How was Jim able to become a doctor on a carpenter's credentials? He's inhabiting Sam's body now, and so Jim's paramedic credentials and educational record wouldn't apply to him. The only way he could have become a doctor during the five years we skip in the flash forward would be if Sam had gone to college and majored in pre-med before dropping out to become a carpenter.
Unless Jim/Sam is using fake credentials... it would be great to see the resulting problems... but since he's doing his work with Jim's knowledge no one is moved to confirm the authenticity of Sam's credentials...
Yes, it’s in S4 E19. Sam had an architecture degree that included math and science courses that Jim needed. This might make sense if he studied Architectural Engineering, for example. (Sam was an architect; he took that knowledge with him when he crossed over. Then Melinda offered Jim/Sam a job fixing her garage.) Since he officially had a Bachelors degree, Jim could apply directly to med school. Since Jim had some basic medical knowledge already, he felt he could study on his own to catch up for med school.
