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So the infinity of integers is (somewhat counter intuitively ) the same as the infinity of even numbers because you can pair 1-2 2-4 etc. However the infinity of infinite decimal expansions is not the same as the infinity of integers as there is no way the two can have a mapping.
Hope that helps.
"Take these three items, some WD-40, a vise grip, and a roll of duct tape. Any man worth his salt can fix almost any problem with this stuff alone." - Walt Kowalski
"Only two things are infinite - the universe, and human stupidity. And I'm not sure about the universe." - Albert Einstein
The argument for decimal expansions is that however you try to count them, you will miss some. This is dine by a clever construction to build a decimal that differs from every single one that you have counted. The really tricky bit is avoiding the problem that 1=0.999999...
other kinds of infinite forms are more interesting to me, rather than lists.
1 + 1 + 1 + 1 + 1 + 1 + ... = infinity
this is clear right? just like
2 + 2 + 2 + 2 + 2 + 2 + ... = infinity
etc. however, not all infinite sums are infinity. here are some sums which do not equal infinity
1 + 1/10 + 1/100 + 1/1000 + 1/10000 + ... = 1.1111...
1 + 1/2 + 1/4 + 1/8 + ... = 2
1 + 1/2^2 + 1/3^2 + 1/4^2 + ... = pi^2 / 6
now here are some infinite sums which look like they should not equal infinity, but they actually do
1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + ... = infinity
WWHHAT? thats right boi. you have no clue how slowly this thing diverges.
1 + 1/2 + 1/3 + 1/5 + 1/7 + ... (primes) = infinity
I could go on forever
Sadly I don't know the answer to your question of how many infinities there are, but the answer has to depend on what axioms of set theory you take, given the resolution of the continuum hypothesis.