TITLE: Uncountably many functions coinciding only finitely many values
QUESTION [10 upvotes]: Is it possible to have uncountably many functions of the type $$f:\mathbb{N}\longrightarrow\{0,1\}$$ pairwise coinciding only on finitely many values?
My guess is no, but I cannot come with a proof of this.

REPLY [3 votes]: Interpreting the functions as indicator functions of subsets of $\mathbb N$ the question can be reformulated as:

Is there an uncountable collection $\mathcal A$ of subsets of $\mathbb N$ such that $(A\Delta B)^{\complement}$ is finite for every pair $A,B\in\mathcal A$?

Or equivalently:

Is there an uncountable collection $\mathcal A$ of subsets of $\mathbb N$ such that $A\cap B$ and $A^{\complement}\cap B^{\complement}$ are both finite for every pair $A,B\in\mathcal A$?

The answer on this is "no" and (as suggested in the comment of bof) for this it is enough to observe that finiteness of $(A\Delta B)^{\complement}$, $(A\Delta C)^{\complement}$ and $(B\Delta C)^{\complement}$ implies the absurd conclusion that $\mathbb N$ is finite on base of: $$\mathbb N\subseteq(A\Delta B)^{\complement}\cup(A\Delta C)^{\complement}\cup(B\Delta C)^{\complement}\tag1$$for any triple $(A,B,C)$ of subsets of $\mathbb N$.

For verification of $(1)$ observe that every $n\in\mathbb N$ we can find sets $U\in\{A,A^{\complement}\}$,$V\in\{B,B^{\complement}\}$,$W\in\{C,C^{\complement}\}$ such that $n\in U\cap V\cap W$ and it is not difficult to verify that: $$U\cap V\cap W\subseteq(A\Delta B)^{\complement}\cup(A\Delta C)^{\complement}\cup(B\Delta C)^{\complement}$$