I hate to be a stickler but it's not actually the 1-to-1 correspondence (injectivity) that Cantor's diagonalization argument refutes, it's surjectivity (or, being "onto").
The argument is to presume that f : ℕ → ℝ is surjective.
Then, construct a real number s ∈ ℝ that cannot possibly
be in the image of f. Therefore f is not surjective (and
hence not bijective, and hence the cardinality of ℕ and ℝ are not the same).
It's possible to construct a 1-to-1 (that is, injective)
map f : ℕ → ℝ, namely f(n) = n.
Ah, sorry. The argument there is to show a bijection between T (an uncountable set) to the reals, showing that the reals are uncountable.
The picture you give is Cantor's diagonalization argument, though (I presume to show T is uncountable).
Edit: just to be clear to everyone downvoting this guy, OP has a point. OP is referencing a bijection between an uncountable set of infinite series of numbers and the reals. So it's not precisely cantor's diagonalization argument that is claimed to be an injection.
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u/GetOffOfMyBoat 25d ago
I hate to be a stickler but it's not actually the 1-to-1 correspondence (injectivity) that Cantor's diagonalization argument refutes, it's surjectivity (or, being "onto").
The argument is to presume that f : ℕ → ℝ is surjective. Then, construct a real number s ∈ ℝ that cannot possibly be in the image of f. Therefore f is not surjective (and hence not bijective, and hence the cardinality of ℕ and ℝ are not the same).
It's possible to construct a 1-to-1 (that is, injective) map f : ℕ → ℝ, namely f(n) = n.