive integers are not countably infinite ( a != a)?
Have I divided by zero or some other basic error here, and if so where?
I had said I was not going to continue posting on this thread, but you are picking at a point I would like more clarificatiion on.
1) Let's assume f(n) for all 1,2,3... infinity contains all the possible values for my set
Ok.
2) I can show that this is not true, by coming up with a member of the set which is not f(n) for any n.
You lost me.
If f(n) where n is in the positive integers maps to ALL POSSIBLE MEMBERS OF YOUR SET then you can not possibly create, devise, make up, calculate, or otherwise have a number in your set that is not mapped using f(n), otherwise you have a number in your set that is not a member of the set of all the possible numbers in your set. A contradiction. Of course, a contradiction means the original premis is false, so you would have proved your point. Or would you...?
I guess my point is that, if I have all the possible members in your set in one of my positive integer buckets, then any manipulation you can do to any single member or group of members of the set will result in a number that is in the set in one of my infinite buckets.
If you do Cantor's diagonal argument and come up with a number that is a valid member of the set of possible numbers in the set, then, by defining the buckets as holding 'all possible numbers in the set' any number you arrive at that is a valid number is going to be in the set. It has to be, if it is a valid member of the set!
You may say "bullshit", it's one of the f(n)'s.
Then I'll say "ok, which one is it"
Then you'll say, f(1085), for example.
Then I'll say, no it's not, because the 1085th "sequence" (or digit or however this is done) of f(1085) is 0, but the 1085th "Sequence" of my new element is 1. They can't be the same.
By the way we generated our "new member" it will not be equal to f(n) for any n because the nth "digit" will be different.
Interesting, but leads to a paradox I think.
create a function on x where x ranges over the positive integers such that every integer maps to itself. f(n) = n.
f(1) = 1 and f(f(1)) = 1 f(2) = 2 and f(f(2)) = 2 f(3) = 3 and f(f(3)) = 3
..
.. f(x) = x and f(f(x)) = x
Do you agree that the function, when applied to the (infinitely large) range of positive integers will give an infinitely large set, and that every integer in the set will, when put through the function give itself as a result, and that every result can be mapped 1-1 to the set of positive intergers? If any of my assumptions or intentions is not correct, then my whole thought is not correct.
We can now list the numbers resulting from the application of the function to the positive integers, pad the results on the left with zeros to make all the entries in the list the same length, and apply Cantor's diagonal argument.
This will result in a number, and because the function is f(number) = number, the entry that maps to the number found in the diagonalization must be arrived at when run through the f(x) = x function.
But you are stating that your number is different than my number, even though my number IS your number.
We have a contradiction.
The definition of a countable infinity is one that can be mapped on a 1-1 basis to the counting number which are the positive integers. Cantor's diagonal argument shows that the set of positive integers CAN NOT BE MAPPED to the set of positive integers! I.e., the set is larger than itself!
Congratulations, you have just proved that a countable infinity is not countable.
Whoops, THAT is a contradiciton!
So either Cantor's diagonal argument proves that the set of real numbers is not countable and so is larger than the set of positive integers, or the diagonal argument proves that BOTH sets, the real numbers and the positive integers, are uncountable and therefore may be the same size after all, or the argument proves nothing at all.
Sure you can. Give me any real number between 1 and 2. Write it down on one of these infinite number of pieces of paper. Put that piece of paper in one of these infinite number of buckets. Repeat for all the real number between 1 and 2 without duplicating any.
The bolded sentence is the problem. You need to be able to specify a way to determine the n'th real number you are going to choose.
I still don't see it.
You seem to be saying it is hard to find a function to determine the real numbers, so therefore the set of real numbers is not countable. I am saying that if you gave me every possible real number - however obtained, guessed, calculated, made up on the spot - those numbers could be assigned to a 'positive integer set' bucket in a 1-1 relationship and are therefore 'countably infinite' and so the set of real numbers is the same size as the set of positive integers.
As I see it, the only way there can be an infinity that is larger than another is if the elements of one set of infinity can not be assigned to another set in a 1-1 correspondence.
Not only can't you find a neat and tidy function that you can write down, but you can prove that it isn't possible for one to exist.
AH HA! A new direction for my investigation! THANK YOU!
As I have posted before, though, Cantor's method using the diagonal does not prove the uncountability to me, as a paradox is created when you claim you have listed all the numbers then claim you find a new number that is not listed. Both can not be true, so either not all the numbers were listed in the beginning or the number 'found' IS on the original list.
I have taken up enough time from everyone, so, while I will monitor the thread, I will not be posting further.
Thanks to all who attempted to educate, it is appreciated!
I appreciate the patience shown by respondents and the willingness to educate a layperson in the given field, a field that can be very difficult!
Thanks for the links - mathworld is very 'information dense', I am more the wikipedia speed.
I will say, though, that I need more education in the area, as Cantor's diagonal argument does not convince me.
Given the interval between 0 and 1, create a function that maps the positive integers to each of the real numbers between 0 and 1. Now do Cantor's diagonal procedure. Either you will get a number that is on the list generated by the function, or the function is faulty as it did not generate the number you arrived at.
The list of all possible real numbers between 0 and 1 would consist of a zero followed by a decimal point, followed by all possible combinations of the digits 0 through 9. I maintain it is not possible to then 'twiddle' one of the digits and arrive at a combination of the digits 0 through 9 that is not already in the set of "all possible combinations of digits 0 through 9."
Cantor's argument says it is possible to find a combination of digits consisting of the numbers 0 through 9 that is not in "all possible combinations of the digits 0 through 9." I have a logical problem with that.
Specifically, using the example on wikipedia, if we "enumerate all numbers in this interval as a sequence" then all numbers between 0.4444...443 and 0.5555...556 are in our list (Zero followed by an infinite string of fours, followed by a three through zero followed by a decimal point followed by an infinite string of fives followed by a 6, giving ALL POSSIBLE COMBINATIONS OF 4s AND 5s). If there is a number between these two numbers not on our list, then we have not "enumerated all numbers in this interval" as required.
I then maintain that, no matter what combination of manipulations done to any other string of digits, as long as a number consisting of a zero followed by a decimal point, followed by any combination of 4s and 5s results, the number WILL be on our list, directly contradicting the statement "However, because of the way we have chosen 4's and 5's as digits in step (6), x differs in the nth decimal place from rn, so x is not in the sequence ( r1, r2, r3,... ). "
Therefore, "This sequence is therefore not an enumeration of the set of all reals in the interval [0,1]. This is a contradiction." is false.
I have been thinking about this, and think there is a problem with the logic.
Which set is larger, the set of all positive integers, or the set of all EVEN positive integers?
The sets are the same size.
Take all the positive integers, multiply their value by 2 and you have all the positive EVEN integers. Therefore there is a 1-1 correspondence between the members of the first set and the members of the second set, so they are the same size.
Because of this, a (the set of all positive integers, even or odd) is a superset of b (the set of all even positive integers) but a IS NOT larger than b.
Interesting that c (the set of all odd positive integers) is also exactly as large as a, which gives (the set of all even positive integers PLUS the set of all odd positive integers gives the set of all positive integers) b + c = a, but the size of b = the size of a and the size of c = the size of a, therefore the size of a + the size of a = the size of a!
I have been following links provided by i2hsu in a prior posting to wikipedia entries and have been working on Cantors Diagional theory and was not getting it. Your example helped me understand the reasoning.
I also am not sure I buy the theory, but I think I understand it better after reading your post.
Based on Cantor's diagonal argument and the arguments and examples given there, the results of the manipulation results in a supposedly new number, one that was not originally in the listing of 'all numbers in the interval.'
Is there not a logical problem with finding a NEW number betweeen two boundaries that is not in a listing of ALL NUMBERS between two boundaries?
What this gives me is a sequence that is guaranteed to be different than every f(n) sequence, because the number in position n is different.
I don't agree. If your f(n) really does give every possible value between the two boundaries, then any number generated by your manipulations WILL be in the set generated by f(n) - BY DEFINITION of 'every possible value'.
I maintain that f(n), where n ranges over ALL the positive integers, if it gives all the numbers between 0 and 1 then you are not able to maipulate the values generated to get a value NOT in the listing but still between 0 and 1. If you can, then your function is not valid or it would have generated the value.
If the functions you describe in your post are guaranteed to give a binary fraction (a zero, followed by a decimal point [binary point?] followed by any number of zeros or ones in every possible combination) and give ALL the possible combinations, then as long as your manipulation gives either a one or a zero you will always wind up with a number in the original list, no matter which digit you flip.
Chose a binary fraction between 0.0000000 and 0.1111111. Chose a position x such that x is between 1 and 7. Whatever the bit in position x is, flip it to the other state (1 -> 0, 0 -> 1). the result will always be a value that is already in the series.
I chose 1010101. Any bit you should chose to flip would give a result that is in the defined range.
I may be guilty of muddy thinking, missing something, or not using the terms in a way that is understood by those in the field,, but if your f(n) is supposed to give EVERY POSSIBLE number between two boundaries, but does not give the number you obtain when you perform Cantor's diagonal argument, then your f(n) is flawed in that it did not give every possible number.
This would seem to me to eliminate the contridiction claimed in Cantor's argument, rendering the claim that some infinities are larger that others false.
Again, I find this interesting, but lack the background for affirmming or refuting what you say.
What makes a set uncountable - the fact that there are an infinite number of members? If that is the definition of an uncountable set then all infinite sets are uncountable.
If the definition is that a countable set can be assigned a 1-1 correspondence to the positive integers ("counted") then I can contend that the real numbers can be assigned a 1-1 correspondence with the positive integers and therefore is countable.
Please point me to definitions of 'countable' and 'uncountable' infinities for my edification.
You are already assuming all infinite sets to be the same size... so it doesn't show anything.
The set of real numbers is larger than the set of natural numbers, you cannot argue against that.
Yes I can. There are an infinite number of natural numbers, and an infinite number of real numbers, and an infinite number of positive integers so UNLESS YOU ASSUME INFINITE SETS ARE NOT all the same size, they must all be identical in size.
I agree that, logically, if I have an infinite number of chickens and each lays two eggs then it would seem to require that there be more eggs than chickens, but there is only an infinity of chickens and an infinity of eggs, not (2*infinity) eggs as the statement (2*infinity) makes no sense.
If you (or anyone else) can give links to information on this topic I would be grateful.
This is, in my estimation, like saying God can do anything, but there are somethings God can't do -God can't make a rock so big God can't lift it - a basic contradiciton found in most religious beliefs in my estimation.
This is where it breaks down. The problem is that when you try and figure out a way to determine how to assign the real numbers to your buckets (heck even try it with just the numbers between zero and one) you realise that you can not do it. Even in theory, it is absolutely impossible to determine a way to do this.
Sure you can. Give me any real number between 1 and 2. Write it down on one of these infinite number of pieces of paper. Put that piece of paper in one of these infinite number of buckets. Repeat for all the real number between 1 and 2 without duplicating any. Because there are an infinite number of pieces of paper, you can not come up with more real numbers than there are pieces of paper to write them down on.
There are far too many real numbers.
More than an infinite number? Interesting concept, but I see a logical problem here.
The typical way of proving that there does not exist any way to assign all the real number rocks to the natural number buckets is to assume that you can.
Sounds like you have information or education I don't have. Can you point me toward this information? reductio ad absurdum is a classical technique, I have noy seen it used in this context, though. Links?
You are mapping between MEMBERS of infinite sets and saying they are different, and I am saying the NUMBER of members in two infinite sets (i.e., mapping both sets to the infinite set of positive integers) is the same - both are exactly equal to the number of positive integers. Two things equal to a third thing are equal to each other.
Did you leave something out, as I am not following the logic here -
Heck, even assume that all of the digits are either 2 or 4. This leaves you intuitively with far less numbers. Certainly no more.
So now, if we can create a real number between zero and one whose only digits are 2 or 4, then we have proven that our list is not complete.
I read this to say that we assume that all the digits are either a 2 or a 4, so creating a number whose only digits are 2 or 4, however accomplished, only proves our assumption, and no contradiction that I can see despite your statement that there is one.
Perhaps if you could point me toward some links it would be a faster way to educate me - I am not a math whiz.
You can measure the sizes of sets using 1-1 correspondance between elements in the set to positive integers.
The set of positive integers is an infinite set, agreed? Any "proposed largerst positive integer" can be topped by adding one to the previous integer, so there is no 'largest positive integer'.
A 1-1 correspondence can be set up between any real number and a number in the set of positive integers.
As well, a 1-1 correspondence can be set up between any of the natural numbers and the positive integers.
Therefore the set of real numbers does not have the same set members as the natural numbers, but both sets have exactly the same number of members - i.e., infinity - and are the same size.
Like I said, there are an infinite number of fractions between each positive integer, but they can still be assigned a 1-1 correspondence to the infinite set that is the positive counting numbers.
If I had a computer that could count an infinite number of objects in one second, I could count all the positive integers (an infinite number) in one second. How long would it take to count all the fractions between 1 and 2 (an infinite number)? One second. How long to count all the possible fractions (an infinite number) between all the positive integers (another infinite number)? One second.
If you think about it, the set of all real numbes contain the set of al natural numbers... but not the other way around.
Agreed, but that does not matter. You can still assign a 1-1 correspondence between the positive integers and each member of the set of real numbers, and a 1-1 correspondence to the members of the set of all natural numbers - proving that both infinitely large sets are the same size.
I still don't understand how there can be larger and smaller infinities. To my mind, there is only one infinity.
If I had an infinite number of bins numbered from 1 to infinity, then you could not give any listing or formula that would give a number that could not be put into one of the bins.
Non-intuitively, if you defined your set as (two rocks per each bin, each rock put into a separate bin) [2*infinity]they would still fit into the existing bins! [2*infinity=infinity]
A common example is the integer counting numbers (1,2,3,4...). They could be assigned, one to a bin, to my infinite number of bins.
There are an infinite number of decimals between the integer counting numbers (1.1, 1.01, 1.001,...) BUT THEY CAN STILL BE ASSIGNED, one to a bin, TO MY INFINITE NUMBER OF BINS. [infinity * infinity = infinity]
I have not had stats or combinatorics, so can not address what you were taught, but unless one infinitely large set can be larger than another infinitely large set then all infinitely large sets are the same size - by definition.
Slashdot Mirror
Isn't this the kind of thing Free Software was supposed to be against?
No. You have confused trademark with copyright - and possibly with patent.
Anyone can distribute their own flavor of Linux and call it Linux without being threatened by lawsuits over trademarks?
No. You have confused trademark with copyright again.
According to your posting, Microsoft should be able to release an operating system and call it Linux. I am glad they are not allowed to do so under TRADEMARK law.
+5, Insightful? Moderators must be on crack again.
Free (as in speech) software is about being able to reuse others code through PRIOR licensing of copyrighted material. However, I can not make my own DISTRIBUTION based on the Linux kernel and call it RedHat, as that NAME is already trademarked. Nothing in the free software philosophy contradicts that, or says that I should be able to use the RedHat name.
Linux is a word that is used in a fairly generic way by many, but the laws require the trademark holder to defend the mark in business - or lose it to 'public domain' as a generic term. Once lost, people are free to use the word in any way they want - we could have linux soap, Windows applications, Linux air conditioners, soda pop, ANYTHING.
(that's a joke, I say, that's a joke son!)
Wish I had moderator points; you would get a -1, Overrated just for being so wrong.
A quick google searsh shows 4,260,000 web sites found using the search phrase 'world vista', 6,870 sites found using '"World Vista"' and 3,940 sites using 'worldvista'
One (obscure? - I don't think the US Government thinks so...) software related site is this sorceforge.net hosted site.
From their site:
WorldVistA is a charitable organization: a nonprofit, 501(c)(3) public-benefit corporation. WorldVistA was incorporated March 18, 2002 to measurably improve health worldwide by making medical software better and more accessible.
Note the specific reference to software. So how is Microsoft able to trademark this combination of words? It is already in use in the same industry, making it ineligable for trademark I thought.
Dirt is also good insulation as well as being great thermal mass. Those two attributes make it not as practical as you would seem to think.
Anything below the frost line is deep enough - some places could require 6-8 feet, I would think 2-3 feet would be more common.
Your efficiency would start out at adequate but, as the temp of the ground immediately surrounding the copper pipe rose the efficiency would drop quickly. At some point temp in would equal temp out, and your 'better idea' no longer works.
Press a button - works for me!
Also is seen by the CPU as a standard keyboard, available in 'ergonomic' version, no special driver, etc.
I have one and recommend it!
OK, so who is working on the phase conjugate mirror? Can't have a super secrete weapon without a phase conjugate mirror...
I agree with your second point - except that I am not sure that the Federal government has any business regulating it, it should be the state government or the pocket book of 'the people' who are doing the regulating.
On the first point, you are just wrong.
The gasoline in the 1920's was used in very low compression engines which were very wasteful of gasoline - but gasoline was cheap and plentiful, so who cared!
The car manufacturers wanted to sell higher compression engines in heavier cars, but the higher compression caused unpleasant (and damaging) knocking when using the gasoline of the day. Many things were tried to reduce or eliminate knocking - including Bromine, Selenium, Iodine, Anine dyes, and ethol alcohol, as well as benzine (sp?), sulpher, and other aromatics.
Tetraethyl lead was chosen for a variety of reasons, including the patentability of the process for making it. Delco labs was a pioneer in anti-knock fuel additives, and was swallowed by GM as their Research Division. After a time, the Ethyl Lead production was spun off as a separate company, the Ethyl Additive Corp.
From internal memos from GM, the process that produced tetraethly lead cost about one penny per gallon and was sold to the fuel distributers at three cents per gallon - for a net profit of two cents per gallon. The memos estimated that GM would have a profit from the production of tetraethyl lead of $60,000,000 per year BACK IN THE 20's! I have no way of calculating the equivalent in todays dollars, but would assume it would be significant!!
The production facilities were toured by professional chemists who were horrified by the lack of safety precautions. Even when 8 workers died from lead poisoning in the first year - the go-live had to be postponed when the trial runs poisoned the workes to the point they were unable to work on the live production line - the main concern (as stated in internal memos) was the bad PR which could reduce the demand for the lead additive, not the safety of the workers.
Bottom line, it DID NOT cost more to produce gasoline without lead, unless you figure the PROFIT from selling the lead additive reduced the cost of producing the gasoline.
Unless you are talking about lately when regular (with lead) was removed from the market in favor of unleaded gasoline. In that case, I would probably claim that the increase in cost (if any) would be in the refinery hardware needed to increase the octane (the anti-knock component of the gasoline), not in the production of the gasoline itself. In addition, the lead was used as a lubricant in engins with soft valve seats and would have to be replaced with another lubricant - or the valve seats would have to be made of harder materials that did not require the lead buffering.
On a related (but not directly) note, my fiancee buys the higher 89 octane gasoline for a '97 Jeep. I have tried to explain that the higher octane is not needed if there is no knocking, but she seems to think 'a higher price means it is better gas'. My thoughts on the subject are that, if there is no knocking, there is no need for additional anti-knock compounds (higher octane), the vast (or maybe half-vast?) majority buy the lowest price gasoline, so the higher octane - and higher priced - gasoline is more likely to be older with the attending greater possibility of 'stuff' (like water, gum, and other crap and crud), as well as a suspicion of pricing motives when there is, and remains, a $0.10 price difference between the lowest grade (87 octane locally) the mid grade (89 octane) and the highest grade (91 octane locally). As long as she is spending her money on gasoline, I don't care which octane she choses, but will continue to try to educate her.
Octane is added to reduce knocking - or premature detonation/ignition of the air/gas mixture. That means higher octane gasoline DOES NOT BURN AS WELL as lower octane gasoline - but we are charged MORE for the gasoline that burns WORSE. Interesting marketing concept.
It wasn't enlightened self-interest which took the lead out saving millions of kids from brain-damage.
You are right, the producers of the lead based paint and gasoline had better (and lead free) products available, but people would not buy them as they were more expensive than the products containing lead.
So what did the paint producers, the gasoline producers, and the insurance companies do? They lobbied for legislation REQUIRING the elimination of lead in the products. When the laws were passed the profits of the paint companies rose, the profits of the gasoline companies rose, the insurance companies paid out less for care of lead-based brain-damage, and everyone was happy - except the end user who was paying more for what they needed.
Interesting that regular gasoline (with lead) was cheaper than unleaded - and that many cars could not run with unleaded because they had valves that would burn up without the buffering of the lead additives.
Interesting that the lead was BOUGHT and added to the gasoline, but the price of gasoline WITHOUT the additive was higher - and when the additive was banned, only the higher priced version was available. The gasoline producers saved money by not having to buy the additive, and made more profit by selling the same thing at a higher price. SWEEEEET!
Also interesting is that you can buy tetraethyle lead gasoline additives in auto performance stores over the counter for those cars that have to have lead to protect their valves.
Or take the case of auto accidents. For decades Detroit couldn't sell safe cars. Few manufacturers tried and they failed.
Good point. Why did 'Few manufacturers tr[y]'? because the buying public was being led by the nose through advertising (by the auto manufacturing companies) to buy the larger, more powerful models (6000 SUX!! YEAH BABY!!), not the safer, more fuel efficient models.
When there was more profit in selling land barges that got terrible gas milage, that weighed several thousands of pounds, and that were capable of speeds their parts (tires, brakes, etc) could not manage, then there was no point in trying to also make safe models.
Remember the cost/benefit rules the auto manufacturers live by - if it is cheaper to settle a few claims for millions of dollars than to re-tool to FIX a problem, the problem doesn't get fixed no matter the number of people killed.
Remember 'Unsafe at Any Speed' by Ralph Nader?
Remember when seat belts were optional and no one would pay extra for them? Remember when they were required in the front seat but not the back seat? If the back seat is so much safer, why not put the driver in the back seat 'for their own safety'?
Also, when an 'optional' extra cost safety feature is made mandatory, the playing field is leveled in that ALL the manufacturers have to include the 'feature' - and the base price can be jacked up to cover the costs.
The results were immediate and today with many more cars on the road, we have fewer deaths than we did in the '60s.
Interesting. You do realize, don't you, that the government allows MORE depreciation against taxes for large, unsafe, non-fuel efficient vehicles than for smaller, fuel efficient vehicles - which means the government seems to be advocating usage of larger, more dangerous, gas-guzzling trucks over the use of safer, more environmentally friendly vehicles.
[D]on't blame the government just because it can't ensure the safety of every citizen.
It has never been and should not now be the job of the government to ensure the safety of ANY citizen. The Federal government sould be most concerned with the interaction between the states - UNITING the states into the UNITED STATES - and with the interaction of this nation with outher nations and coutries. The states should be most concerend with the providing of infrastructure of needed services at the best cost to their citizens, not to the highest campaign contributer or the company most willing to hire the office holder when they leave office.
Damn, I am long winded! I am sure I had a point in there somewhere, if anyone finds it, could you point it out for me?
Thanks!
Michigan isn't satisfied and is proposing banning all over-the-net wine orders on the flimsy reasoning that kids will be able to buy booze without government control.
Interesting thought process; before the ruling it was OK for Michigan kids to buy Michigan booze over the internet, but it was NOT OK for Michigan kids to buy OUT-OF-STATE booze over the internet. Now either Michigan booze is so waterlike that it is suitable for children and so can be sold over the internet - which should put QUITE a damper on internet sales of Michigan alcoholic beverages to ANYONE, adults as well as children - or the only reason for the previous rules was protectionism and the kids be damned.
Claiming now that the reason for the previous rules was for the protection of children is so obviously false it is pathetic.
I agree with the ruling - it either is OK to buy wine over the internet, or it isn't. The "won't somebody PLEASE think of the children" argument is stupid and reflects badly on those using it as an arguing point.
But it can't have been emitted at both times because it's just a single electron
Whoa!!
I was following up to this point.
According to the article, these are NOT "a single electron" at various points in time, it is DIFFERENT electrons at various point in time.
Take a classic sine wave, varying from +1 to -1 over time.
Take any two slices at varying times and add them together - you will get a value between +2 and -2 - i.e., constructive or destructive reinforcement. This is with ONE electron sampled at varying times.
Point being, if you take ANY two electrons at different points in the sine wave cycle and add their values, you will get the same effect. The two-slit experiment is interesting (in one aspect) because if you use one photon - as you point out - it still seems to pass through both slits AT THE SAME TIME. Part of the 'gee whiz' factor goes away if multiple photons are used.
As I read TFA, they are looking at multiple electrons at multiple times - and getting the results that wuold seem to be expected. I don't see the 'gee whiz' factor in that.
If there were two 'things' with nothing between them then the distance could be stated as being one unit. Now move the objects so that there is a distance of two units between them.
Is there more 'something' between them? Yes, more distance, more 'units' (was one, now two), etc., but if empty space has a physical existance then you would seem to be saying that there is "more nothing" between them as well.
You can describe the space between as having volume, and moving the objects increases the volume between, but it does not increase the contents of that volume - an empty 5 gallon bucket holds as much water as an empty 55 gallon barrel - both hold exactly no water.
I agree the 55 gallon POTENTALLY holds more than the 5 gallon bucket, but potential 'something' does not have a physical existance - that is why it is 'potential'.
I am not saying you are wrong, I am saying I have great difficulty getting my mind around the concept of "more nothing".
I also do not see how this is time rather than space.
;-D ) I don't see that conclusions can be validly drawn from this data.
The laser is creating a maxima, then a minima, then a maxima. The interaction between the electrons means the time of the maxima causes the electrons to be deflected to the (for example) left and then the minima causes a deflection to the right, then the second maxima causes a deflection to the left again.
Therefore there is a difference in time of arrival at one of the detectors - which is their 'slits in time' - and the defraction seen on the detector is a result (supposedly) of the differences based on time, not on differences in spacial location.
I say supposedly because I am not convinced this is a valid experiment relating to electrons interacting through time.
To my thoughts, if you are using one electron and checking it at varying times then you could get different information/results - I am not set up to perform quantum physics experiments at home!
It would seem to me that you could take a representation of a sine wave, take the value at an arbitrary time, then take another value at another arbitrary time, and analyse the results and get results that should equate to a single electron at varying points in time.
If the results of tfa experiments are not in line with those results, then I would suspect the experiment is not showing what it is purporting to show.
I find it hard to understand how looking at one electron at one time, and another electron at some other time gives useful information - At 10 pm last evening I was at the gas station, where were you at noon today, and what possible conclusions can you derive from those two pieces of information?
Even with large numbers of examples (there must have been thousands of people at service stations sat 10 pm last evening, and I assume there were thousands of people still in bed at noon...
It's not like their restricting you from running Windows on a competing platform.
What does this have to do with anything, and who said this is what was happening?
I read the article, and unless I missed something, this is NOT the complaint.
I don't use WINE to run Windows(c) OS, I run it to run some (work required) Office apps and some games.
The Office apps were purchased and presumably have rights to be updated the same as any other user of Office apps. Same with the games.
But Microsoft is saying that, because I am using a valid purchased version of their software on an OS other than Windows (by using WINE) they will not allow updates from their servers.
This is the mirror image of their antitrust loss - they were accused of using their market possition (monopoly) in the OS to maintain and grow their market position in other markets, while here they are using their market possition in the other areas to maintain their possition in the OS market.
You say you were a victim of the DR-DOS 'trick', where a competiting product was specifically checked for and then bogus 'error' messages were given, or the applications just didn't work as expected - not because of a problem with DR-DOS, but because the app was PROGRAMMED to work differently when used with DR-DOS. Like is happening here?
You say you worked at WordPerfect. Isn't that the company that worked with Microsoft to be compatable and competitive, then Microsoft changed the APIs and didn't publish them to competitors of their Office (specifically Word(c)) and royally screwed WordPerfect over?
Novell - didn't I hear their networking applications were deliberately 'broken' by Microsoft so that Microsofts' market share of networking would not be threatened? Like here?
They're just saying "don't expect to be able to use our bandwidth and download from us without being a customer first".
No, they are just saying "don't expect to be able to use our bandwidth and download from us without being a Microsoft Windows OS customer first (even if you are a valid Microsoft Office customer)." Very different than what you posted.
I'm beginning to understand. I just wish I knew how to stop it.
You, me, and George Soros
Money doesn't seem to help.
Except for the provisions in section 110 (from your linked web site.
.
(1) performance or display of a work by instructors or pupils in the course of face-to-face teaching activities of a nonprofit educational institution, in a classroom or similar place devoted to instruction, unless, in the case of a motion picture or other audiovisual work, the performance, or the display of individual images, is given by means of a copy that was not lawfully made under this title, and that the person responsible for the performance knew or had reason to believe was not lawfully made;
So assuming you can get a legal copy and you are showing it to students of a specific class in a setting where the general public is not permitted while the showing is going on - as the parent states "in class, as part of the class" it would seem to be covered by Fair Use exclusions
There are a LOT of ifs, ands and buts included in section 110, including the requirement for maximum SqFt area of estabilshments (2000 not including parking area) for non-food or drinking establishments and using not more than 6 loudspeakers, with not more than 4 in any one room...
Laws are complex, for actual legal advise, consult a shyster! (um, I mean a lawyer)
Thank you.
The "famous proof of Cantor" you refer to, is that the diagonal argument or another proof?
If it is the diagonal argument, can you help me to understand why it does not also prove that the set of positive integers is not uncountably infinite, even though there exists a 1-1 relationship with the members of the set of positive integers and the definition of a countable infinity I found states that a countable infinity is one with a 1-1 correspondence with the positive integers?
The setup is this:
Create a function f such that f(x) = x, i.e., f(1) = 1 and f(f(1)) = 1, f(2) = 2 and f(f(2)) = 2, etc.
Apply this function to each of the positive integers, with the results being a new set WITH EXACTLY THE SAME ELEMENTS as the set of positive integers. Because the set of positive integers is countably infinite, the new set should be also countably infinite unless I am missing something.
List the elements of the result set one to a line and pad the left with zeros to make the lines all the same length.
Apply Cantor's diagonal argument to the listing created above, resulting in a number i. My understanding is that i is claimed to be different in the ith place from any number in the result set which means we have determined a number in the set of f(n) = n where f(i) != i.
Based on Cantor's argument being a proof that the set of real numbers is not countably infinite, haven't we just proven that the countably infinite set of positive integers are not countably infinite ( a != a)?
Have I divided by zero or some other basic error here, and if so where?
I had said I was not going to continue posting on this thread, but you are picking at a point I would like more clarificatiion on.
. .
1) Let's assume f(n) for all 1,2,3... infinity contains all the possible values for my set
Ok.
2) I can show that this is not true, by coming up with a member of the set which is not f(n) for any n.
You lost me.
If f(n) where n is in the positive integers maps to ALL POSSIBLE MEMBERS OF YOUR SET then you can not possibly create, devise, make up, calculate, or otherwise have a number in your set that is not mapped using f(n), otherwise you have a number in your set that is not a member of the set of all the possible numbers in your set. A contradiction. Of course, a contradiction means the original premis is false, so you would have proved your point. Or would you...?
I guess my point is that, if I have all the possible members in your set in one of my positive integer buckets, then any manipulation you can do to any single member or group of members of the set will result in a number that is in the set in one of my infinite buckets.
If you do Cantor's diagonal argument and come up with a number that is a valid member of the set of possible numbers in the set, then, by defining the buckets as holding 'all possible numbers in the set' any number you arrive at that is a valid number is going to be in the set. It has to be, if it is a valid member of the set!
You may say "bullshit", it's one of the f(n)'s.
Then I'll say "ok, which one is it"
Then you'll say, f(1085), for example.
Then I'll say, no it's not, because the 1085th "sequence" (or digit or however this is done) of f(1085) is 0, but the 1085th "Sequence" of my new element is 1. They can't be the same.
By the way we generated our "new member" it will not be equal to f(n) for any n because the nth "digit" will be different.
Interesting, but leads to a paradox I think.
create a function on x where x ranges over the positive integers such that every integer maps to itself. f(n) = n.
f(1) = 1 and f(f(1)) = 1
f(2) = 2 and f(f(2)) = 2
f(3) = 3 and f(f(3)) = 3
.
.
f(x) = x and f(f(x)) = x
Do you agree that the function, when applied to the (infinitely large) range of positive integers will give an infinitely large set, and that every integer in the set will, when put through the function give itself as a result, and that every result can be mapped 1-1 to the set of positive intergers? If any of my assumptions or intentions is not correct, then my whole thought is not correct.
We can now list the numbers resulting from the application of the function to the positive integers, pad the results on the left with zeros to make all the entries in the list the same length, and apply Cantor's diagonal argument.
This will result in a number, and because the function is f(number) = number, the entry that maps to the number found in the diagonalization must be arrived at when run through the f(x) = x function.
But you are stating that your number is different than my number, even though my number IS your number.
We have a contradiction.
The definition of a countable infinity is one that can be mapped on a 1-1 basis to the counting number which are the positive integers. Cantor's diagonal argument shows that the set of positive integers CAN NOT BE MAPPED to the set of positive integers! I.e., the set is larger than itself!
Congratulations, you have just proved that a countable infinity is not countable.
Whoops, THAT is a contradiciton!
So either Cantor's diagonal argument proves that the set of real numbers is not countable and so is larger than the set of positive integers, or the diagonal argument proves that BOTH sets, the real numbers and the positive integers, are uncountable and therefore may be the same size after all, or the argument proves nothing at all.
Sure you can. Give me any real number between 1 and 2. Write it down on one of these infinite number of pieces of paper. Put that piece of paper in one of these infinite number of buckets. Repeat for all the real number between 1 and 2 without duplicating any.
The bolded sentence is the problem.
You need to be able to specify a way to determine the n'th real number you are going to choose.
I still don't see it.
You seem to be saying it is hard to find a function to determine the real numbers, so therefore the set of real numbers is not countable. I am saying that if you gave me every possible real number - however obtained, guessed, calculated, made up on the spot - those numbers could be assigned to a 'positive integer set' bucket in a 1-1 relationship and are therefore 'countably infinite' and so the set of real numbers is the same size as the set of positive integers.
As I see it, the only way there can be an infinity that is larger than another is if the elements of one set of infinity can not be assigned to another set in a 1-1 correspondence.
Not only can't you find a neat and tidy function that you can write down, but you can prove that it isn't possible for one to exist.
AH HA! A new direction for my investigation! THANK YOU!
As I have posted before, though, Cantor's method using the diagonal does not prove the uncountability to me, as a paradox is created when you claim you have listed all the numbers then claim you find a new number that is not listed. Both can not be true, so either not all the numbers were listed in the beginning or the number 'found' IS on the original list.
I have taken up enough time from everyone, so, while I will monitor the thread, I will not be posting further.
Thanks to all who attempted to educate, it is appreciated!
I appreciate the patience shown by respondents and the willingness to educate a layperson in the given field, a field that can be very difficult!
... ). "
Thanks for the links - mathworld is very 'information dense', I am more the wikipedia speed.
I will say, though, that I need more education in the area, as Cantor's diagonal argument does not convince me.
Given the interval between 0 and 1, create a function that maps the positive integers to each of the real numbers between 0 and 1. Now do Cantor's diagonal procedure. Either you will get a number that is on the list generated by the function, or the function is faulty as it did not generate the number you arrived at.
The list of all possible real numbers between 0 and 1 would consist of a zero followed by a decimal point, followed by all possible combinations of the digits 0 through 9. I maintain it is not possible to then 'twiddle' one of the digits and arrive at a combination of the digits 0 through 9 that is not already in the set of "all possible combinations of digits 0 through 9."
Cantor's argument says it is possible to find a combination of digits consisting of the numbers 0 through 9 that is not in "all possible combinations of the digits 0 through 9." I have a logical problem with that.
Specifically, using the example on wikipedia, if we "enumerate all numbers in this interval as a sequence" then all numbers between 0.4444...443 and 0.5555...556 are in our list (Zero followed by an infinite string of fours, followed by a three through zero followed by a decimal point followed by an infinite string of fives followed by a 6, giving ALL POSSIBLE COMBINATIONS OF 4s AND 5s). If there is a number between these two numbers not on our list, then we have not "enumerated all numbers in this interval" as required.
I then maintain that, no matter what combination of manipulations done to any other string of digits, as long as a number consisting of a zero followed by a decimal point, followed by any combination of 4s and 5s results, the number WILL be on our list, directly contradicting the statement "However, because of the way we have chosen 4's and 5's as digits in step (6), x differs in the nth decimal place from rn, so x is not in the sequence ( r1, r2, r3,
Therefore, "This sequence is therefore not an enumeration of the set of all reals in the interval [0,1]. This is a contradiction." is false.
I have been thinking about this, and think there is a problem with the logic.
Which set is larger, the set of all positive integers, or the set of all EVEN positive integers?
The sets are the same size.
Take all the positive integers, multiply their value by 2 and you have all the positive EVEN integers. Therefore there is a 1-1 correspondence between the members of the first set and the members of the second set, so they are the same size.
Because of this, a (the set of all positive integers, even or odd) is a superset of b (the set of all even positive integers) but a IS NOT larger than b.
Interesting that c (the set of all odd positive integers) is also exactly as large as a, which gives (the set of all even positive integers PLUS the set of all odd positive integers gives the set of all positive integers) b + c = a, but the size of b = the size of a and the size of c = the size of a, therefore the size of a + the size of a = the size of a!
Thanks, this helped me a lot.
I have been following links provided by i2hsu in a prior posting to wikipedia entries and have been working on Cantors Diagional theory and was not getting it. Your example helped me understand the reasoning.
I also am not sure I buy the theory, but I think I understand it better after reading your post.
Based on Cantor's diagonal argument and the arguments and examples given there, the results of the manipulation results in a supposedly new number, one that was not originally in the listing of 'all numbers in the interval.'
Is there not a logical problem with finding a NEW number betweeen two boundaries that is not in a listing of ALL NUMBERS between two boundaries?
What this gives me is a sequence that is guaranteed to be different than every f(n) sequence, because the number in position n is different.
I don't agree. If your f(n) really does give every possible value between the two boundaries, then any number generated by your manipulations WILL be in the set generated by f(n) - BY DEFINITION of 'every possible value'.
I maintain that f(n), where n ranges over ALL the positive integers, if it gives all the numbers between 0 and 1 then you are not able to maipulate the values generated to get a value NOT in the listing but still between 0 and 1. If you can, then your function is not valid or it would have generated the value.
If the functions you describe in your post are guaranteed to give a binary fraction (a zero, followed by a decimal point [binary point?] followed by any number of zeros or ones in every possible combination) and give ALL the possible combinations, then as long as your manipulation gives either a one or a zero you will always wind up with a number in the original list, no matter which digit you flip.
Chose a binary fraction between 0.0000000 and 0.1111111. Chose a position x such that x is between 1 and 7. Whatever the bit in position x is, flip it to the other state (1 -> 0, 0 -> 1). the result will always be a value that is already in the series.
I chose 1010101. Any bit you should chose to flip would give a result that is in the defined range.
1) 0.1010101
1) 0.0010101
2) 0.1110101
3) 0.1000101
4) 0.1011101
5) 0.1010001
6) 0.1010111
7) 0.1010100
I may be guilty of muddy thinking, missing something, or not using the terms in a way that is understood by those in the field,, but if your f(n) is supposed to give EVERY POSSIBLE number between two boundaries, but does not give the number you obtain when you perform Cantor's diagonal argument, then your f(n) is flawed in that it did not give every possible number.
This would seem to me to eliminate the contridiction claimed in Cantor's argument, rendering the claim that some infinities are larger that others false.
Again, I find this interesting, but lack the background for affirmming or refuting what you say.
What makes a set uncountable - the fact that there are an infinite number of members? If that is the definition of an uncountable set then all infinite sets are uncountable.
If the definition is that a countable set can be assigned a 1-1 correspondence to the positive integers ("counted") then I can contend that the real numbers can be assigned a 1-1 correspondence with the positive integers and therefore is countable.
Please point me to definitions of 'countable' and 'uncountable' infinities for my edification.
You are already assuming all infinite sets to be the same size... so it doesn't show anything.
The set of real numbers is larger than the set of natural numbers, you cannot argue against that.
Yes I can. There are an infinite number of natural numbers, and an infinite number of real numbers, and an infinite number of positive integers so UNLESS YOU ASSUME INFINITE SETS ARE NOT all the same size, they must all be identical in size.
I agree that, logically, if I have an infinite number of chickens and each lays two eggs then it would seem to require that there be more eggs than chickens, but there is only an infinity of chickens and an infinity of eggs, not (2*infinity) eggs as the statement (2*infinity) makes no sense.
If you (or anyone else) can give links to information on this topic I would be grateful.
VERY interesting!
I still don't buy it, though.
This is, in my estimation, like saying God can do anything, but there are somethings God can't do -God can't make a rock so big God can't lift it - a basic contradiciton found in most religious beliefs in my estimation.
This is where it breaks down.
The problem is that when you try and figure out a way to determine how to assign the real numbers to your buckets (heck even try it with just the numbers between zero and one) you realise that you can not do it.
Even in theory, it is absolutely impossible to determine a way to do this.
Sure you can. Give me any real number between 1 and 2. Write it down on one of these infinite number of pieces of paper. Put that piece of paper in one of these infinite number of buckets. Repeat for all the real number between 1 and 2 without duplicating any. Because there are an infinite number of pieces of paper, you can not come up with more real numbers than there are pieces of paper to write them down on.
There are far too many real numbers.
More than an infinite number? Interesting concept, but I see a logical problem here.
The typical way of proving that there does not exist any way to assign all the real number rocks to the natural number buckets is to assume that you can.
Sounds like you have information or education I don't have. Can you point me toward this information? reductio ad absurdum is a classical technique, I have noy seen it used in this context, though. Links?
You are mapping between MEMBERS of infinite sets and saying they are different, and I am saying the NUMBER of members in two infinite sets (i.e., mapping both sets to the infinite set of positive integers) is the same - both are exactly equal to the number of positive integers. Two things equal to a third thing are equal to each other.
Did you leave something out, as I am not following the logic here -
Heck, even assume that all of the digits are either 2 or 4. This leaves you intuitively with far less numbers. Certainly no more.
So now, if we can create a real number between zero and one whose only digits are 2 or 4, then we have proven that our list is not complete.
I read this to say that we assume that all the digits are either a 2 or a 4, so creating a number whose only digits are 2 or 4, however accomplished, only proves our assumption, and no contradiction that I can see despite your statement that there is one.
Perhaps if you could point me toward some links it would be a faster way to educate me - I am not a math whiz.
Think about what you are saying!
You can measure the sizes of sets using 1-1 correspondance between elements in the set to positive integers.
The set of positive integers is an infinite set, agreed? Any "proposed largerst positive integer" can be topped by adding one to the previous integer, so there is no 'largest positive integer'.
A 1-1 correspondence can be set up between any real number and a number in the set of positive integers.
As well, a 1-1 correspondence can be set up between any of the natural numbers and the positive integers.
Therefore the set of real numbers does not have the same set members as the natural numbers, but both sets have exactly the same number of members - i.e., infinity - and are the same size.
Like I said, there are an infinite number of fractions between each positive integer, but they can still be assigned a 1-1 correspondence to the infinite set that is the positive counting numbers.
If I had a computer that could count an infinite number of objects in one second, I could count all the positive integers (an infinite number) in one second. How long would it take to count all the fractions between 1 and 2 (an infinite number)? One second. How long to count all the possible fractions (an infinite number) between all the positive integers (another infinite number)? One second.
If you think about it, the set of all real numbes contain the set of al natural numbers... but not the other way around.
Agreed, but that does not matter. You can still assign a 1-1 correspondence between the positive integers and each member of the set of real numbers, and a 1-1 correspondence to the members of the set of all natural numbers - proving that both infinitely large sets are the same size.
SIZE(a) = SIZE(b) and SIZE(b) = SIZE(c), therefore SIZE(a) = SIZE(c).
I still don't understand how there can be larger and smaller infinities. To my mind, there is only one infinity.
If I had an infinite number of bins numbered from 1 to infinity, then you could not give any listing or formula that would give a number that could not be put into one of the bins.
Non-intuitively, if you defined your set as (two rocks per each bin, each rock put into a separate bin) [2*infinity] they would still fit into the existing bins! [2*infinity=infinity]
A common example is the integer counting numbers (1,2,3,4...). They could be assigned, one to a bin, to my infinite number of bins.
There are an infinite number of decimals between the integer counting numbers (1.1, 1.01, 1.001,...) BUT THEY CAN STILL BE ASSIGNED, one to a bin, TO MY INFINITE NUMBER OF BINS . [infinity * infinity = infinity]
I have not had stats or combinatorics, so can not address what you were taught, but unless one infinitely large set can be larger than another infinitely large set then all infinitely large sets are the same size - by definition.
Any other contention is amphigory, as I see it.