I have just come across the concept that there are different types of infinite numbers (whole integers, irrational numbers etc.) and the concept that for example, the amount of irrational numbers between 0 and 1 is higher than the amount of whole numbers from 1 to infinity.
I guess I just don’t understand how an infinite amount of something can be bigger/smaller than an infinite amount of something else…
Please un-f**k my brain 😀
In: 4
Step 1: What does it mean for two finite groups of objects to be the same size i.e. to have the same number of objects in them?
Even if we don’t have the concept of counting, we can find out. We pair off one object from the first group with one object from the second, and remove them. Then repeat the process with the new, smaller, groups. When one group is empty we look at the other group. If it is also empty, the two groups were the same size. If there are still some objects left then that group was bigger.
Step 2: we extend this method to infinite sets. This is trickier because if we do it one by one, we never end. But we do something similar. Here is an example. Consider the set of all positive integers: 1,2,3,4… etc, and the set of all positive even integers 2,4,6,8…. Etc.
It might look like the first set has more objects in it, because it have all the same objects as the second set, plus all the odd integers. But we can map each object in the first set to exactly one object in the second set so that ever object in each set appears in one of the mappings:
1 <->2
2<->4
3<->6
And in general
n<->2n
If we can do this in some way with two infinite sets, then we say that they are the same size.
Step 3: Suppose we have two infinite sets and we can *prove* that we can’t make such a mapping between them. I.e. however we try there will always be some members of one of the sets unmapped. In that case we would say that the set with unmapped members is bigger. I.e. it’s size is a bigger infinity than the other’s size.
Here’s an outline of a proof that that set of real numbers between 0 and 1 is a bigger infinity than the positive integers:
Any real number between 0 and 1 can be expressed as an infinite decimal 0.abcdef… etc
Now Imagine we had a mapping between the two sets it will look something like this
1 <-> 0.3742851….
2 <-> 0.4188803….
3 <-> 0.9013246….
Etc
Now we can construct a real number that can not be in the list.
We choose the first decimal place to be different to the first decimal place in the first number. So here the first decimal place is 3, so we pick, say, 7.
We choose the second decimal place to be different to the second decimal place in the second number. Here that is 1, so we can pick any other digit. Eg 9
We choose the third decimal place to be different to the the third decimal place in the third number. Here that is again 1, so we can pick eg 5
And we continue for ever.
The resulting decimal is definitely between 0 and 1, but is not in the mapping because for every number n it differs from the nth number in the list in the nth decimal place.
This process would work no matter how we tried to list the real numbers, ie. No mapping is possible between them and the integers, and so the set of them must be larger.
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