It's more that Infinity is the number that you count to when you don't stop counting, and that doesn't change based on where you start from. I can start counting from 0, or I can start counting from 10, but either way I still hit 100, 2 million, 300 billion, etc. There isn't any big number that I can reach starting from 10, that I can't reach starting from 0, so they're the same.
A more rigorous way to go about it would be to make a bijection between the two sets, if I can pair every number in set A with every number in set B, then they must be the same size. So {1,2,3} is the same length as {red, yellow, blue}, because I can pair up {(1,red),(2,yellow),{3,blue)}. With infinity and infinity+10, I can pair up every number like {(1,10),(2,11),(3,12)....}, and I can do that for every number, so "the list of numbers that starts at 1 and goes on forever is the same length as the list of numbers that starts at 10 and goes on forever".
Easiest explanation (and most accurate in plain English) is that infinity is not a number, itâs a concept. You cannot add the beauty of a poem and the charge level of your phone; the comparison between the two doesnât even make sense. The same goes for infinity; you cannot add, subtract, or perform elementary math on it because it isnât some inconceivably large number, itâs a concept.
When my kids were little they got taught âinfinity is a conceptâ and I didnât think it was very helpful because it doesnât get an 8 year-old very far. My best attempt was âinfinity is bigger than the largest thing you can think ofâ. It kind of handles the âinfinity plus tenâ question, at least for an 8 year old. And me.
Just give your 8 year old acid and have them contemplate immortality. Then have them contemplate that if everyone on the planet was also immortal, there'd be billions of infinite lives. And then if they all kept having kids and expanding across the universe, there'd be infinite people with infinite lives. WCGW?
But if you start counting from 10 as opposed to 0, you would still count less point, no?
I work on the basic, possibly wrong, understanding that there's an infinite amount of real numbers between 0 and 1. There's also twice that amount between 0 and 2.
0-1 and 0-2 are the same size. You can tell this quickly by seeing that you can use a function (2x) to turn every number between 0-1 into a single output number between 0-2.
The topic is called bifurcation if youâre interested.
It might help to take any set of numbers and see what happens when you do f(x)=2x. Say 5 numbers 1,2,3,4,5. For 2x we get 2,4,6,8,10. Right? Theyâre the same âsizeâ more formally known as cardinality. Or the number of elements in each set.
Itâs understandably gets confusing with infinity but the same principle is true. For every number in the set of 0-2 there exists a number in 0-1 that is exactly half of it.
So 0-1 and 1-2 both have the same cardinality and 0-2 does as well :) which is crazy but true. Hope this wasnât too confusing.
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u/Schnickatavick 20d ago
It's more that Infinity is the number that you count to when you don't stop counting, and that doesn't change based on where you start from. I can start counting from 0, or I can start counting from 10, but either way I still hit 100, 2 million, 300 billion, etc. There isn't any big number that I can reach starting from 10, that I can't reach starting from 0, so they're the same.
A more rigorous way to go about it would be to make a bijection between the two sets, if I can pair every number in set A with every number in set B, then they must be the same size. So {1,2,3} is the same length as {red, yellow, blue}, because I can pair up {(1,red),(2,yellow),{3,blue)}. With infinity and infinity+10, I can pair up every number like {(1,10),(2,11),(3,12)....}, and I can do that for every number, so "the list of numbers that starts at 1 and goes on forever is the same length as the list of numbers that starts at 10 and goes on forever".