I read a book on the philosophy of set theory—and I get lost right at the point where classical infinite thought was replaced by modern infinite thought. IIRC the problem was paradoxes based on infinite recursion (Zeno et. all) and finding mathematical foundations to satisfy calculus limits. Then something about Cantor, cardinality and some hand wavy ‘infinite sets are real!’.
1.999… is just an infinite set summation of finite numbers 1 + 0.9 + 0.09 + …
Now, how an infinite process on an infinite set can equal an integer is a problem I still grapple with. Classical theory said that this was nonsense since one would never finish the summation (if one were to begin). I tend to agree and I suppose one could say I see infinity as a verb and not a noun.
I suggest anyone who believes 1.999… === 2 really looks into what that means. The root of the argument isn’t “What is the number between 1.999… and 2?” but rather “Can we say that 1.999… is a sensible theoretical concept?”
I read a book on the philosophy of set theory—and I get lost right at the point where classical infinite thought was replaced by modern infinite thought. IIRC the problem was paradoxes based on infinite recursion (Zeno et. all) and finding mathematical foundations to satisfy calculus limits. Then something about Cantor, cardinality and some hand wavy ‘infinite sets are real!’.
1.999… is just an infinite set summation of finite numbers 1 + 0.9 + 0.09 + …
Now, how an infinite process on an infinite set can equal an integer is a problem I still grapple with. Classical theory said that this was nonsense since one would never finish the summation (if one were to begin). I tend to agree and I suppose one could say I see infinity as a verb and not a noun.
I suggest anyone who believes 1.999… === 2 really looks into what that means. The root of the argument isn’t “What is the number between 1.999… and 2?” but rather “Can we say that 1.999… is a sensible theoretical concept?”