If you have seen the proof for some infinities being larger than others it goes something like this...
You start with a set of numbers, something like...
314
111
789
And then to show that there is an set of numbers that is provably NOT inside the set of numbers you already have here,
you take the first digit of the first number(314) in this case 3, and change it to any number other than what it originally is... i'll use 4,
then you take the second digit of the second number and change it too, then the third of the third and so on...
So in this example I'll change 3 to 4, 1 to 2, 9 to 0, and end up with the number 420.
Now the conventional proof goes that the number 420 is definitively not in the first set of numbers I started with...
because it cannot be the first number since the first digit is different...
it cannot be the second because the second is different,
it cannot be the third because the number in the third place is different,
and you can continue doing this for as long a number as you want,
infact you can do this for an infinitly long number if you like.
So the preposed conclousion is that even if you have an infinitly big set of infinitly long numbers, you can using this method always create a new number that is not in that set.
Using the assumption that infinity is like an infinit set of numbers,
and this conclusion that you can create a new number that is not in this infinit set of numbers,
it is said that this new number has to be outside of this first infinity,
so therefor it is commonly conclouded that there is some numbers not containable inside a regular infinity,
and that there are possibly an infinit ammount of these numbers,
and finally we get to the conclousion that there are multiple different infinit numbers that cannot be contained within eachother,
and therefor that there are multiple different infinities.
So this is a sloppy version of the usual proof for the existance of multiple different types of infinity.
Here's some links to other explinations of it in perhaps better understanable formats,(TODO add links to bottom of page and a jump link to links here)
now this is all fine and dandy, but don't you feel like we missed something?
and i'm not talking about that feeling i cant quite put my fingure on that we perhaps had a jump in logic in the second half of the proof,
in the first part about creating a new number that is not within the original set of number we have.
if you like you can pause here and try to spot it, we made an unstated assumption.
it's about the number we created, it is not in the set of numbers we checked.
but what set of number did we start with, and what set of numbers did we check?
and importantly are they the same set of numbers?
now lets go over the first part creating a 'new' number again
but just a bit more carefully
this time lets fill our starting set like an infinit one would be
so our first digits will be
0
1
2
3
4
5
6
7
8
9
same with our second, third and so on...
now do you see the problem
if we take the first digit and change it, the new number is still in the set of numbers we have.
same with the second and so on
it is eazier to show in bianary spacewise, if you understand bianay that is
i'll link something for that here
so now se see that the new number wemake is always already in the set of numbers we have
it just didnt show up becouse in the initial proof we only check the first small section of the set for it,
and not he whole set so we missed it.
so where do we go from here?
well in my humble opinion first of all double check my work, and check the original proof for yourself
and look for more proofs if they exist and check for their validity or lack thereof.
My hypothises (link)? is that there is only one infinty, or if you prefere a different wording that all infinit sets can be mapped onto eachother.
I have started a draft of a proof (link?) that shows that shows this,
by proving that any definable set can be mapped onto regular rational number infinity.
that should do it for
... i'll finish this in a bit hehe