Is it possible to compare two different infinities and say which one is larger?? Or else infinities as a whole are uncountable??
To state the point, one has to say it is not possible to count the number of numbers in the infinite series. but it is possible to compare two infinities and figure out which one is larger.
To stress the point, if we can pair the objects of two infinite groups so that each object of one infinite collection pairs with each object of another infinite collection, and no objects in either group are left alone, the two infinities are equal. If, however, such arrangement is impossible and in one of the collections some unpaired objects are left, we say that the infinity of objects in this collection is larger, or we can say stronger, than the infinity of objects in the other collection.
To understand this better, we can take the example of the infinity of all even numbers and the infinity of all odd numbers and try to figure out which one is larger??
That is,
1+3+5+7+9+11+....+∞
and
2+4+6+8+10+....+∞
You feel, of course, intuitively that there are as many even numbers as there are odd, and this is in complete agreement with the above rule, since a one-to-one correspondence of these numbers can be arranged:
1 3 5 7 9 11 13 15 17 19
^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
2 4 6 8 10 12 14 16 18 20
There is an even number to correspond with each odd number in this table, and vice versa; hence the infinity of even numbers is equal to the infinity of odd numbers. Seems quite simple and natural indeed!
But wait a moment. Which do you think is larger: the number of all numbers, both even and odd, or the number of even numbers only? Of course you would say the number of all numbers is larger because it contains in itself all even numbers and in addition to all odd ones. But that is just your impression, and in order to get the exact answer you must use the above rule for comparing two infinities. And if you use it you will find to your surprise that your impression was wrong. In fact here is the table of one-to-one correspondence of all numbers on one side, and even numbers only on the other:
1 2 3 4 5 6 7 8 etc..
^ ^ ^ ^ ^ ^ ^ ^^
2 4 6 8 10 12 14 etc..
Here again, writing it in the form of the series,
1+2+3+4+5+6+7+8....+∞ - a series of natural numbers
2+4+6+8+10+12+14+...+∞ - a series of even numbers alone
But again the series of even numbers can be re-written in the following manner by taking the common factor, 2 , out of the series.
which re-arranges to,
2(1+2+3+4+5+6+7+8+...+∞)
and this implies that the infinite even number series is twice as large the infinite natural numbers series. Mind-boggling. What seemed so clear on the outset has taken a complete turn-over.
Now, it becomes even easier to find out which is larger - an infinite even number series or an infinite odd number series. Since odd numbers are only a subset of the natural number series, it becomes crystal clear that the infinite even number series are a way larger than the odd series.
1+3+5+7+9+11+...+∞ is a subset of
1+2+3+4+5+6+7+...+∞ which in turn is a subset of
2+4+6+8+10+...+∞.
QED.
N.B: QED stands for Quod Erat Demonstratum in Latin, it is nothing but when we say as "Hence Proved" in English.
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To state the point, one has to say it is not possible to count the number of numbers in the infinite series. but it is possible to compare two infinities and figure out which one is larger.
To stress the point, if we can pair the objects of two infinite groups so that each object of one infinite collection pairs with each object of another infinite collection, and no objects in either group are left alone, the two infinities are equal. If, however, such arrangement is impossible and in one of the collections some unpaired objects are left, we say that the infinity of objects in this collection is larger, or we can say stronger, than the infinity of objects in the other collection.
To understand this better, we can take the example of the infinity of all even numbers and the infinity of all odd numbers and try to figure out which one is larger??
That is,
1+3+5+7+9+11+....+∞
and
2+4+6+8+10+....+∞
You feel, of course, intuitively that there are as many even numbers as there are odd, and this is in complete agreement with the above rule, since a one-to-one correspondence of these numbers can be arranged:
1 3 5 7 9 11 13 15 17 19
^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
2 4 6 8 10 12 14 16 18 20
There is an even number to correspond with each odd number in this table, and vice versa; hence the infinity of even numbers is equal to the infinity of odd numbers. Seems quite simple and natural indeed!
But wait a moment. Which do you think is larger: the number of all numbers, both even and odd, or the number of even numbers only? Of course you would say the number of all numbers is larger because it contains in itself all even numbers and in addition to all odd ones. But that is just your impression, and in order to get the exact answer you must use the above rule for comparing two infinities. And if you use it you will find to your surprise that your impression was wrong. In fact here is the table of one-to-one correspondence of all numbers on one side, and even numbers only on the other:
1 2 3 4 5 6 7 8 etc..
^ ^ ^ ^ ^ ^ ^ ^^
2 4 6 8 10 12 14 etc..
Here again, writing it in the form of the series,
1+2+3+4+5+6+7+8....+∞ - a series of natural numbers
2+4+6+8+10+12+14+...+∞ - a series of even numbers alone
But again the series of even numbers can be re-written in the following manner by taking the common factor, 2 , out of the series.
which re-arranges to,
2(1+2+3+4+5+6+7+8+...+∞)
and this implies that the infinite even number series is twice as large the infinite natural numbers series. Mind-boggling. What seemed so clear on the outset has taken a complete turn-over.
Now, it becomes even easier to find out which is larger - an infinite even number series or an infinite odd number series. Since odd numbers are only a subset of the natural number series, it becomes crystal clear that the infinite even number series are a way larger than the odd series.
1+3+5+7+9+11+...+∞ is a subset of
1+2+3+4+5+6+7+...+∞ which in turn is a subset of
2+4+6+8+10+...+∞.
QED.
N.B: QED stands for Quod Erat Demonstratum in Latin, it is nothing but when we say as "Hence Proved" in English.
