(in reference to, I assume, my post)
Of course; mathematics is all "theory". You can't actually have an infinite number of anything in the real world.
That doesn't change the results. All infinite sets are equal in size. Of course, then you can get into arguments about different kinds of infinities (countable, uncountable, etc.)
Originally posted by: gsellis
Here is the only part of all this you want to keep, "My fiance". All the rest is immaterial. Since you argued with her, I can assume that you are marrying "up".- grins, ducks, runs.
Due to a learning disability and a long period of drug use, it took me six years of community college to make up for pissing away high school. I was one of those "gifted kids" on track to be a Nobel Prize winner or whatever when I was young. Not quite one of those kids who learns five languages by the age of 4, but I sure wasn't supposed to do what I did.
As a result, I am a strange hybrid between a reformed homeless whino and a genius (I learned to read French in a summer and can read French, Spanish, Swedish, Czech, Russian and I get by in Italian). I have degrees from top universities, but I went the path of least resistance, opting for a bullshit major that would allow me to graduate with honors without having to so any sort of reading or anything else. The end result is a useless degree. I have the ability to learn anything, but the things I have applied that ability to are essentially useless (unless you want to know about the NHL Draft or urban music history)
That's the long answer (as if any of you care).
The short answer is yes, I am marrying up. She is everything I am not, and I thank God everyday that she thinks I am good enough for her.
I love arguments with science people.
I understand the theory. I understand the concept. But it doesn't work.
Define "doesn't work". Doing it any other way breaks most of discrete mathematics and set theory, and doesn't make any sense given any reasonable definition of "infinite".
I can't go toe to toe with you in a math discussion. That should be obvious.
You're talking in math theory and I am talking in common sense.
You're talking in math theory and I am talking in common sense.
The cardinality of the continuim is greater than that of N.We can map all of the integers to any (a,b) where a,b elements of R.
I can see why you get into a lot of arguments about stuff like this, then...
Like I said, it's counterintuitive. Just because it doesn't seem obviously correct to you doesn't mean it's not true.
Right -- that's the whole 'countable' versus 'uncountable' thing (aleph-1 versus aleph-noll). The 'number' of real numbers is, in some sense, not 'equal' to the number of integers (depending on what exactly you mean by 'equal'; of course, then you have to start defining what you mean by 'number', and it just gets ugly). What I can say is there are an infinite number of real numbers in any bounded subset of R, which is not true for N. My knowledge of detailed interactions between them sort of breaks down not a whole lot further than here, since my background is more in computer science and discrete math.
more discussion along these lines
Well, I'm getting the feeling that you don't want to "get it". So maybe we'll just leave it at that...
DrPizza
Administrator Elite Member Goat Whisperer
Fortunately, mathematicians threw out "common sense" a while ago and instead are going with valid proofs. Common sense leads to errors and contradictions - our intuitions often fail us.
Take .9999999.... (repeating forever) - those who understand the mathematics know that it's exactly equal to one... the proofs are relatively simple. Those relying on "common sense" persist in believing it's "slightly less than" or "insignificantly less than" 1. It's *exactly* equal to 1.
Another one...
sqrt(0 + sqrt(0 + sqrt(0 + sqrt(0 + sqrt(0 +....)
(endlessly nested square roots of zero)
= 0
However, the limit as x approaches 0 (becomes infinitesimally close to zer) of
sqrt(x + sqrt(x + sqrt(x + sqrt(x + sqrt(x +....
=1
not 0.
If I had more, I'd give some more examples of problems that "common sense" frequently misses.
DrPizza
Administrator Elite Member Goat Whisperer
Also, in regard to the different sizes of infinity.
The size of infinity for even numbers is the same as the size of infinity for counting numbers (as someone said above)
Some people would think there are twice as many counting numbers, since
1,2,3,4,5,6,7,8,9,10
2,4,6,8,10
It seems like there are twice as many counting numbers...
But, no matter what counting number you list, there corresponds exactly 1 even number.
Name a counting number... any counting number...
There exists a corresponding even number which is 2 times the counting number.
Similarly, there is an exact 1 to 1 mapping of the counting numbers for infinity +1 and infinity...
Thus, they are the same size. (not insignicantly different, but the *same* size of infinity)
The size of infinity for rational numbers is the same size as the infinity of counting numbers (but requires a diagram that I can't reproduce here to illustrate this)
However, this is frequently used to illustrate that there are different sizes of infinity:
There are more real numbers than counting numbers.
Proof:
Assume there is a 1 to 1 mapping between the two sets.
1 corresponds to .191253897102...
2 corresponds to .159827301957...
3 corresponds to .769434573102...
4 corresponds to .982876501957...
5 corresponds to .362754347102...
6 corresponds to .159827301957...
Now, I'm going to create a new number:
For the 1st decimal place, take the 1st decimal place of the 1st number
For the 2nd decimal place, take the 2nd decimal place of the 2nd number
For the 3rd decimal place, take the 3rd decimal place of the 3rd number
and so on... this gives us:
.159857...
Now, add 1 to each digit (nine becomes a zero)
The new number is .260968...
This new number is guaranteed to be different from the 1st number in the 1st decimal place
and different from the 2nd number in the 2nd decimal place
and different from the 3rd number in the 3rd decimal place,
and so on... Sooooo, the new number cannot possibly be on the 1 to 1 list because it differs from every single number by at least 1 decimal place.
This is a contradiction of our assumption: that a 1 to 1 correspondence exists. Therefore, a 1 to 1 correspondance doesn't exist.
In fact, there are infinitely many more real numbers than counting numbers - a new size of infinity.
The size of infinity for even numbers is the same as the size of infinity for counting numbers (as someone said above)
Some people would think there are twice as many counting numbers, since
1,2,3,4,5,6,7,8,9,10
2,4,6,8,10
It seems like there are twice as many counting numbers...
But, no matter what counting number you list, there corresponds exactly 1 even number.
Name a counting number... any counting number...
There exists a corresponding even number which is 2 times the counting number.
Similarly, there is an exact 1 to 1 mapping of the counting numbers for infinity +1 and infinity...
Thus, they are the same size. (not insignicantly different, but the *same* size of infinity)
The size of infinity for rational numbers is the same size as the infinity of counting numbers (but requires a diagram that I can't reproduce here to illustrate this)
However, this is frequently used to illustrate that there are different sizes of infinity:
There are more real numbers than counting numbers.
Proof:
Assume there is a 1 to 1 mapping between the two sets.
1 corresponds to .191253897102...
2 corresponds to .159827301957...
3 corresponds to .769434573102...
4 corresponds to .982876501957...
5 corresponds to .362754347102...
6 corresponds to .159827301957...
Now, I'm going to create a new number:
For the 1st decimal place, take the 1st decimal place of the 1st number
For the 2nd decimal place, take the 2nd decimal place of the 2nd number
For the 3rd decimal place, take the 3rd decimal place of the 3rd number
and so on... this gives us:
.159857...
Now, add 1 to each digit (nine becomes a zero)
The new number is .260968...
This new number is guaranteed to be different from the 1st number in the 1st decimal place
and different from the 2nd number in the 2nd decimal place
and different from the 3rd number in the 3rd decimal place,
and so on... Sooooo, the new number cannot possibly be on the 1 to 1 list because it differs from every single number by at least 1 decimal place.
This is a contradiction of our assumption: that a 1 to 1 correspondence exists. Therefore, a 1 to 1 correspondance doesn't exist.
In fact, there are infinitely many more real numbers than counting numbers - a new size of infinity.
Actually I think this comes down to advanced set theory, of which I know a little, but not nearly as much as I would like.
Point 1:
Set theory, amoung other things, can predicate that nothing is completely "discrete" and that things need to be looked at on a continuum.
For example, I am asked "how many motorcycles do you have?" If I own a fully functional motorcycle, my answer should be "I own 1 motorcycle".
But motorcycles are generally seen as being counted in wholes, with little implicit agreement as to how a "motorcycle" is to be determined to be a "motorcycle" (you either have a motorcylce, or you do not: you are rarely asked "how many functional and how many partial motorcycles do you have").
But if the motorcyle is in parts in my garage, and I know I'm missing a couple of small parts, then, when asked "how many motorcylces do you have", I really have less then a motorcycle, but it is still perfectly acceptable to say "I have 1 motorcycle" (since saying "I have 7/8ths of a motorcycle makes little or no sense for the sake of the discussion). So in this case, motorcycles = 1 for a very low value of 1. Assuming, I buy a brand new motorcycle and keep the other in peices on the floor, and am asked this question again, I can reasonably answer either, "I have 1 motorcycle" for a very high value of "1" or I can answer "I have 2 motorcycles" for very small values of 2.
Therefor, if I have 2 complete motorcycles and most of a third, then buy 2 more complete motorcycles from a dealer, and she throws in a big box of old parts to get it them off his hands, it's not inconceivable that the two junkers put together could equal a fully functioning motorcycle. Now if I were asked, it would not be inapproprate for me to say "I had 2 motorcylces, then I bought 2 motorcycles, and now I have 5 motorcycles." In effect "2+2" =5 for extremely high values of "2".
Point 2:
From a set theory perspective even framing a question like infinity+1 > infinity makes no sense. If the set is "all things" then there can be no "thing" to add to a set of "all things". The shear act of it existing includes it in the set of "all things". The fact that it is not included in the set means it is not "a thing". So the phrase "infinity+1" is literally non-sense (e.g., [all pizzas] + [this pizza] still = [all pizzas]).
What I think IS interesting though is that though "infinity+1" as an expression is non-sense, "infinity-1" is a very concrete concept that we have all intuitively understood since before we could probably understand language.
For examle, children understand very young there are lots of people in the world, but that their mother unique and special, so of all the humans in the entire world (all things[human]) MY mother is the only one that I love. Stated another way all things[human]-1 [my mother]= 2 discrete sets:
- all things [human - my mother], and
- [my mother].
before they can even say ma-ma. I think that's neat.
Hopefully this has made some sense.
AMB
Point 1:
Set theory, amoung other things, can predicate that nothing is completely "discrete" and that things need to be looked at on a continuum.
For example, I am asked "how many motorcycles do you have?" If I own a fully functional motorcycle, my answer should be "I own 1 motorcycle".
But motorcycles are generally seen as being counted in wholes, with little implicit agreement as to how a "motorcycle" is to be determined to be a "motorcycle" (you either have a motorcylce, or you do not: you are rarely asked "how many functional and how many partial motorcycles do you have").
But if the motorcyle is in parts in my garage, and I know I'm missing a couple of small parts, then, when asked "how many motorcylces do you have", I really have less then a motorcycle, but it is still perfectly acceptable to say "I have 1 motorcycle" (since saying "I have 7/8ths of a motorcycle makes little or no sense for the sake of the discussion). So in this case, motorcycles = 1 for a very low value of 1. Assuming, I buy a brand new motorcycle and keep the other in peices on the floor, and am asked this question again, I can reasonably answer either, "I have 1 motorcycle" for a very high value of "1" or I can answer "I have 2 motorcycles" for very small values of 2.
Therefor, if I have 2 complete motorcycles and most of a third, then buy 2 more complete motorcycles from a dealer, and she throws in a big box of old parts to get it them off his hands, it's not inconceivable that the two junkers put together could equal a fully functioning motorcycle. Now if I were asked, it would not be inapproprate for me to say "I had 2 motorcylces, then I bought 2 motorcycles, and now I have 5 motorcycles." In effect "2+2" =5 for extremely high values of "2".
Point 2:
From a set theory perspective even framing a question like infinity+1 > infinity makes no sense. If the set is "all things" then there can be no "thing" to add to a set of "all things". The shear act of it existing includes it in the set of "all things". The fact that it is not included in the set means it is not "a thing". So the phrase "infinity+1" is literally non-sense (e.g., [all pizzas] + [this pizza] still = [all pizzas]).
What I think IS interesting though is that though "infinity+1" as an expression is non-sense, "infinity-1" is a very concrete concept that we have all intuitively understood since before we could probably understand language.
For examle, children understand very young there are lots of people in the world, but that their mother unique and special, so of all the humans in the entire world (all things[human]) MY mother is the only one that I love. Stated another way all things[human]-1 [my mother]= 2 discrete sets:
- all things [human - my mother], and
- [my mother].
before they can even say ma-ma. I think that's neat.
Hopefully this has made some sense.
AMB
You could have saved a lot of time and stated both points with brevity:
1. The set of all things + some other thing still = all things. (Infinity + 1 = Infinity)
2. The set of all things - one thing are two different sets. (Infinity - 1 = Infinity + that 1 thing)
What does this have to do with 2+2 = 5?
Here's one way: Remember that .99999999 = 1
2.499999 + 2.499999 = 5
(2.4 + .099999) + (2.4 + .099999) = 5
(2.4 + .1) + (2.4 + .1) = 5
2.5 + 2.5 = 5
5=5
Therefore, 2+2 rounded down = 5
Actually, this isn't the real way to do it, however.
Also, this has been over by the philo people, from "1984", when Winston Smith is taught to believe he must accept a known lie as the truth.
2.499999 + 2.499999 = 5
(2.4 + .099999) + (2.4 + .099999) = 5
(2.4 + .1) + (2.4 + .1) = 5
2.5 + 2.5 = 5
5=5
Therefore, 2+2 rounded down = 5
Actually, this isn't the real way to do it, however.
Also, this has been over by the philo people, from "1984", when Winston Smith is taught to believe he must accept a known lie as the truth.
Apparently I wasn't clear in my post, as I was trying to demonstrate that if you have a "set of all things" you can't have "+ some other things": by definition the "other things" can't exist.
As for the infinity +1 = Infinity question, it was brought up by twharry earlier in the thread.
AMB
when do you learn that .999... is *exactly* zero?
it seems like you guys are rounding or using undefined functions to explain the theory.
or i'm just too remedial compared to you guys. i'll be quiet now
It is a set theory problem...
Take some set of numbers, if you add them across vs. adding them vertically the answer should be the same.
However, in some cases, the total sum will be different depending on which way you add up the number.
(for example, sum by adding the number across = 0, sum by adding the number vertically = 5)
Take some set of numbers, if you add them across vs. adding them vertically the answer should be the same.
However, in some cases, the total sum will be different depending on which way you add up the number.
(for example, sum by adding the number across = 0, sum by adding the number vertically = 5)
Uh, never. Unless you meant "*exactly* one".
it seems like you guys are rounding or using undefined functions to explain the theory.
or i'm just too remedial compared to you guys. i'll be quiet now
If you search the archives, you might be able to find some of the extremely long and heated threads from a while back about this topic. Suffice it to say that the limit of the sum from 1 to x of (9 * 10^(-x)) as x goes to infinity (more commonly written as ".999...") is exactly equal to "1.000...". Anyone who says otherwise just doesn't quite grasp limits and/or first-year calculus.
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