This one has got me.
I’m reading a book at the moment that mentioned very briefly that some infinities are bigger than others. The book is unrelated and the author dedicated all of one sentence to the fact – but it’s blown my mind!
I’ve always thought (pretty sure I was always taught) that infinity just is. Something is infinite if it goes on forever, but how can something go on forever more than another?
I’ve tried to google but I’m just not grasping it. How is it that one infinity can be bigger than another?
In: Mathematics
If you can subtract infinity from both sides of the equation, you can still negate the difference regardless if it’s logical.
ie: ♾️+2 > ♾️-3
If you look at the set of Real Numbers (all numbers that aren’t imaginary) vs the set of Natural Numbers (the countable numbers: 1, 2, 3, …), they are both infinite, but the set of Real Numbers contains all of the Natural Numbers and more (the set of Real Numbers includes non-integer numbers such as 1.273, as well as negative numbers). So they’re both infinite sets of numbers, but there’s more Real Numbers than Natural Numbers.
10100100010000….. is infinite
9999999999999…. is also infinite
But the 2nd one will always be bigger
Is it The Fault in Our Stars?
Lets start simple.
Consider a set made up of all the numbers x such that x > 0. Since there are infinite positive numbers, the number of elements in this set is infinite. Lets call this set A.
Now, lets consider a set made up of of all the numbers x such that x >= 0. Since there are infinite positive numbers, the number of elements in this set is infinite. Lets call this set B.
Now, set B will contain all of the numbers in set A, but it will contain 1 number that set A doesn’t: 0. So, the size of set B is 1 + size of set A. Now, since set A is infinite, set B is infinite too, but it is clearly a larger infinite than A.
Lets consider another set, made up of all the numbers x such that x < 0. Since there are infinite negative numbers, the number of elements in this set is infinite. Lets call this set C.
Set C has the same size as set A, since every positive number has a negative equivalent, and thus, it’s a smaller set that B.
If we combine sets C & B together, we get the set of all real numbers, which is more than twice the size of set A, but less than twice the size of set B.
Or, consider a line starting at one point and going out to infinity. If we extend that line backwards from the starting point, we have created a larger infinity. If we choose some new point along that line, and cut away from the starting point to that new point, we have created a smaller infinity. If we draw new lines at different angles from that starting point out to infinity, we have multiplied that infinity. We can do that an infinite number of times, as there is a infinite number of angles in the arc.
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There’s a rule that set theorists use when talking about sets with infinitely many objects in them. They say that such sets are the same size if there exists some way to pair up members of one set with members of the other set so that neither set has any members left out. For every item in set A you could ever name, you should be able to name one, and only one, counterpart in set B, and vice versa.
So one weird result of this rule, is that the set of positive even numbers, has the same size as the set of positive whole numbers (aka the natural numbers, or N.) You might initially guess that one is twice as big as the other. But you can easily come up with a rule between these sets which matches their members up so none is left out. For every member of the whole numbers, multiply it by 2 to find its partner in the even numbers. For every member of the even numbers, divide by 2 to find its partner in the whole numbers.
When you can devise a rule like this which perfectly assigns partners to partners between two sets, we call this a “one to one ~~mapping~~correspondence”, or a “bijection”.
Now there’s a tricky question to ask: Between any 2 infinite sets, is there always a bijection?
A mathematician named Georg Cantor proved that sometimes there isn’t, using his now-famous “diagonal argument.” Specifically, he proved that there’s no bijection between the natural numbers N and the real numbers R. No matter what scheme you might propose to assign every whole whole number a partner in the real numbers, you can always come up with more real numbers which are unpartnered. This means that the size of R is strictly greater than the size of N.
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You can get infinitely small-grained between 1 and 2 (like 1.000000000000000000000000000001). You can do the same thing between 1 and 3, which gives more options.
Infinity isn’t a number, it’s a placeholder that represents the result of some mathematical function. You can’t just take two infinities in isolation and compare them, because we don’t have the language or notation to determine any difference between them. But it’s possible for the functions behind those infinities to be compared and to mathematically prove that one will always be larger than the other.
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