Should be simple for those trained. If there are infinite possible numbers between 0 and 1, and there are also infinite numbers between 0 and 2, then the possible numbers between 0 and 1 cannot be less than those between 0 and 2. Only their sum values can be less. Are these assumptions correct?
Your question is a bit mind-blowing, and so shall my answer be. As I understand your question: It seems to me that, while you are correct that there is a set of infinite numbers between 0 and 1 and another set of infinite numbers between 0 and 2…the latter set can still contain elements greater than those in the former set.
Correct me if I am wrong, but I think (in the back of your mind) you are equating “infinite numbers” to an infinite set. And, while at first it may seem one infinite set’s as good as the next, your question demonstrates that the individual elements of two distinct infinite sets can vary greatly. (Hmm, such a curious thing that an infinite set of numbers does not need to contain the realm of all numbers.)
There is the same quantity of real numbers between 0 and 1 as the quantity of real numbers between 0 and 2. This is easily demonstrated by the existence of a 1 to 1 correspondence between them.
However, this quantity is far greater than the quantity of integers. In other words, while one can assign to each integer a distinct real number between 0 and 1, one cannot assign to each real number between 0 and 1 a distinct integer. The demonstration of this is a bit more complicated, but still accessible to most laymen.
I think the one-to-one correspondence does it. There is such a thing as “greater than infinite” but it follows some very strict assumptions.
It is true that there are equally many numbers in the interval (0,1) and the interval (0,2). The way mathematicians would say this is that (0,1) is a set with the same cardinality as (0,2). As Chronos says, this fact relies on the existence of a correspondence between them. The function f(x) = 2x is a one-to-one and onto function (the mathematical jargon is bijective function) that maps every element in (0,1) to an elements in (0,2) and hits every element in (0,2).
The question of the sum of the numbers in (0,1) or (0,2) cannot be answered, however, at least not in a mathematically rigorous way. To put it bluntly, there are to many real numbers in those intervals, so there’s no way to add them up and get a sum.
This I can imagine.
This is just two different definitions of infinite sets. So what?
Mind not blown.
Also. The original post says “the possible numbers between 0 and 1 cannot be less than those between 0 and 2.”
The number 2.1 is not possible between 0 and 1.
Better stated as “the number of numbers between 0 and 1 cannot be less than those between 0 and 2.”
But still not right. The number of numbers between 0 and 1 is at least minus the number 2.1 as well as an infinite number of other numbers.
In fact, a functional definition of what it means for a set to be “infinite” is that there is a bijective function between itself and a proper subset of itself.
Some infinities are bigger than others.
Nothing is greater than infinity, strictly speaking. There are an infinity of infinities, and it can be shown that each iteration upward contains more elements than those below, but every one, starting with the counting numbers, is infinite.
Don’t try to use common sense or everyday language around infinities. They require special, carefully-drawn rules.
OK, my last foray into tough math (for now?) Since the number of possible numbers in a set 1-2 is infinite, then the sum of their values is likewise infinite. One can’t argue it’s greater than infinite? What about a positive number that’s less than 0? when used as a divisor to a positive number, will it result in something greater than infinite?
The OP didn’t specify real numbers, though. We might be talking about rational numbers instead. The same 1-to-1 correspondence holds, of course, but it also holds with the integers. Therefore, there are the same number of integers as there are rational numbers between 0 and 1.
Something “greater than infinite” is still infinite. But it might be a different kind of infinite. Any interval of the real number line (such as the interval (1,2)) has an uncountably-infinite amount of real numbers in it. By contrast the whole set of integers is countably-infinite. Infinite just means something that continues for ever, and both the real numbers and the integers go on and on forever. But the nature of their infinite sizes is different.
A positive number can’t be less than zero, because the definition of a positive number is something greater than zero. So that’s not a question that can be answered in a meaningful way, unless you want to propose some weird new number system where a number could simultaneously be positive and less than zero. (Nothing wrong with inventing weird new number systems, but I’m not aware of any that have this property.)
Summation is defined for sequences of numbers, not sets. However it is easy to show that when the set of terms is all the rational numbers in a given interval (or indeed any dense set) then the sum diverges.
There can be different infinities… You can compare them…
if y = 2x, even if x is infinity, then y/x =2
You can’t really do this, because multiplication and division by an infinity is not well defined, because infinities are not numbers.
There are ways to compare the sizes of infinite sets as mentioned above, but that’s not the same thing.
Freido is right. But then, infinity x 2 = infinity
Like zero (i.e. 1/0 = not defined) , infinity has its own rules.
If it weren’t for the Dope, my knowledge of infinity will stay stagnant as it has for the past 33 years.
As you may have gathered from the replies so far, OP question leads us into the huge (infinite?) gray area where mathematics and logic part ways. From a mathematical pov, the fallacy is - as already said - that ‘infinite’ can not be treated simply as just another number: it cant be manipulated (add, sustract etc) like 3, 12.43 or pi.
Btw: Some time ago Cecil or the staff wrote a column about a related question - is .999~ the same as 1? http://www.straightdope.com/columns/read/2459/an-infinite-question-why-doesnt-999-1