I can't understand what is wrong with this paradox. How we should strictly mathematically explain it?
Mathematical induction:
1. The basis:
$n=1,n=2$. Through any two (one) points on a plane we can draw a straight line.
2. The inductive step:
$n=k$. Through any $k$ distinct points on a plane we can draw a straight line.
3. Fake-Paradox:
We have an arbitrary $k+1$ points on the plane: $P_1, P_2, ..., P_{k+1}$. From $2)$ (inductive step) we can draw a straight line $L_1$ through $k$ points $P_1, P_2, ..., P_{k}$ and line $L_2$ through $k$ points $P_2, ..., P_k, P_{k+1}$. Lines $L_1$ and $L_2$ have at least two common points $P_2$ and $P_k$. But any two distinct points of a straight line completely determine that line $\Rightarrow L_1=L_2$ and $P_{k+1} \in L_1$. And we prove that
Through any $n$ distinct points on a plane we can draw a straight line.