I hope everyone will pardon the deletion and repost of my two previous posts but they were inaccurate in certain place.
@Euler13 I trust you to verify, if you want. (I notably previously mixed rational, real, and complex numbers, it’s been a while)
@Urben The best example of two very different set of infinity is :
- N set, the natural set (ie “full numbers” used for counting, 1 2 3 4 5…) : there is obviously an infinity of them
- Q set, the rational set (ie decimals 0,1 0,01 0,001…) : also an infinity of them, but well there is an infinity of them more than the natural one since there is a infinity of them between each natural number.
Both are infinite. One is seemingly larger, but both are countable and in the end as.
Even more both can be expressed with the other.
@Hichkas Something else you could have for the hotel is to imagine it incrementaly.
Imagine the following :
- You have an hotel with an infinity of room.
- Arrive an infinity of clients
- The hotel is full
Suddenly an new client arrives, can you have more than the infinity already here ? yes
- you wake every client and ask them to go to the next room
- the client in room 1 goes to room 2, the client in room 2 goes to room 3, the client in rrom 3 goes to 4, …
- you give the key to room 1 now unoccupied to the new client
- the hotel is full, all client are served
So infinity +1 is still the same infinity
Even more suddenly an infinity of clients arrive, can you have that much more ? yes
- you wake every client and ask them to go to the room with twice the number
- the client in room 1 goes to room 2, the client in room 2 goes to room 4, the client in room 3 goes to 6, …
- every client is in an even room
- you give the new clients the uneven rooms, there is an infinity of them
- the hotel is full, all clients are served
So two time the infinity is still the same infinity
An infinity of bus arrive with an infinity of passengers each, can you have that much more ? yes
- you do a double entry table with at the top the bus number and on the side the passenger number
- you fill it in diagonal for an easy match for everyone
| - || Bus 1 |Bus 2|Bus 3| Bus 4 |Bus 5|…|
|—|—|—|—|—|—|—|—|—|—|
| P1 ||1|3|6|10|15|…|
| P2 ||2|5|9|14|20|…|
| P3 ||4|8|13|19|26|…|
| P4 ||7|12|18|25|33|…|
| P5 ||11|17|24|32|41|…|
|…||…|…|…|…|…|…|…|…|
- the hotel is full, all the clients are served
An infinity of infinity is still the same infinity
The importance in set of infinity is to know if you can express one with an other.
In other term can we transform the numbers in the set with mathematics into a number that also is in the same set.
Can we transfer a number in another number place, a client in another room
If so they are as big/identical, if not one is larger than the other.
(I just realised I might have just resaid what was told in the video, I can’t watch it now)
But now is when things start to be tricky.
As I said, the important part is to know if each set of infinity can be expressed from another one.
In the previous examples, in mathematics all of them are part of the Q set. The rational set.
The definition of the Q set is any numbers that can be expressed as a fraction of two natural number
- 1 is a rational number.
- 0.5 is a rational number because it also is 1 / 2
- minus 2.27465892247 is a rational number because it’s also -53872153 / 23683618
- …
But then you have numbers like pi or the square root of 2.
They are part of the R set, the real set.
It’s definition is that it’s any numbers. Like that’s it. Any numbers that are in reality (hence the name). (alright I simplify, there is obviously more, much more, because mathematicians are a bunch of rude people, but that will be enough for us here)
As a consequence they also includes numbers that cannot express themself as a fraction of two natural numbers.
- for the funny anecdote, in ancient greece there once was a sect
- this sect was one of mathematicians (yes, I know)
- it was called the pythagorian
- the largest tenet of the sect was the beauty of the world and how everything can be expressed through fraction
- the most holy number was what we now call pi, because it was the beautifull number that could express through a fraction a circle.
- they then discovered that pi itself could not be expressed by a fraction
- it did not end well
- madness
- like murder over it to hide the shame kind of madness
Remember when I said the importance is if that the infinity set need to be capable of expressing themself with another ?
Well, that’s it.
In the metaphor of the hotel, you have a rational number of room, so if a real number of clients arrive then you won’t be able to accomodate them all.
Because there is no way to transform all the possibility of the real set into the rational set.
You will always have some left, all the one that are not a fraction.
You would be capable to do the reverse though.
An hotel with a real number of room, some named pi, some other name squareroot of 2, some just named 42.
A list of client named after rational number.
With a lot of room to spare
)
