I have to reduce $L_\text{empty}$ to $L_\text{inf}$where $\qquad L_\text{empty} = \{e \mid |L(P_e)| = 0 \}$ and$\qquad L_\text{inf}= \{e \mid |L(P_e)| = \infty \}$.where $P_e$ is the program with Gdel number $e$, and $L(P_e)$ the language it accepts.So can I define a program $P$ like thisinput nRun Pe on 1...n for n stepsif Pe accepts any number then   enter an infinite loop and start accepting infinite numberif Pe does not accept any number then   accept 1Now if I can define program P like above then asking the question whether the index of $P$ is in $L_\text{inf}$ will tell me whether the language accepted by $P_e$ is empty or not.
Can I use infinite numbers in a reduction?
computability;reductions;undecidability