This chapter marks the start of our development of the algebraic approach to thermodynamics, which continues until the end of Chap. 9. Here we describe the classification of thermodynamic variables; between global and local variables; between thermal, deformation and mass variables; between extensive and intensive variables; between controlled and uncontrolled variables. We define thermodynamic systems, systems of fixed mass and families of systems. The latter being generated by forming Cartesian products and by scaling. What is involved in producing and controlling a thermodynamic process \({\pmb {X}}\rightarrow {\pmb {X}}'\) , taking the system from state \({\pmb {X}}\) to state \({\pmb {X}}'\) , is then discussed. This leads to the introduction of the General-Accessibility Axioms A–I to A–VI which define the algebra of the accessibility relationship \({\pmb {X}}\prec {\pmb {X}}'\) between states \({\pmb {X}}\) and \({\pmb {X}}'\) . The problematic concepts of equilibrium and quasi-static processes are examined. Further discussion of general accessibility and recoverability provides a basis for understanding the significance of the restricted class of adiabatic processes.