Does a severed limb/finger experience pain after its separated?
Pain is a signal emitted from a nervous pain receptor cell (called a nociceptor) to the central nervous system (CNS, i.e. the brain). If I hurt myself causing a small cut in my finger, nociceptors around the cut will send the pain signal to the CNS, which in turn processes it and causes the physical feeling of pain in that region.

If a finger is severed, nociceptors just above the cut will be extremely active, interacting with the CNS and causing a lot of pain. On the other hand, nociceptors in the finger itself have nowhere to "push" the pain signal to, so in this sense they cannot measure any pain - there's not a CNS connected to the finger anymore that is able to process the signal emitted by any nociceptors in the finger.