Let's define $S(n)$ as $S(n) = \sum_{i=0}^n 2^{-i}$. Obviously, $\lim_{n \to \infty} S(n) = 2$ and also $\forall n \in \mathbb{N}, S(n)<2$. Now my questions are about $Q = \sum_{i=0}^\infty 2^{-i}$.
- First, is $Q$ a valid mathematical object? Does it exist? Can we assign mathematical propositions to it or is that not feasible because it includes the infinity? If $Q$ exists, then can we assign a value to it? Can we say $Q = 2$ or is that wrong?
- What is the relationship between $S(n)$ and $Q$? It seems that $S(n)$ is "building" $Q$. Yet $S(n)$ is always smaller than $2$. If we assume $Q = 2$ (which might be a completely wrong assumption), then we have this seemingly paradoxical situation in which we are building an object (building $Q$ using $S(n)$) with a certain property ($Q = 2$) although in the construction steps we never acquire that property ($S(n)$ is never equal to $2$ for $\forall n \in \mathbb{N}$). So what went wrong? Is assuming $Q = 2$ a wrong assumption? Or does induction not applicable when we are dealing with infinity itself?
Now these questions are giving me a hard time understanding Cantor's arguments, mainly because he is working directly with infinity. For example, in his diagonal proof he first lists real numbers between $0$ and $1$, e.g.:
$$ r_1 = 0.9128987234\ldots \\ r_2 = 0.2342312343\ldots \\ \vdots $$
and then shows we can build a new number $p$ where $p(i) \neq r_i(i)$ and he then shows that $\forall r_i, p \neq r_i$ and hence the cardinality of real numbers is higher than the cardinality of natural numbers. But then comparing this approach with the issues rising from working with $S(n)$ and $Q$ leads to new questions:
- Can we write down $r_i$ as an actual mathematical object, in this case a number, with infinite digits? Or maybe in the same way that we avoid working directly with $Q$ and instead we work with $\lim_{n \to \infty} S(n)$, we should avoid working with actual real numbers with infinite expansions and only work with either their constructors (in cases in which we have a Turing Machine for constructing that number) or with ranges encompassing them.
- In the case of $S(n)$ and $Q$, we saw that induction reserves a certain property for $S(n)$ and $\forall n \in \mathbb{N}$, yet that property may not hold for the infinity itself (i.e. $Q$). Yet in Cantor's proof, the construction of $p$ from $r_i(i)$ is similar to constructing $Q$ from $S(n)$. Now here, how do we know that the fact that some property does hold during the construction (e.g. $p(i) \neq r_i(i)$), would mean that it would also hold for the final infinite object too? (meaning that $\forall r_i, p \neq r_i$)
$Q$ is is simply a convergent series. $S(n)$ is a partial sum and by definition $$\sum_{i=0}^\infty 2^{-i}=\lim_{n\to\infty}\sum_{i=0}^n 2^{-i}.$$ There is nothing paradoxical here.
About the Cantor proof:
(1) Any real number can be written as a convergent series (how?) and $$Q=\lim_{n \to \infty} S(n).$$
(2) The limit of a sequence isn't necessarily a member of the sequence.