Confusion in Thermodynamics

For a long time now, confusion has existed in the minds of many over the meaning of various concepts in thermodynamics. Recently, this point has been brought to people's attention by two articles appearing on the well-known archive (arxiv) web site. The content of these two pieces serves to illustrate many of the problems and has occasioned the construction of this answer to at least some of them. The position of the axiom proposed by Carath\'eodory is central in this matter and here its position is clarified and secured within the framework of thermodynamics. In particular, its relation to the First Law is examined and justified.