He says, “I love you more than a billion!”
She says, “I love you a trillion much!”
He says, “I love you a billion raised to the millionth power much!”
She pulls the clincher. “I love you an infinite much!”
And there’s the word. “Infinite” and its cousin “infinity.” What do we mean when we say them? Big. Really big. Perhaps a number larger than the largest number we can think of. Maybe a place further away than any other place.
Funny things can happen when one discusses infinity, and we’ll look at just one of them.
Fortunately, we don’t have to deal with infinity in our daily lives. When we go shopping we don’t ever need an infinite amount of cash. We just hope that the finite amount we have will pay for the finite amount of groceries we need. When we take a car trip we are going only a finite distance. That’s a good thing because otherwise we would need an unlimited amount of gas which would cost an unlimited amount of dollars. And an unlimited amount of time to make the trip.
So infinity is a vague concept for most of us.
Mathematicians don’t like the word “vague” and have discovered concrete ways of dealing with the concept. And one of the. weird things that can happen is this.
Suppose I give you 100 pieces of paper and on them I write each of the positive integers from 1 to 100, one on each piece of paper.. If I asked you how many positive integers were written you’d reply 100. If I said how many of these were even—you know, 2, 4, 6, etc.—you’d correctly say 50. If I gave you a million pieces of paper your answers would have been a million and 500,000, respectively. The number of even integers written is half the total number of integers written,
Now here’s the weirdness. Suppose I gave you an infinite number of pieces of paper, something I can’t do, of course. Then I claim the number of positive integers, and we know there are an infinite number of them because I can always find one bigger than any you can name, maybe by adding 1 to yours, is the same as the number of positive even integers! Mathematicians talk about equal cardinality instead of numbers, but that’s not necessary here.
Pick any positive integer you want, say 17. Then multiply it by 2 to get 34. Multiplying it by 2 always gives you an even integer. So, for any positive integer, there is a unique positive even integer associated with it. If you start with 26, for example, you get 52 after multiplying. And if you start with different integers you will wind up with different integers after multiplying. And this is always possible no matter how large the starting integer is because there is no limit to the integers so the one twice as big will always be available. Thus, for every integer there is an associated even integer so in some sense there must be at least as many positive even integers as there are positive integers. If so, this means the positive even integers are at least as many as the positive integers.
Now go the other way. Start with a positive even integer. Then there is a unique positive integer that is half of its value. Hence, there must be at least as many positive integers as there are positive even integers.
Think about those two conclusions. Doesn’t it mean the number of integers is the same as the number of even integers?
And that’s what we mean when we say the set of all positive integers has the same cardinality as the set of positive even integers.
And that concludes our introduction to the concept of infinity where weird things happen.
She says, “I love you a trillion much!”
He says, “I love you a billion raised to the millionth power much!”
She pulls the clincher. “I love you an infinite much!”
And there’s the word. “Infinite” and its cousin “infinity.” What do we mean when we say them? Big. Really big. Perhaps a number larger than the largest number we can think of. Maybe a place further away than any other place.
Funny things can happen when one discusses infinity, and we’ll look at just one of them.
Fortunately, we don’t have to deal with infinity in our daily lives. When we go shopping we don’t ever need an infinite amount of cash. We just hope that the finite amount we have will pay for the finite amount of groceries we need. When we take a car trip we are going only a finite distance. That’s a good thing because otherwise we would need an unlimited amount of gas which would cost an unlimited amount of dollars. And an unlimited amount of time to make the trip.
So infinity is a vague concept for most of us.
Mathematicians don’t like the word “vague” and have discovered concrete ways of dealing with the concept. And one of the. weird things that can happen is this.
Suppose I give you 100 pieces of paper and on them I write each of the positive integers from 1 to 100, one on each piece of paper.. If I asked you how many positive integers were written you’d reply 100. If I said how many of these were even—you know, 2, 4, 6, etc.—you’d correctly say 50. If I gave you a million pieces of paper your answers would have been a million and 500,000, respectively. The number of even integers written is half the total number of integers written,
Now here’s the weirdness. Suppose I gave you an infinite number of pieces of paper, something I can’t do, of course. Then I claim the number of positive integers, and we know there are an infinite number of them because I can always find one bigger than any you can name, maybe by adding 1 to yours, is the same as the number of positive even integers! Mathematicians talk about equal cardinality instead of numbers, but that’s not necessary here.
Pick any positive integer you want, say 17. Then multiply it by 2 to get 34. Multiplying it by 2 always gives you an even integer. So, for any positive integer, there is a unique positive even integer associated with it. If you start with 26, for example, you get 52 after multiplying. And if you start with different integers you will wind up with different integers after multiplying. And this is always possible no matter how large the starting integer is because there is no limit to the integers so the one twice as big will always be available. Thus, for every integer there is an associated even integer so in some sense there must be at least as many positive even integers as there are positive integers. If so, this means the positive even integers are at least as many as the positive integers.
Now go the other way. Start with a positive even integer. Then there is a unique positive integer that is half of its value. Hence, there must be at least as many positive integers as there are positive even integers.
Think about those two conclusions. Doesn’t it mean the number of integers is the same as the number of even integers?
And that’s what we mean when we say the set of all positive integers has the same cardinality as the set of positive even integers.
And that concludes our introduction to the concept of infinity where weird things happen.
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