TITLE: Induction without a base case
QUESTION [9 upvotes]: I am looking for an example where you have $P(n)$ implying $P(n+1)$. However there is no base case. For which there is therefore no solution at all for the induction problem even though the inductive step itself works.

REPLY [2 votes]: Let $P(n)$  be the statement "$2^{\aleph_0}=n+\aleph_1$"(Continuum Hypothesis). The inductive step itself works but  it gives nothing. We are not able  to check (in the theory $ZFC$) the statement $P(0)$ is false or true.