What is the contemporary opinion on the problem of induction? It seems that no justification can be given, according to the SEP and an Oxford lecturer. It seems that the SEP does not provide any justification that does not rely on appealing to faith of one sort or another, and the Oxford lecturer says, "...the ultimate principles of science are ones that we cannot hope to base on pure reason." (from 18:40 - 19:10 in the video)
Does this mean that most philosophers reject induction as a proper justification for knowledge? This seems to be a rather extreme conclusion, but I have difficulty seeing how we can avoid this. Is the only option to declare that rationality itself requires induction? This does not seem to be a very satisfying solution.
Question: What is the contemporary status of the problem of induction? Is it considered a true problem in epistemology? Is it dismissed as not important pragmatically? Or something else entirely?