I have watched many videos on YouTube about it from people like vsauce, veratasium and others and even my math tutor a few years ago but still don’t understand.
Infinity is just infinity it doesn’t end so how can there be larger than that.
It’s like saying there are 4s greater than 4 which I don’t know what that means. If they both equal and are four how is one four larger.
In: Mathematics
In your day-to-day life you are mostly confronted with normal numbers and by that I mean so called “real numbers” (reals). There is a way to determine if one real is larger than another real – this way doesn’t apply to infinity, because infinity isn’t a real number.
For example the length of a stick can be modelled as a real. A stick has a “greater” length than another stick, when you lay them next to each other and the longer stick goes on, while the shorter stick already stopped. long – short > 0
For numbers or number-like things that include the real numbers, you need a different way of thinking about the “greater-than” relation, that nevertheless has to be compatible with the “greater-than” of the reals. That’s called a “generalization”.
The way mathematicians came up with a more general “greater-than” – that notably might not align with your intuitive real-based “greater-than” – is to have two sets of elements A and B and if you can make a pair of each element in A with one element of B and some As are left over, the number of elements in A is defined to be greater.
For example if there are some men and some women in an old-fashioned dance event where only mixed-gender pairs are allowed and they all pair up, but some men are left over – then we know the number of men was “greater”.
This allows us to compare two sets that are both infinite and it still can turn out that one set is larger. Not in the lay-stick-besides-each-other-sense, but in the pair-up-sense.
When you have the set of all natural numbers (1, 2, 3, …) and the set of all integers (…, -2, -1, 0, 1, 2, …) then they can be paired up, so both infinities are equal. One way to pair them up would be: 1&1, 2&0, 3&2, 4&-1, 5&3, 6&-2 and so on.
When you have the set of all integers and the set of all rationals (= fractions), then they can be paired up as well.
But when you have the set of all rationals and the set of all reals, then no matter how you pair them up, some reals will still be left over. Both numbers are infinite, but one is larger than the other.
Think of “infinite” as a property like “even” in this case and not of an identity. *Of course* two numbers have to be equal if they are identical.
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